Basic Statistical Analysis, 9E Richard C Sprinthall Test Bank & Instructor Manuals

Instructor’s Manual and Test Bank

For

Basic Statistical Analysis

Ninth Edition

Richard C. Sprinthall

American International College

i

Introduction This Instructional Aid has been prepared to accompany Sprinthall’s “Basic Statistical Analysis”, ninth edition, and was written by Richard C. Sprinthall. The purpose of this manual is to aid the busy instructor. Large student enrollments, students with varying educational backgrounds, and a course that students often approach with trepidation….all combine to present the instructor in elementary statistics with a most difficult job. In fact, there is probably no other college course which is more demanding of the instructor's time than elementary statistics. Your students need time and attention, office-hour visits, extra sessions, and perhaps most of all, lots of problems to work out and immediate feedback. This all takes time, and in many cases, the instructor who has perhaps three (or more) other courses to prepare, simply doesn't have enough of that precious commodity ... time.

Hopefully~ this manual will help ease some of the instructor's burden. Each chapter is divided into four parts. Part A involves suggestions for classroom activities. I certainly do not presume to tell any instructor how to teach this course, especially since most of you have had years of experience, not only in teaching statistics but also specifically teaching from previous editions of this text. There may some of you, however, who haven't taught the course before and these suggestions may help your preparations for class and hopefully provide a few ideas for keeping the class involved and interested. Too many students look forward to this course as though it were akin to a public flogging, but once you get them into it, once they recognize that these concepts have practical value in their lives, the course should then take on a higher interest level. There will always be some students who claim to have math blocks, and are even willing to proudly proclaim their innumeracy. These may be the same students who would never admit to even a smidgen of illiteracy, but somehow take perverse pride in their lack of numerical sophistication. "I may be innumerate, but I'm not illiterate". This is the same student who will fret over some technical grammatical point, but won't know how to find a percentage. One brief suggestion is to ask these students to clip out and bring to class newspaper or magazine articles that are devoted to proving some statistical or research point. Once they learn to become research critics, they may start carrying their calculators with a new sense of self esteem, and may even look forward to their visits to the computer lab. They may also discover that all newspaper accounts, even of the same story, aren't covered with the same attention to detail. In one analysis of how scientific stories are covered by the press, Evans et al. compared those in the New York Times and Philadelphia Inquirer on the one hand, with those in the National Enquirer or Star on the other (Evans et aI, 2009). The Times and Inquirer offered significantly more details than did the tabloids, but, alas, even the prestige newspapers omitted relevant information in the majority of their reports.

We all know that feedback and constant practice are the essential ingredients for student success. The student who puts things off, and "lets it get away", may find it impossible to later catch up. Unlike a history course, where the student can spend an entire weekend ii

doing catch-up reading, the student in statistics needs daily involvement. And daily student involvement translates into more work for the instructor, making up problems, doing the problems, and correcting the problems.

The manual contains over 2000 test items, over 1200 multiple-choice items, over 400 true-false items and well over 400 problems requiring calculated solutions. Except for Chapters 1 and 11, which do not demand any numerical calculations, each chapter has a large number of problems where actual values are supplied. These may be used as homework problems, or as exam items.

Further, both the multiple-choice and true-false items are odd-even balanced throughout each chapter. If the even-numbered items are used for the main exam, the odd-numbered items may be used for students needing make-up exams. The items are designed to be straight-forward and unambiguous. They are NOT intended to trick the student, but instead to provide an honest evaluation of the student's ability. There are no "curve balls" or "sliders"; instead, the questions all come "right down the middle", and although some may require a very careful reading, the student who is thoughtful and prepared should not find them overly difficult. One of the criticisms of the multiple-choice and true-false formats is that they reward rote memorization rather than true understanding. Although this can certainly at times happen, especially when the items are poorly designed, a carefully prepared multiple-choice test, such as the one presented in this manual, can assess a student’s ability to apply concepts to problem-solving situations. Rather than breaking up the unity of knowledge and isolating the pieces, as critics typically charge, a well- designed multiple-choice test demands that the student be able to understand concepts and bring facts together (Aiken, 1988). One fairly recent report showed that multiple-choice items may actually be more culturally fair than are essay questions.

Verbally adept but uninformed students can sometimes bluff their way through essay exams, a tactic that is used less often by minorities (Science Agenda, 1990). Although some poorly prepared students will always insist on calling multiple-choice items, "multiple-guess tests", a careful analysis of student responses allows the instructor to illuminate a student's thought process in rather sharp detail (often to the amazement of the complaining student) .

Finally, the manual also includes the answers to all the even numbered text problems (the odd-numbered answers are presented in the text itself).

Aiken, L.R. (1988). Psychological testing and assessment (6th ed.).

Boston: Allyn and Bacon.

Evans,W. A., Krippendorf,M.,Yoon,J.H.,Posluszny,P. & Thomas, S. (2009).

Science in the prestige and national tabloid presses. Social Science Quarterly, 71, 105-117.

Science Agenda. (1990). New directions in educational testing. Science Agenda. (August-September), 14. iii

CONTENTS TOPIC

PAGE

Introduction:

i-xii

CHAPTER 1: INTRODUCTION TO STATISTICS

1

Suggestions for Class Activities Statistics and the Media Multiple Choice Items True-False

1 3 7

CHAPTER 2: PERCENTAGES, GRAPHS AND MEASURES OF CENTRAL TENDENCY Suggestions for Class Activities The Mean is still the Mean Assessing Central Tendency Your Students as Statistical Consultants Averaging Averages Evaluating Percentages The Mean and Adding a Constant The Mean and Multiplying by a Constant The Mean and Independent Measures Multiple Choice Items True-False Calculate the Values

8

CHAPTER 3: MEASURES OF VARIABILITY Suggestions for Class Activities Quartiles Anyone?

How Normal is Normal?

Skewing the Distribution Show Your Cash What About Constants?

The SD and Multiplying by a Constant Multiple Choice Items True-False Calculate the Values

22 22 22 22 23 23 23 24 25 33 35

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8 8 8 8 9 9 10 11 11 12 20 21

CHAPTER 4: THE NORMAL CURVE AND z SCORES Suggestions for Class Activities Means Z Scores and Skewness Small Samples and Z Scores Remembering Z Scores Multiple Choice Items True-False Calculate the Values

38 38 38 38 39 40 49 51

CHAPTER 5: z SCORES, T SCORES AND OTHER NORMAL CURVE TRANSFORMATIONS Suggestions for Class Activities Curing the Algebra Phobe Translating Into T Scores Using Table 5.1 : Finding Stanines Multiple Choice Items True-False Calculate the Values

54 54 54 55 56 57 65 67

CHAPTER 6: PROBABILITY Suggestions for Class Activities The Lottery and Combinations Low Probabilities and the Impossible Multiple Choice Items True-False Calculate the Values

70 70 70 70 72 79 81

CHAPTER 7: STATISTICS AND PARAMETERS Suggestions for Class Activities Ask Me, and I Will Tell You Multiple Choice Items True-False Calculate the Values

84 84 84 85 93 95

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CHAPTER 8: PARAMETER ESTIMATES Suggestions for Class Activities Interval Estimates and Precision Interval Estimates, Hypothesis Testing and the Alpha Level Multiple Choice Items True-False Calculate the Values

98 98 98 98 100 109 112

CHAPTER 9: THE FUNDAMENTALS OF RESEARCH METHODOLOGY Suggestions for Class Activities The Hawthorne Effect and John Henry Sensitization and Causal Inference A Research Example Reviewing the Literature The Student Researcher Multiple Choice Items True-False

116 116 116 117 117 118 121 123 145

CHAPTER 10:THE HYPOTHESIS OF DIFFERENCE Suggestions for Class Activities Student’s t as a Ratio Increasing the Variability Within Getting Creative with the Standard Deviation Multiple Choice Items True-False Calculate the Values

149 149 149 150 150 152 162 165

CHAPTER 11: THE HYPOTHESIS OF ASSOCIATION: CORRELATION Suggestions for Class Activities Correlation and the Cause-And-Effect Trap The Spearman rs (rho) Hard Data?

Multiple Choice Items True-False Calculate the Values

171 171 171 172 172 174 185 187

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CHAPTER 12: ANALYSIS OF VARIANCE Suggestions for Class Activities Comparing t and F F, t and the Direction of the Difference ANOVA and Independence ANOVA and Post-Facto Research The Levene Test Multiple Choice Items True-False Calculate the Values

197 197 197 197 197 198 199 200 211 212

CHAPTER 13: NOMINAL DATA AND THE CHI SQUARE Suggestions for Class Activities Identifying Nominal Data Nominal and Interval: A Comparison Chi Square and the Hypothesis of Difference Chi Square and Correlation Multiple Choice Items True-False Calculate the Values

217 217 217 218 218 218 220 229 231

CHAPTER 14: REGRESSION ANALYSIS Suggestions for Class Activities Regression and Secular Trend Secular Trend and the Computer Trend Analysis and the Rear-View Mirror Multiple Choice Items True-False Calculate the Values

233 233 233 234 234 236 245 247

CHAPTER 15: REPEATED-MEASURES AND MATCHED-SUBJECTS DESIGNS (Interval Data) Suggestions for Class Activities 252 Degrees of Freedom and Marital Bliss 252 The Paired t and the Pearson r 252 Multiple Choice Items 255 True-False 265 Calculate the Values 267 vii

CHAPTER 16: NONPARAMETRICS REVISITED - THE ORDINAL CASE Suggestions for Class Activities When Normality Fails Parametric Restrictions Non-Parametrics and their Parametric Counterparts Multiple Choice Items True-False Calculate the Values

275 275 275 275 278 287 289

CHAPTER 17: TESTS AND MEASUREMENTS Suggestions for Class Activities Library Sources and the Internet Creating a Test Multiple Choice Items True-False Calculate the Values

294 294 294 295 296 301 302

SPECIAL UNIT 2 Multiple Choice Items

314

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ANSWERS TO INSTRUCTOR TEST BANK (multiple choice, true-false and calculate the values).

Chapter 1 317 Chapter 2

318

Chapter 3

319

Chapter 4

320

Chapter 5

322

Chapter 6

323

Chapter 7

324

Chapter 8

325

Chapter 9

327

Chapter 10

329

Chapter 11

332

Chapter 12

335

Chapter 13

337

Chapter 14

338

Chapter 15

339

Chapter 16

341

Chapter 17

343

Special Unit 2

345

ix

COMPUTER PROBLEMS Chapter 2

346

Chapter 3

348

Chapter 8

349

Chapter 10

351

Chapter 11

356

Chapter 12

364

Chapter 13

370

Chapter 14

375

Chapter 15

381

Chapter 16

390

Chapter 17

393

x

ANSWERS TO COMPUTER PROBLEMS Chapter 2

397

Chapter 3

398

Chapter 8

399

Chapter 10

400

Chapter 11

401

Chapter 12

402

Chapter 13

404

Chapter 14

405

Chapter 15

406

Chapter 16

407

Chapter 17

408

xi

ANSWERS TO EVEN NUMBERED TEXT PROBLEMS Chapter 1

409

Chapter 2

411

Chapter 3

412

Chapter 4

413

Chapter 5

414

Chapter 6

415

Chapter 7

416

Chapter 8

417

Chapter 9

418

Chapter 10

420

Chapter 11

422

Chapter 12

424

Chapter 13

426

Chapter 14

427

Chapter 15

429

Chapter 16

430

Chapter 17

431

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CHAPTER 1

INTRODUCTION TO STATISTICS A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Statistics and the Media Try to encourage your students to feel that statistics can help them understand current events as well as their own areas of specializations. Some of your students will devoutly believe that they are math dim-wits and will never understand these arcane statistical scribblings that you are putting up on the board. Impress your students with the fact that although the field of statistics has its detractors, and in many instances rightfully so, it is also an area that offers rich rewards to those with some degree of quantitative sophistication. At meetings, in journals and at conventions, students may find themselves bombarded with the latest findings of social research studies, and the results of these studies can often appear bewildering and at times even misleading if the student is totally unfamiliar with the general rules of statistical analysis and research design. Statistics and statistical research should be an aid to judgment, not a replacement for it.

The kinds of topics covered in this first chapter lend themselves to lively discussions, usually based on current events reported in almost any newspaper or magazine. Ask the students to be on the alert for articles that attempt to use statistical persuasion to push for a point of view that may not really be backed up by the numbers. As one example tell them that a recent US administration took credit for creating wealth for American citizens during its eight years in office. The percentage of individuals earning over $50,000 a year, they said, had increased from 8% to 10%. But, by not factoring in inflation, that may not be (and isn't) any gain at all. $50,000 today does not have the purchasing power that $50,000 would have had eight years earlier. To stretch the point, tell them that 50 years ago very few Americans earned over $10,000, but does that mean we are all that much richer today?

In the text, there is a discussion of a research study that attempted to prove that as women left the kitchen and entered the work force, terrible consequences ensued, such as dramatic increases in the prison population, small business failures and reported felonies. Students must be constantly reminded that just because various events might occur together they are not necessarily linked and one should not be led into the trap of a cause-and-effect explanation. On June 20, 1993 the New York Times ran an ad that advised its readers that virtually all of society's problems can be blamed on TV. The ad referred to a study by B.S. Centerwall at the Univ. of Washington that showed that in the 30 year period following the introduction of TV in the United 1

States and Canada, white homicide rates have increased by 93%, and in Canada by 92%, whereas in South Africa, which had no TV until the mid 1970's, the white homicide rate decreased by 7%. Perhaps your students might comment on this artful juxtaposition of statistics, and you might point out that a whole host of things may have changed during that time span. And as for South Africa, maybe whites were not being killed at the same rate, but that in itself doesn't prove that there were fewer overall numbers of homicides, with or without TV. Finally, it might be mentioned that some of history's most notorious murderers, Hitler, Stalin, Al Capone and Jack the Ripper, never saw one second of TV.

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B. Multiple Choice Items 1-1. The inherent fallacy in the argument that capital punishment is not a deterrent (since pockets were being picked even at the public hanging of a pickpocket) is that a. pickpockets were never hanged b. you can't compare groups of unequal sizes c. there was no comparison or control group d. you can't make comparisons with only nominal data 1-2. Whenever two events occur simultaneously, such as increasing numbers of joint bank accounts and increasing numbers of felonies, one must be careful not to a. assume that one event is the cause of the other b. assume that any correlation exists between them c. assume that the data have been reported honestly d. all of these 1-3. The founding of probability theory is popularly credited to a. the Chevalier de Mere b. Sir Francis Galton c. Karl Pearson d. Blaise Pascal 1-4. William Sealy Gossett, the statistician at the Guinness Brewing Company, published his works using the pen name a. Professor b. Blaise Pascal c. Cicero d. Student 1-5. Statistics as a general field is divided into two sub-areas. They are a. predictive and inferential b. descriptive and inferential c. nominal and ordinal d. none of these 1-6. When we say that the actual average height of all adult females in the U.S. is known to be 5' 6", we are using a. predictive statistics b. probability statistics c. inferential statistics d. descriptive statistics

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1-7. Techniques which are used for describing small or large amounts of data in abbreviated form, are called a. descriptive statistics b. inferential statistics c. predictive statistics d. probability estimates 1-8. The difference between inferential and predictive statistics is that a. inferential extrapolates, whereas predictive does not b. predictive extrapolates, whereas inferential does not c. inferential assumes that a sample has been measured, whereas predictive makes no such assumption d. there is no difference between them 1-9. When a researcher attempts to estimate the characteristics of an entire population on the basis of sample measures, the techniques employed are called a. inferential statistics b. descriptive statistics c. skewed statistics d. b and c, but not a 1-10. The researcher using inferential statistics always makes predictions that are based on having measured a. a population b. a sample c. the entire group being predicted d. in inferential statistics predictions are never made 1-11. The goal of the researcher using inferential statistics is to a. make better-than-chance predictions b. predict the characteristics of the entire group, based on measures taken on a smaller group c. describe the difference between the highest and lowest score d. a and b, but not c 1-12. An example of a variable would be a. height b. weight c. the number of inches in a foot d. a and b, but not c

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1-13. Anything that can be measured and observed to vary is called a. a constant b. a variable c. an integer d. all of these 1-14.The statement that one can prove anything with statistics is only true when a. the data have been faked b. the reader is naive regarding statistical procedures c. the statement is always true d. the statement is never true 1-15. The first attempt to employ statistical techniques for estimating population parameters was designed in order to a. estimate beer-drinking tastes in Dublin, Ireland b. calculate the odds for casino gambling in France c. calculate the probability of certain exponential functions d. all of these 1-16. The first attempt to employ probability theory in a practical setting was developed in order to a. predict the results of presidential elections b. predict economic forces in the market place c. predict how consumers will evaluate new products d. make winning bets in the gambling casinos 1-17. One author suggests that the best way to compare the safety of autos versus planes is on the basis of a. the total number of deaths per year for each mode of travel b. the total number of individuals who drive versus fly. c. the per-hour death rate for the two modes of travel d. none of these, since travel rates are not comparable.

1-18. In comparing the safety of allowing children to visit friends whose parents own a gun versus those parents who have a swimming pool, it was found that a. there was no safety difference between the two b. children were safer at the homes with swimming pools c. children were safer at the homes with guns d. more parents had guns than swimming pools

5

1-19. Since poll results are based on a sample size which is less than the population size, generalizing from the sample to the population demands a. data graphing b. inferential statistics c. descriptive statistics only d. leap of faith 1-20. The commercial that specifies that a certain brand of aspirin should be used, since no other brand prevents more heart attacks proves that a. the brand mentioned prevents more heart attacks than does its competition b. the brand mentioned has been proven to prevent heart attacks c. the brand mentioned has only been shown effective when combined with exercise d. none of the above

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C. True or False: For the following, indicate T (true) of F (false) 1-21. Descriptive statistics are designed to infer population values.

1-22. Descriptive statistics provide symbolic techniques for describing both large and small groups of data.

1-23. Inferential statistics involve the measuring of a sample and then using this value for predicting to the population.

1-24. Inferential statistical techniques and predictive statistical techniques are identical.

1-25. Descriptive statistics are especially important in using polling data to infer population characteristics.

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CHAPTER 2 Percentages, Graphs and Measures of Central Tendency A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: The Mean is still the Mean Although in the text the decision was made to use the symbol M for the arithmetic mean, point out to the class that this is not the only way the mean can be expressed. Although most of the Education and all of the Psychology journals now being published, use M for the arithmetic mean, there continues to be a small number of statistics texts that are still using X with a bar across the top. Students may at first be confused by this inconsistency, but once it is pointed out that both symbols mean exactly the same thing, the mean is the mean is the mean, the level of possible frustration should be reduced. In fact, you may wish to accept either symbol.

Activity: Assessing Central Tendency: What do both the Median and Mean mean?

Perhaps the most glaring trap awaiting students who misunderstand central tendency is the confusion that arises between the use of the mean and the median. Both measures, to be sure, provide information regarding how the average or typical subject performed, but in certain situations the use of one of these measures rather than the other can create an extremely inaccurate portrayal of centrality. .

Activity: Your Students as Statistical Consultants Ask your students to assume that have been selected as statistical consultants and have been given the following scores on a standardized test of reading ability test (where a score of 100 indicated normal progress): X 110 109 109 108 107 107 107 15 X=772 For this distribution, then, X = 772, and the mean of X = 772/8 = 96.50.

On the basis of this mean value of only 96.50 it would seem that on average the group was not performing up to the standards of normal progress, even though 8

every single student in the group, except for one, was scoring well above the average. In this case, as with all skewed distributions, the median, which for this distribution is 107.50, is a far more accurate indicator of true centrality than was the mean. Point out that when the distribution is skewed to the left, as in the above scores, the mean is going to severely underestimate the true centrality.

Have the students graph the above distribution to again reinforce what the shape of a skewed distribution looks like. If equal intervals are chosen for the base line (abscissa), the graph will have to be extremely wide to fit all the values in. Also, point out that the median remains at 107.50 whether the low score had been 15 or 105. However if the low score were changed to 105 the mean would then jump to 107.75. Changing that one score caused the mean to gyrate, but the median remained rock steady at 107.5.

Activity: Averaging Averages?

Sometimes students will intuitively assume that to get the mean of two sets of scores, all they have to do is average the two means. This of course is only true if both sets of scores have equal numbers of cases. But show them that with unequal numbers of cases, averaging the means can be a big mistake. For example, the following distribution 16,12,12,11,10,9,9,7,4 adds up to 90, with a mean of 10. A second distribution, 10,8,8,8,8,6 adds up to 48, with a mean of 8. The mean of the two distributions combined is 9.20, not the average of the two means which would have been 9. You can, however, teach students to do this correctly without going back and adding all the scores. Show them that since the mean, M, is = to X/N, then X = the M times N, or (M)(N). X for the first distribution is then equal to (M)(N) or (10)(9) = 90. Similarly for the second distribution, X = (M)(N) = (8)(6) = 48. They will quickly see that the mean of both distributions combined can easily be found by adding the two Xs (48+90) = 138 and dividing by the total N of 15, to get 138/15 = 9.20. If they think this is complicated and would rather just put all the scores together and add them up, explain that with a large data base, the technique you're showing them is far more efficient.

Activity: Evaluating Percentages The same concerns also tend to show up when evaluating percentages. Too often students want to average percentages, even when the totals in the various percentage categories are not the same. Even faculty members have been known to have difficulty accepting the fact that means and percentages cannot always simply be averaged. On a Master's Comprehensive exam at a small eastern college, the passing grade on the objective section of the test was determined to be 80% correct. This section was composed of 300 multiple choice items, covering seven different content areas, but the seven content areas were not all composed of the same number of items. For example, the exam could have had 9

100 items devoted to Learning, 100 to Systems and Theories, and then 20 items each in Cognitive Psychology, Psychological Assessment, WISC-Assessment, Statistical Analysis and, finally, Learning Disabilities. A student could then have received scores of 95% in Learning, 85% in Systems, 60% in Cognitive, 60% in Assessment, 60% in WISC, 60% in Statistics and 60% in LD. The faculty group challenged the fact that the student's overall score could have resulted in a passing grade of 80%. The scoring breakdown was as follows, 95 out of a hundred in Learning, 85 out of a hundred in Systems, and then 12 out of 20 for the other five sections. This resulted in a total of 240 correct responses out of 300 items, or 80% correct. Thus, the student could have passed the exam, even though failing in 5 of the 7 sections.

Or cite the example of the baseball player who hit .300 in day games and only .200 in night games. This player wondered why he was being sent down to the minors when his day-night average was a seemingly adequate .250. The problem was that the team had only played 5 day games but had already played over a 100 night games, and for all 105 games his average was a mere .215. By the way, this situation was seriously argued on a sports, call-in radio show.

Activity: The Mean and Adding a Constant Explain that the effect on the mean of adding a constant to every value is to simply change the mean by the amount of that constant. Thus, the new mean = the old mean + the constant.

In the following set of scores: X 14 12 3 9 8 10 5 2 63 Mean = 63/8 = 7.875 Now we will add the constant 10 to each of the previous values X+10 24 22 13 19 18 20 15 12 143 Mean = 143/8 = 17.875 (or 7.875 plus the constant 10).

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Activity: The Mean and Multiplying by a Constant Show your students that multiplying by a constant has the effect of changing the mean by a function of that constant, such that the new mean equals the old mean times the constant. Using that first distribution shown above, each value will be multiplied by the constant 10 (X)(10) 140 120 30 90 80 100 50 20 630 Mean = 630/8 = 78.750 (or 7.875 times the constant 10).

Activity: The Mean and Independent Measures Let the students see what happens to the mean when two independent measures are summed. When the mean is being found for the sum of two measures, for example if you have two independent measures on each subject, and these measures are added, then X1 + X2 = X1+X2 15 + 11 = 26 14 + 10 = 24 12 + 9 = 21 11 + 5 = 16 10 + 5 = 15 9 + 4 = 13 7 + 3 = 10 2 + 1 = 3 80 48 128 m1 = (80/8 = 10.00) + m2= (48/8 = 6.00) = M for 128/8 = 16.00 Thus, the mean of the sums (16.00) is equal to the sum of the two means (10.00+6.00).

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B. Multiple Choice Items 2-1. When scores are arranged in order of magnitude, the researcher has formed a a. histogram b. measure of centrality c. measure of dispersion d. distribution 2-2. Traditionally, the researcher indicates frequency of occurrence on the graph's a. ordinate b. abscissa c. line of ascent d. horizontal axis 2-3. When single points are used to designate the frequency of each score, the points being connected by a series of straight lines, this is called a a. frequency polygon b. frequency rectangle c. scatter plot d. histogram 2-4. The mean, median, and mode are all measures of a. dispersion b. variability c. central tendency d. all of these 2-5. When a graph is constructed using a series of rectangles indicating the frequency of occurrence for each score, it is called a a. frequency polygon b. frequency rectangle c. scatter plot d. histogram 2-6. The measurement which occurs most often in a distribution is called the a. median b. percentile c. mean d. mode

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2-7. When a distribution is skewed, the researcher who is interested in central tendency should use the a. mean b. median c. mode d. all of these are appropriate 2-8. When a distribution shows a large majority of very low scores and a few very high scores, the distribution is said to be a. skewed to the right b. skewed to the left c. skewed to the middle d. bimodal 2-9. The influence of a few extreme scores in one direction is most pronounced on the value of the a. mean b. median c. mode d. percentile 2-10. Using the mean to indicate centrality on a distribution of income scores usually results in a. a false image of poverty b. an accurate portrayal of income c. a false image of prosperity d. income scores never lend themselves to centrality 2-11. When each score is listed in order of magnitude, together with the number of individuals receiving each score, the researcher has set up a. a unimodal distribution b. a bimodal distribution c. a skewed distribution d. a frequency distribution 2-12. The abscissa is a. the horizontal axis b. the vertical axis c. the connected points on a polygon d. a measure of central tendency

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2-13. On a frequency distribution, raw scores are plotted on the a. abscissa b. ordinate c. vertical axis d. all of these, depending on the size of the group being measured 2-14. When graphing data, it is traditional to make the length of the ordinate equal to a. the length of the abscissa b. twice the length of the abscissa c. three-quarters of the length of the abscissa d. one-half of the length of the abscissa 2-15. With a frequency polygon, scores are always presented on a. the X axis b. the Y axis c. the Z axis d. the frequency polygon may never be used to represent scores 2-16. The more separate scores there are in a given distribution, the higher will be the value of the a. the mean b. the median c. the mode d. none of these 2-17. The ordinate is identical to the a. X axis b. Y axis c. mean d. none of these 2-18. The so-called "wow" graph is always possible whenever a. scores are presented on the X axis b. the abscissa does not begin with zero c. the base of the ordinate is not set at zero d. two distributions are being presented simultaneously 2-19. Perhaps the most serious flaw in graphing data is due to a. not placing frequencies on the abscissa b. not placing raw scores on the ordinate c. not placing the ordinate on the X axis d. not setting the base of the ordinate at zero 14

2-20. The following are all measures of central tendency, except a. the mean b. the median c. the range d. the mode 2-21. The arithmetic average defines the a. mean b. median c. sigma d. mode 2-22. The point above which half the scores fall and below which half the scores fall, defines the a. mean b. median c. sigma d. mode 2-23. The most frequently occurring score in the distribution defines the a. mean b. median c. sigma d. mode 2-24. The mean is not overly affected by extreme scores, unless a. the extreme scores are all in one direction b. the extreme scores are in both directions c. the number of extreme scores is fewer than 5 d. all of these 2-25. The fact that the mean IQ of college seniors is higher than that of freshmen is probably due to a. the fact that going to college increases the IQ b. the fact that there is a big IQ gain between the junior and senior years c. an incorrect interpretation of the data d. the fact that the lower IQ freshmen tend to drop out of college and, therefore, never become seniors 2-26. Adding just one or two extreme scores to the high end of a distribution, has a great effect on a. the median, but not the mode b. the mode, but not the mean c. the mean, but not the median d. none of these 15

2-27. Adding just one or two extreme scores to the low end of a distribution, has a great effect on a. the median but not the mode b. the mode, but not the median c. the mean, but not the median d. none of these 2-28. When the majority of scores are at the high end of the distribution, but there are a few extremely low scores, the distribution is a. bimodal b. multimodal c. skewed left d. skewed right 2-29. When the mean lies to the right of the median, the distribution is probably a. bimodal b. multimodal c. skewed left d. skewed right 2-30. When the median lies to the right of the mean, the distribution is probably a. bimodal b. multimodal c. skewed left d. skewed right 2-31. When a distribution is skewed to the right, a. the mode will be to the left of the median b. the mode will be to the right of the median c. the mode will be to the right of the mean d. the mode will always be identical to the mean 2-32 Percentages are based on a standardized denominator of a. 100 b. 10 c. 50 d. 0 2-33 In order to read a percentage a. only the numerator of the percentage needs to be shown b. the percentage is always shown in fraction form c. the percentages shown are always in the form of inferential statistics d. to establish a percentage for a specific event, the total number of events need not be known 16

2-34 When comparing percentage rate increases with decreases, the same absolute difference yields a. the same percentage difference b. the percentage increase calculates out as larger than the decrease c. the percentage decrease calculates out as larger than the increase d. comparing percentage increases with decreases cannot be done 2-35 The FBI’s Uniform Crime Reports provide per capita data based on a rate per a. 100,000 b. 50,000 c 25,000 d. one million 2-36 Bar charts are used instead of histograms when the data are a. Continuous b. Non-continuous c. In the form of values that may fall at any point along an unseparated scale of points d. None of these since bar charts and histograms are synonymous.

Questions 37 through 42 are based on the following: In a certain community, the median per-family annual income is $80,000. The Mean per-family income is $100,000, whereas the mode is $71,000.

2-37. the distribution of income scores is A. skewed right B. skewed left C. skewed to the middle D. not skewed 2-38. the most appropriate measure of central tendency in this distribution Would yield a value of A. $80,000 B. $100,000 C. $71,000 D. none of these values could yield a measure of central tendency 2-39. if a new family were to move into the community with an annual income of $295,000, this would most affect A. the mean B. the median C. the mode D. all of these

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2-40. the annual income achieved by most of the families is A. $71,000 B. $80,000 C. $100,000 D. half way between the mean and the mode 2-41. The annual income which is surpassed by 50% of the families is a. $80,000 b. $71,000 c. $100,000 d. cannot tell from these data 2-42. The annual income which is surpassed by 90% of the families is a. $100,000 b. $71,000 c. $80,000 d. cannot tell from these data 2-43. Whenever a distribution is skewed left, the measure yielding the highest numerical value is always the a. mean b. median c. mode d. percentile 2-44. When a skewed distribution tails off to the right, the distribution is a. skewed right b. skewed left c. skewed to the center d. not skewed at all 2-45. In a histogram, the mode is always located a. under the shortest bar b. under the tallest bar c. under the last bar to the right d. under the last bar to the left 2-46. A bimodal distribution often indicates a. that there will be two means b. that there will be two medians c. that the mean, median and mode have the same value d. that two separate sub-groups may have probably been measured

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2-47. The most appropriate measure of central tendency in a bimodal distribution is (are) the a. mean b. median c. modes d. ordinate 2-48. When a distribution has two separate and distinct medians, then a. it is skewed right b. it is skewed left c. it is probably bimodal d. a distribution can never have more than one median 2-49. With a fairly balanced distribution of (neither skewed nor bimodal), the most appropriate measure of central tendency is the a. mean b. median c. mode d. none of these

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C. True or False: For the following, indicate T (True) or F (False) 2-50. A skewed right distribution has the mean lower than the mode.

2-51. The median is always exactly half-way numerically between the highest and lowest scores.

2-52. The most appropriate measure of central tendency in a skewed right distribution is the median.

2-53. A positively skewed distribution is identical to a skewed right distribution.

2-54. Other things being equal, the mean is the most stable measure when the data form is skewed..

2-55. With a skewed left distribution, the median is always to the right of the mean.

2-56. With a skewed left distribution, the mode is never to the left of the mean.

2-57. All three measures of central tendency can be calculated when the data are in interval form.

2-58. On a frequency distribution curve, frequency of occurrence is always plotted on the abscissa.

2-59. One should expect a distribution of personal income measures to be skewed to the right.

2-60. When the median is being calculated, it makes no difference whether one starts counting from the bottom or the top of the distribution.

2-61. If a positively skewed and negatively skewed distribution were combined, the resulting distribution would probably be bimodal.

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D. For the following questions, calculate the values.

2-62. For the following set of scores, calculate the mean, median and mode: 11, 2, 3, 3, 7, 6.

2-63. For the following set of scores, calculate the mean, median and mode: 20, 8, 18, 10, 15, 10, 13, 11.

2-64. For the following set of scores, calculate the mean, median and mode: 3, 4, 7, 7, 5, 9.

2-65. For the following set of scores, calculate the mean, median and mode: 5, 7, 3, 9, 4, 5, 10, 9.

2-66. For the following set of scores, calculate the mean, median and mode: 8, 8, 6, 7, 5, 9.

2-67. For the following set of scores, calculate the mean, median and mode: 12, 1, 9, 7, 2, 4.

2-68. For the following set of scores, calculate the mean, median and mode: 10, 12, 9, 10, 10.

Questions 69 through 75 are based on the following: Thirty-four members of a certain sorority were selected, and asked to indicate how many hours each had spent reading (for pleasure, not school work) during the previous week. The data are as follows: 50, 4, 10, 5, 5, 6, 7, 3, 5, 4, 4, 5, 6, 6, 7, 5, 8, 1, 8, 7, 5, 6, 10, 6, 8, 7, 7, 6, 5, 5, 4, 3, 4, 5.

2-69. Find the mean.

2-70. Find the median.

2-71. Find the mode.

2-72. Which measure of central tendency yielded the highest numerical value?

2-73. Which measure of central tendency yielded the lowest numerical value?

2-74. If the distribution is skewed, in what direction is the skew?.

2-75. What would the mean and median have been if the highest score had been a 12 instead of a 50?

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CHAPTER 3 MEASURES OF VARIABILITY A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Quartiles Anyone?

Ask the students to raise their hands if they graduated from high school in the first quartile of their class. Many, too many, will assume that the first quartile is the top quarter of the class, even if they have read the chapter. With this quick demonstration, the point can be dramatized by telling them that if they really were in the first quartile (the bottom 25%), they probably wouldn't be in college at all. Remind them again that Q1, which falls at exactly the 25th percentile, therefore includes the bottom 25% of the distribution, not the top 25%. Sometimes even percentiles are misread in this same way, where someone proudly proclaims that he/she was in the 5th percentile on some test. Such a person has to be asked to put the victory dance on hold, because the 5th percentile is at the wrong end of the distribution, unless it's some non-esteemed distribution, like the number of felony arrests.

Activity: How Normal is Normal?

Ask your students to assume that a certain class had 25 students, and each student was asked to indicate how much money he/she was carrying at that moment. The amounts were as follows (in dollars): 2.00,2.00,2.10,2.10,2.10,2.20,2.20,2.50,3.00,1.90,2.00,1.00,2.00,2.00,2.00.1.80, 2.00,2.00,2.00,1.90,1.90,1.80,2.00,1.50,2.00.

Have the class calculate the mean, median, mode, SD, and the standard value for the SD had the distribution been normal (R/6).

The mean is 2.00 The median is 2.00 The mode is 2.00 The SD is .33 The SV is 2/6 = .33 Describe the distribution. Show that it is neatly in balance and, since all the measures of central tendency are the same that it can't be skewed. Also, show them that for a small distribution of this size, only 25, it seems to be fairly normal, with its true SD approximating the standard value of R/6. Explain again that for a distribution to be perfectly normal, it has to coincide with a theoretical distribution that assumes an enormous number of scores (actually infinite). Explain that with the small number of scores shown above, the percentage areas will not conform to those of the perfectly normal curve when z scores are later found.

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Activity: Skewing the Distribution Now suppose that one of the students who was carrying 2.00 left the room and his place was taken by another student who had 40.00 in her purse. Ask how this would impact the distribution. Although the median and mode would not be changed, the distribution would quickly become skewed. The mean would jump to 3.52, which is more than anyone in the room had (other than the new person with the 40.00). Also, the SD would jump to 7.45, which is another feature of a skewed distribution. Unless the scores remain in balance, which didn't happen here, a large SD doesn't necessarily indicate a platykurtic distribution. It also occurs in skewed distributions. Unless the new value was considered to be an outlier, it will change the distribution dramatically, skewing it severely to the right.

Activity: Show Your Cash If your class isn't too large, you may want to try this out yourself, by literally asking the students how much money they are carrying and creating a distribution from these values.

Activity: What About Constants?

Now let's visit the faculty lounge, where the School of Business faculty is gathering for coffee between classes. Each of them is polled as to how much money he/she is carrying and the results are (in dollars): 20,20,21,21,21,22,22,25,30,19,20,10,20,20,20,18,20,20,20,19,19,18,20,15,20.

If the class doesn't notice, don't tell them immediately that these are simply the same values used before but now multiplied by ten.

Mean = 20 Median = 20 Mode = 20 SD = 3.30 Indicate that the variability remains relatively the same when a constant is either added to or multiplied by all the scores in the distribution. In this case, all the resulting values are multiples of 10 for the values shown above.

Now, let's have one of the Business Professors leave the faculty lounge and the President of the College enters carrying $400 in her purse. Notice that this new value jumps the mean to 74.50 and the SD to 35.20, exactly ten times what they had been with the skewed distribution shown previously.

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Activity: The SD and Multiplying by a Constant Illustrate again what happens to the SD when each value in the distribution is multiplied by a constant. Let us multiply each value in the first distribution by 10.

X X² (X)(10) [(X)(10)]² 14 196 140 19600 12 144 120 14400 3 9 30 900 9 81 90 8100 8 64 80 6400 10 100 100 10000 5 25 50 2500 2 4 20 400 Σ X=63 ΣX2=623 Σ(X)(10)=630 Σ [(X)(10)]²=62300 SD = √623/8 - 7.875² SD = 3.982

SD = √62300/8 - 78.750² SD = 39.82

Thus, when a constant is multiplied to each value, the new standard deviation = the old standard deviation times the constant, 3.982 times 10 = 39.82.

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B: Multiple Choice Items 3-1. The concept of "individual differences" may best be described by a. the mode b. the median c. the measures of central tendency d. the measures of variability 3-2. The difference between the highest and lowest scores in any distribution is called a. the deviation score b. the standard deviation c. the range d. none of these 3-3. Of the following, the measure(s) of variability is (are), a. the range b. the standard deviation c. the mode d. a and b, but not c 3-4. One major defect of the range is that a. it includes the entire width of the distribution b. it is based on only two scores c. it is not a measure of variability d. all of these 3-5. The measure of the entire width of the distribution is called the a. range b. percentile c. quartile d. b and c, but not a 3-6. When new scores that are close to the mean are added to any distribution, there is a fairly high probability that the range will a. remain unaffected b. be reduced c. be increased d. new scores may never be added to a distribution 3-7. Specific points may be located on any distribution by using the a. percentile b. quartile c. decile d. all of these 25

3-8. A score which is at the 95th percentile must exceed a. at least 5% of the scores b. at least 95% of the scores c. at least 45% of the scores d. at least the lowest 47.50% of the scores 3-9. When a student takes a 600-item standardized test and is told he/she is at the 70th percentile, it means that the student a. answered 30% of the items correctly b. answered more than 70% of the items correctly c. answered exactly 420 items correctly d. none of these 3-10. A score which falls at the 50th percentile must always be at the a. mean b. mode c. median d. none of these 3-11. If 50% of the cases fall above a given point, then 50% must also fall a. below that point b. between the mean and median c. between the first and fifth deciles d. none of these, since the distribution can never account for 100% of the cases

3-12. The first quartile coincides with the a. first percentile b. twenty-fifth percentile c. seventy-fifth percentile d. median 3-13. The interquartile range lies between a. the median and the third quartile b. the median and the first quartile c. the median and the lowest score in the distribution d. the first and third quartiles 3-14. The fifth decile coincides with the a. 50th percentile b. 2nd quartile c. median d. all of these 26

3-15. The third quartile begins between the a. 1st and 5th deciles b. 2nd and 3rd deciles c. 4th and 6th deciles d. 7th and 8th deciles 3-16. The interdecile range lies between the a. 3rd and 7th deciles b. 2nd and 8th deciles c. 1st and 9th deciles d. median and the 10th decile 3-17. The interquartile range includes the middle-most a. 25% of the cases b. 50% of the cases c. 75% of the cases d. 95% of the cases 3-18. Of the following, which should not be calculated when the data are in ordinal form? a. median b. quartile c. decile d. mean 3-19. Of the following, which can be calculated even though the data are in ordinal form? a. interquartile range b. interdecile range c. the middle-most 50% of the scores d. all of these 3-20. The standard deviation is a measure of a. how far all the scores vary from the mean b. how far the middle-most 50% of cases vary from the mean c. how far the highest and lowest scores vary from each other d. central tendency

3-21. The standard deviation must always be calculated with reference to the a. decile b. median c. mode d. mean 27

3-22. The more heterogeneous the distribution of scores, a. the smaller the standard deviation b. the larger the standard deviation c. the more the distribution tends to be skewed d. the greater the chance of excluding extreme scores 3-23. The deviation score is a. exactly the same as the standard deviation b. only the same as the standard deviation when variability is small c. only the same as the standard deviation when variability is large d. the difference between the raw score and the mean 3-24. When a deviation score has a negative value, then a. the raw score is below the mean b. the raw score is above the mean c. the raw score equals the mean d. none of these, since deviation scores can never be negative 3-25. When the standard deviation has a negative value, then a. most scores were above the mean b. most scores were below the mean c. the distribution is badly skewed d. none of these, since the standard deviation can never be negative 3-26. If the value of the standard deviation is greater than the value of the range, then a. the distribution is heterogeneous b. the distribution is homogeneous c. the distribution is skewed d. none of these, since the standard deviation must always be less than the range 3-27. If two football running-backs have the same yards-per-carry average, but player A has a larger standard deviation than player B, then a. player A has a greater chance of making a long gain b. player A has a greater chance of being caught for a loss c. player A is the more consistent runner d. a and b, but not c 3-28. Of two golfers, each of whom drives the ball the same average distance, the one with the smaller standard deviation would be a. the more consistent b. the less consistent c. the one more apt, on a given shot, to drive the ball further d. none of these, since they both drive the same average distance 28

3-29. The higher the value of the standard deviation, the higher the a. percentile b. decile c. quartile d. none of these 3-30. If two distributions have the same mean, then they must also have the same a. median b. mode c. standard deviation d. none of these 3-31. A distribution with a large standard deviation must also have a large a. number of scores near the mean b. number of scores near the median c. variance d. none of these 3-32. The standard deviation demands that the data form be at least a. nominal b. ordinal c. interval d. none of these, since the standard deviation has no data-scale requirements 3-33. The variance is equal to a. the standard deviation squared b. the square-root of the standard deviation c. twice the standard deviation d. none of these 3-34. You know you've made a math error whenever the standard deviation is a. smaller than the variance b. negative c. smaller than the range d. b and c, but not a 3-35. The only time the standard deviation may equal zero, is when a. the range is less than 50 b. every score in the distribution is the same c. the range is negative d. none of these, since the standard deviation may never be equal to zero

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3-36. The only time the range may equal zero, is when a. the standard deviation is less than 1.00 b. every score in the distribution is the same c. the standard deviation is negative d. the interdecile range is skewed to the left 3-37. A platykurtic distribution always has a. a relatively small standard deviation b. a relatively large standard deviation c. a standard deviation of zero d. a few extreme scores at just one end of the distribution

3-38. The term which defines the flatness or peakedness of a curve is a. kurtosis b. range-values c. central tendency d. homoscedasticity 3-39. When most scores cluster closely around the mean, the curve is a. asymmetrical b. platykurtic c. shaped like a U d. leptokurtic 3-40. The smaller the standard deviation a. the more leptokurtic the resulting curve b. the more mesokurtic the resulting curve c. the more platykurtic the resulting curve d. the higher the chance of the curve being bimodal 3-41. When the range of a unimodal distribution equals approximately 6 times the standard deviation, then the curve is most likely to be a. skewed b. leptokurtic c. platykurtic d. mesokurtic 3-42. If two distributions have the same mean and same range, then they must also a. have the same standard deviation b. when graphed, have the same kurtosis c. have the same variance d. none of these

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3-43. When an extremely homogeneous set of scores is graphed, then the resulting frequency distribution curve is a. platykurtic b. leptokurtic c. mesokurtic d. bimodal 3-44. If the range on a mesokurtic distribution is equal to 60, then the standard deviation will be approximately equal to a. 10 b. 360 c. .17 d. impossible to determine from the facts given 3-45. Of the following, indicate which is not a measure of variability a. range b. percentile c. standard deviation d. variance

Questions 46 through 54 will be based on the following: A researcher gives IQ tests to a large group of adults, and discovers that the mean, median and mode all equal 100. The highest score in the group was 130 and the lowest was 70. Between IQ's of 75 and 125, 80% of the group was found, and 50% were found between IQ's of 90 and 110. The shape of the graphed distribution was mesokurtic.

3-46. The range was equal to a. 60 b. 50 c. 20 d. 0 3-47. The interdecile range was equal to a. 60 b. 50 c. 20 d. 0 3-48. The interquartile range was equal to a. 60 b. 50 c. 20 d. 0 31

3-49. The fifth decile coincided with an IQ of a. 70 b. 75 c. 100 d. 125 3-50. The second quartile coincided with an IQ of a. 70 b. 75 c. 100 d. 125 3-51. The ninth decile coincided with an IQ of a. 75 b. 100 c. 125 d. 130 3-52. The first decile coincided with an IQ of a. 75 b. 90 c. 100 d. 130 3-53. The third quartile coincided with an IQ of a. 75 b. 90 c. 110 d. 125

3-54. The standard deviation was equal to approximately a. 0 b. 10 c. 15 d. cannot tell, since not enough information is provided 3-55. The variance was equal to a. 0 b. 100 c. 3.87 d. 225

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C. True or False: For the following, indicate T (True) or F (False) 3-56. The distribution with the largest standard deviation necessarily has the largest range.

3-57. Two distributions may have the same mean, and yet have different standard deviations.

3-58. A negative standard deviation indicates that most of the scores were below the mean.

3-59. The standard deviation may only be computed with reference to the mean.

3-60. A leptokurtic distribution has a relatively small standard deviation.

3-61. The interquartile range includes only the middle-most 25% of the distribution.

3-62. The standard deviation of any distribution must always equal one-sixth of the range.

3-63. The median must always coincide with the fifth decile.

3-64. If you were to receive a score on an exam which placed you at the 90th percentile, then 90% of those taking the test scored below you.

3-65. The higher the value of the standard deviation, then the more the scores are spread out around the mean.

3-66. The higher the value of the mean, then the higher the value of the range.

3-67. When the vast majority of scores cluster tightly around the mean, the graphed distribution is leptokurtic.

3-68. The interdecile range includes the middle-most 90% of all the scores.

3-69. The smallest possible standard deviation is equal to zero.

3-70. Kurtosis refers to the flatness or peakedness of the curve.

3-71. The standard deviation may never be larger than the range.

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3-72. If a student takes a test and gets 70% of the items right, then the student's score is at the 70th percentile.

3-73. If a student takes a test and gets a raw score which is at a point that is greater than 70% of all students taking the same test then the student's score is at the 70th percentile.

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D. For the following questions, calculate the values.

Questions 74 through 82 will be based on the following: The Children's Depression Inventory (CDI), developed from Beck's Depression Inventory is a self-report instrument with 27 sets of statements. Higher scores indicate higher levels of depression. Scores above 13 are said to indicate a Major Depressive Disorder (MDD) and few false positives have been reported. A random sample of 20 fifth grade students was selected and given the test. Their scores were as follows: 9,10,11,9,8,9,17,2,7,11,6,12,9,10,9,9,8,10,14,1.

3-74. Find the mean score for the group.

3-75. Find the median score for the group.

3-76. Find the mode for the group.

3-77. Find the range for the group.

3-78. Find the standard deviation for the group.

3-79. Is the distribution skewed?

3-80. If the distribution had been absolutely normal, what standard deviation (standard value) would you have predicted for a group of scores with this range value?

3-81. In comparing your predicted value with the actual value for the standard deviation, what conclusions might you draw regarding the shape of the distribution.

3-82. What percentage of the group scored in the danger zone (13 or above)?

For questions 83 through 88, use the following data: A researcher wished to evaluate the operating life of a certain brand of storage battery. A group of 30, brand-new batteries was selected and placed in 30 identical automobiles. The lights in each automobile were then turned on (without running the engines), and the length of time before the batteries were fully discharged (lights went out) was recorded. The scores, in hours, were: 15, 10, 20, 18, 18, 11, 11, 11, 11, 12, 12, 12, 10, 9, 9, 8, 9, 9, 11, 11, 11, 15, 12, 12, 12, 10, 12, 12, 15, 12.

3-83. Calculate the mean.

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3-84. Calculate the median.

3-85. Calculate the mode.

3-86. Calculate the range.

3-87. Calculate the standard deviation.

3-88. Calculate the variance.

3-89. For the following set of scores, calculate the mean, range and standard deviation: 12, 2, 5, 9, 11, 10, 10, 4.

3-90. For the following set of scores, calculate the standard deviation: 10, 8, 12, 3, 6, 7, 2, 10

3-91. For the following set of scores, calculate the standard deviation:.

13, 14, 12, 15, 10, 7, 12

3-92. For the following set of scores, calculate the standard deviation: 5, 10, 10, 5, 5, 2, 3, 5, 9, 7.

3-93. For the following set of scores, calculate the standard deviation; 5,8,10,10,12,12,12,14,14,16,19.

3-94. For the following set of scores, calculate the standard deviation: 8,9,10,12,10,10,16,11,24.

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For the following, use SPSS: A forensic psychologist was hired to test a group of police recruits and weed out those who might have any anti-social tendencies. The psychologist chose to test them on the MMPI2 and use their scores on the ASP Scale (Anti Social Practices) to identify any possible problems among the group. Their scores were as follows: 50,53,53,47,48,55,44,46,47,45,68,43,50,51.

3-95. Find the mean

3-96. Find the median 3-97. Evaluate the distribution for Skewness 3-98. Evaluate the distribution for Kurtosis

A group of inmates at a State House of Correction were tested during orientation on the LSI (Level of Security Inventory) to assess their possible security risk while incarcerated.

Their scores were as follows: 4,8,2,4,6,5,7,3,4,6,5,7,4,5,4,3,7,6,5,5.

3-99. Find the mean

3-100. Find the median 3-101. Evaluate the distribution for Skewness 3-102. Evaluate the distribution for Kurtosis

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CHAPTER 4 THE NORMAL CURVE AND Z SCORES A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Means, Z Scores and Skewness Remind students that the percentage areas for assessing z scores are only appropriate when the distribution is indeed normal. For example when a distribution is skewed, or leptokurtic or platykurtic or bimodal these z percentage areas are no longer valid. Let's take a skewed distribution, with a mode of 100, a median of 110 and a mean of 120. In this case an individual who scored right at 120 would have a z score of 0, regardless of the size of the standard deviation, but would not be at the 50th percentile. The 50th percentile would coincide with a score of 110, the median of this skewed distribution. In fact, the mean might be at the 84th percentile, which under the normal curve would convert to a z of +1.00. Stress the fact that with any skewed right distribution, the mean must be above the 50th percentile, and with a skewed left distribution, the mean would have to be below the 50th percentile. Also, point out that with a bimodal distribution that is perfectly balanced, the mean and median may coincide, both at the 50th percentile, but that since the distribution is not unimodal, it cannot be normal and the z percentages will then be useless. Finally point out that for platykurtic and leptokurtic distributions, both of which fail to conform to the R/6 rule for mesokurtosis, the z scores would again tell a very misleading and inaccurate story.

Activity: Small Samples and Z Scores Some students will attempt to calculate z scores on small sets of sample data, and then ask why the table reports one result when their eyes tell them a different story. Take the following sample of twenty IQ scores: 145,113,105,104,103,102,101,101,100,100,100,100,99,99,98,97,96,95,87,55.

The mean is 100, as is the median and mode, and the SD is 15, which in this case is indeed 1/6 of the range. It is unimodal and it's symmetrical. By those standards, it sure looks normal. But, taking a score of say 113, which converts to a z of .87, we find that 30.78% of all the cases should fall between that z score and the mean, and that therefore 19.22% should fall above that score. Yet by simply eye-balling the data, it's obvious that there's only one score (the 145) above the 113, and one score out of 20 is only 5%, not 19.22%. Small data sets can never be perfectly NORMAL. Statisticians talk about at least 1000 cases before the distribution can approach normality, and an infinite number of scores to be absolutely normal.

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Activity: Remembering z Scores The two z scores that students should probably put into their long-term memory banks are +1.96 (which include the middle-most 95% of the area under the curve) and + 2.58 (which include the middle-most 99% of the area under the curve). These two are used so often that a few moments spent in their memorization can pay time-saving dividends later in the course. Memory experts typically use encoding devices known as mnemonics (named after the Greek goddess of memory, Mnemosyne) and several mathematicians, who also happen to be memory freaks, have shown what seem to be amazing feats with regard to their skills at retrieving enormously long strings of digits. Some have demonstrated, for example, the ability to recall the value of pi to hundreds of decimal places, a spectacle that must be of enormous interest to the audience.

Yeah, right! To show how this is done, write the value of pi to ten places on the chalk board as follows: 3.1415926536 Explain to your students that although it seems to be impossible to chunk these digits into groups and then memorize them as numbers. An easier technique is to learn a simple series of sentences. Then, count each letter of each word and retrieve the digits on the basis of the number of letters per word. An example for pi might be: "How I want a drink. Yesterday my thirst would not quench".

Breaking it down, we have how (three letters), I (one letter), want (four letters), a (one letter), drink (five letters) and so on. Within a few minutes anyone who can count may have the seemingly astonishing ability to rattle off pi to ten places. Now, using the same technique, let's take the z score 1.96 and learn it as: "A memorized number" And for 2.58, we could use "My other mnemonic" Or for students who lose confidence over the spelling of mnemonic "My other favorite" This demonstration often results in lively interactions in the classroom, as students vie with each other over the creation of more interesting, and often far more provocative mnemonic sentences.

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B: Multiple Choice Items 4-1. The creation of the normal curve concept is popularly credited to a. Blaise Pascal b. Henry Gossett c. Sir Bryant McKenna d. Karl Gauss 4-2. The normal curve is a. symmetrical b. unimodal c. asymptotic to the abscissa d. all of these 4-3. A bimodal distribution may never be a. symmetrical b. a frequency distribution c. normal d. all of these 4-4. A normal curve always has a. a greater frequency of scores around the center than in the tails b. a greater frequency of scores in the tails than around the center c. a greater frequency of scores above the mean than below the mean d. a greater frequency of scores below the mean than above the mean 4-5. Under the normal curve a. the mean lies to the right of the median b. the mode lies to the left of the mean c. the median lies to the left of the mode d. none of these 4-6. The normal curve is a. a frequency distribution curve b. leptokurtic c. platykurtic d. skewed to the left

4-7. When the normal curve is plotted according to standard deviation units, each having a value of 1.00, it is called a. platykurtic b. leptokurtic c. the standard normal curve d. the deviation curve 40

4-8. Under the normal curve, 68% of the cases must always fall a. above the mean b. between + 1 standard deviation units from the mean c. between + 2 standard deviation units from the mean d. all of these, depending on the particular shape of the curve 4-9. Under the normal curve, 50% of the cases must always fall a. below the mean b. above the mean c. below the median d. all of these 4-10. When more than 34% of the cases under the curve fall between the mean and a z score of + 1, then a. 68% of the cases fall below the mean b. the curve cannot be symmetrical c. the curve cannot be normal d. none of these 4-11. When the mean and median do not coincide, then a. the curve cannot be normal b. the curve cannot be skewed c. the curve cannot be unimodal d. the curve cannot be a frequency distribution curve 4-12. Under the normal curve the 50th percentile always falls at the a. mean b. median c. mode d. all of these 4-13. Under the normal curve, between the z scores of + 1 and + 2, there are always a. 68% of the cases b. 95% of the cases c. 13.50% of the cases d. sometimes a, and sometimes b, but never c 4-14. Under the normal curve, the percentage of cases falling above a z score of + 3, is a. 68% b. 95% c. more than 1% d. less than 1%

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4-15. Under the normal curve, between z scores of + 10, there are always a. 100% of the cases b. 95% of the cases c. 68% of the cases d. more than 99% but less than 100% of the cases 4-16. When a z score falls to the left of the mean a. it must always be given a minus sign b. it must always be greater than 1 c. it must always be less than 1 d. none of these, since z scores never fall to the left of the mean 4-17. If an IQ distribution is normal and has a mean of 100 and a standard deviation of 15, then 68% of all those taking the test scored between IQ's of a. 100 and 115 b. 85 and 100 c. 92.5 and 107.5 d. 85 and 115 4-18. If an IQ distribution is normal and has a mean of 100 and a standard deviation of 15, then 99% of all those taking the test scored between IQ's of a. 0 and 150 b. 55 and 145 c. 85 and 115 d. 92.5 and 107.5 4-19. Under the normal curve, when the z score is equal to + 1, then a. the standard deviation of the distribution of raw scores must equal 15 b. the standard deviation of the distribution of raw scores must equal 10 c. the standard deviation of the distribution of raw scores must equal 0 d. none of these 4-20. Under the normal curve, if the mean of the distribution of raw scores is equal to 68, then its equivalent z score is equal to a. 10 b. 1 c. 0 d. cannot tell, since the SD is not given 4-21. Assume a normal distribution of height scores, with a mean of 68" and a standard deviation of 3", then a. 68% of the cases must fall between 65" and 71" b. 50% of the cases must fall below 68" c. 68% of the cases must fall at exactly 68" d. a and b, but not c 42

4-22. The larger the absolute value of the z score (regardless of its sign), then a. the higher its equivalent raw score b. the lower its equivalent raw score c. the further it is from the mean d. the closer its equivalent raw score must be to the mean 4-23. The z score provides direct information regarding how far a given raw score is a. from the mean in units of standard deviation b. from the mean in percentage units c. from the lowest score in percentile units d. from the highest score in percentile units 4-24. On any normal distribution the 50th percentile corresponds with a z score of a. 0 b. 50 c. 68 d. + 1 4-25. Under the normal curve, if a given raw score falls at the 84th percentile, then its equivalent z score must be equal to a. the mean b. 0 c. + 1 d. + 2 4-26. Under the normal curve, any percentile of less than 50 must correspond to a a. negative z score b. raw score of less than 50 c. z score of less than 1 d. z score of 0 4-27. The closer a given z score is to 0, then the closer it is a. to the tails of the curve b. to the mean of the distribution c. to the 68th percentile d. to a 0% frequency 4-28. Under the normal curve, the only point where the frequency of cases is zero is a. between the mean and a z score of + 1 b. beyond a z score of + 1 c. beyond a z score of + 3 d. none of these 43

4-29. The higher a given point on the ordinate of the normal curve, then a. the higher the frequency of scores below that point b. the lower the frequency of scores below that point c. the closer the point is to the abscissa d. the higher the resulting raw score 4-30. A negative z score means that a. the raw score is below the mean b. the mean is below the raw score c. the distribution is skewed left d. none of these, since it is impossible to get a negative z score 4-31. Under the normal curve, the mean falls below the 50th percentile only when a. the mean of the raw score distribution is equal to 50 b. the mean of the raw score distribution is equal to 100 c. the mean of the raw score distribution is unknown d. none of these, since under the normal curve the mean must fall at the 50th percentile

4-32. Under the normal curve, the relationship between a specific z score and the percentage of cases falling between it and the mean a. depends on the size of the standard deviation b. depends on the value of the mean c. depends on the value of the raw score d. is a constant For questions 33 through 42 use the following: The mean on a given normal distribution of raw scores is 500 with a standard deviation of 100 4-33. A z score of + 1 corresponds with a raw score of a. 200 b. 400 c. 500 d. 600 4-34. The 84th percentile corresponds with a raw score of a. 250 b. 500 c. 600 d. 800

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4-35. The 49th percentile must be below a raw score of a. 500 b. 400 c. 300 d. 200 4-36. Between the raw scores of 400 and 600, there must be a. 50% of the scores b. 34% of the scores c. 68% of the scores d. 95% of the scores 4-37. A raw score of 200 would translate into a z score of a. -3 b. -1 c. +1 d. +2 4-38. A raw score of 500 would translate into a z score of a. -3 b. -1 c. -.01 d. 0 4-39. Approximately 68% of all the scores would fall a. between raw scores of 400 and 600 b. between raw scores of 200 and 500 c. between raw scores of 500 and 600 d. above a raw score of 700 4-40. The 50th percentile would yield a raw score of a. 400 b. 450 c. 500 d. 568 4-41. The 50th percentile would yield a z score of a. -1 b. +500 c. above +1 d. 0

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4-42. Over 50% of all the scores would fall above a raw score of a. 568 b. 534 c. 501 d. 499 4-43. When a given point under the normal curve is 1.54 standard deviation units above the mean, then a. its percentile is 15.4 b. its percentile is 1.54 c. its z score is +1.54 d. both a and c, but not b 4-44. All percentiles of less than 50 must necessarily a. fall below the raw score mean b. yield negative z scores c. fall to the left of the z score mean d. all of these 4-45. A positive z score must always yield a percentile which is a. above 50 b. below 50 c. between 40 and 60 d. above 100 4-46. Percentiles always indicate a. the percentage of cases falling below a given point b. the percentage of cases falling above a given point c. the percentage of cases falling between a given z score and the mean d. the percentage of cases falling above a given z score 4-47. A z score of +3 corresponds with a a. percentile of 3 b. percentile of 53 c. percentile of 103 d. raw score which is 3 standard deviation units above the mean 4-48. For a z score of 1.35 the z score table indicates a percentage value of 41.15. This means that a. 41.15% of the cases fell below a z score of +1.35 b. 41.15% of the cases fell above a z score of +1.35 c. 41.15% of the cases fell between z scores of +1.35 d. 41.15% of the cases fell between a z score of +1.35 and the mean

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4-49. The z score table always gives, as a direct read-out, the percentage of cases falling a. between a given z score and the mean b. below a negative z score c. above a positive z score d. none of these

4-50. The percentage values on the z score table are only valid when a. the z scores are positive b. the raw score mean equals 50 c. the raw score mean is greater than 50 d. the distribution is normal 4-51. The difference between z scores and standard scores is that a. z scores can never be negative, whereas standard deviation scores always are b. z scores can only be calculated with reference to the mean, whereas standard scores relate to the median c. z scores can only be calculated with reference to the standard deviation, whereas standard scores relate to the variance d. none of these, since z scores are standard scores 4-52. Every distribution of z scores has a a. mean of zero and a standard deviation of 1 b. mean of 50 and a standard deviation of 10 c. mean of 100 and a standard deviation of 15 d. none of these, since every distribution of z scores is different 4-53. The fact that the normal distribution is asymptotic to the abscissa means that a. its curve must be symmetrical b. its curve must yield both positive and negative values c. its curve may never touch the abscissa d. its curve must always touch the abscissa 4-54. The numerical difference between the raw score and the mean, when divided by the standard deviation, yields the a. normal curve b. percentile c. first moment of the curve d. z score

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4-55. The only time a z score can be negative is when a. the raw score is higher than the mean b. the raw score is lower than the mean c. the standard deviation is negative d. the standard deviation is equal to 0 4-56. When the standard deviation of the raw score distribution is negative, the resulting z score must be a. negative b. equal to 0 c. positive d. none of these, since the standard deviation can never be negative 4-57. Between two negative z scores there must always be a. fewer than 50% of the cases b. more than 50% of the cases c. exactly 50% of the cases d. none of these, since 0% must always fall between negative z scores

4-58. Anytime a z score yields a percentile which is greater than 50, then a. the z score must be positive b. the raw score must be higher than the mean c. the z score must be greater than 0 d. all of these 4-59. The z score for any raw score may be found by a. subtracting the mean from the raw score and dividing by the standard deviation b. subtracting the raw score from the mean and dividing by the standard deviation c. subtracting the raw score from 100 and dividing by 15 d. none of these, since raw scores cannot be converted into z scores 4-60. Whenever a given raw score coincides exactly with the mean, the resulting z score, regardless of the size of the standard deviation, must be a. equal to 0 b. positive c. equal to 50 d. none of these, since in this case, the z score cannot be found

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C. True or False: For the following, indicate T (True) or F (False) 4-61. Between z scores of + 3 over 99% of the cases will be found.

4-62. A negative z score always yields a percentile of less than 50.

4-63. A negative z score always means that the raw score is less than the mean.

4-64. Fewer than 50% of the cases may ever fall between two z scores of the same sign.

4-65. Whenever the mean of a raw score distribution is less than 50, the z score must be negative.

4-66. The mean of the z score distribution must always equal 0.

4-67. A frequency distribution curve may be symmetrical without being normal.

4-68. On a normal distribution the mean always lies to the right of the mode.

4-69. There are more cases which fall above a z score of +2 than there are falling below it.

4-70. Under the normal curve, the further to the right on the abscissa, the greater the number of cases.

4-71. When any frequency distribution curve is symmetrical, it must, by definition, also be normal.

4-72. The normal curve is mesokurtic.

4-73. All percentiles above 50 must yield negative z scores.

4-74. The normal distribution curve is unimodal.

4-75. A positive z score always results when the raw score is greater than the mean.

4-76. A positive z score may only occur when the mean of the raw score distribution is greater than 50.

4-77. A percentile of greater than 50 may only occur when the mean of the raw score distribution is greater than 50.

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4-78. Regardless of the size of the raw score mean of standard deviation, a z score of +1 always means that the raw score is exactly one, full standard deviation unit about the mean.

4-79. Raw scores may never be converted into z scores unless the raw score mean is equal to 0.

4-80. The standard score is synonymous with the raw score.

4-81. On a normal distribution the mean and median coincide.

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D: For the following questions, calculate the values.

4-82. Percentage of scores falling between a z of 1.54 and the mean

4-83. Percentage of scores falling between a z of -1.73 and the mean

4-84. Percentage of scores falling between a z of .56 and the mean

4-85. Percentage of scores falling below a z of 2.56

4-86. Percentage of scores falling below a z of 1.44

4-87. Percentage of scores falling below a z of -.24

4-88. Percentage of scores falling above a z of 1.44

4-89. Percentage of scores falling above a z of -.67

4-90. Percentage of scores falling between z's of -.80 and +.80

4-91. Percentage of scores falling between z's of .50 and 1.25

4-92. Percentage of scores falling between z's of -.53 and +.84

4-93. Percentage of scores falling between z's of -.15 and -.44

4-94. Percentage of scores falling between z's of -.80 and -2.00

4-95. On a normal distribution with a mean of 200 and a standard deviation of 50, what percentage of cases will fall between raw scores of 185 and 195?

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4-96. On a normal distribution with a mean of 500 and a standard deviation of 97, what percentage of cases will fall between 470 and 550?

4-97. On a normal distribution with a mean of 100 and a standard deviation of 17, what percentage of cases will fall between 84 and 93?

4-98. On a normal distribution with a mean of 22.50 and a standard deviation of 6.20, what percentage of cases will fall between 24 and 28?

4-99. On a normal distribution with a mean of 150 and a standard deviation of 17, what percentage of cases will fall between 129 and 145?

4-100. On a normal distribution with a mean of 100 and a standard deviation of 7, what percentage of cases will fall between 129 and 145?

4-101. On a normal distribution with a mean of 150 and a standard deviation of 15, what percentage of cases will fall at 140 or higher?

4-102. On a normal distribution with a mean of 160 and a standard deviation of 16.28, what percentage of cases will fall between 145 and 166?

For questions 103 through 108, use the following: The mean weight among a group of female high school seniors is 120 lbs., with a standard deviation of 16.49 lbs. Assuming a normal distribution, find the percentiles for the following weight scores.

4-103. 125 lbs.

4-104. 140 lbs.

4-105. 115 lbs.

4-106. 90 lbs.

4-107. 107 lbs.

4-108. 165 lbs.

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4-109. Find the percentage of cases falling between z scores of + .80 and -.80..

For questions 110 through 115, find the percentiles for the following: the mean score on a test of psychopathy among a group of incarcerated inmates is 50, with a standard deviation of 10. Assume a normal distribution, and find the Percentile for the following psychopathy scores: 4-110. 57

4-111 60

4-112 48 4-113 55 4-114 68

5-115. 75 116. Using SPSS, find the z scores for all the scores in the following distribution of ASP (Anti-Social Personality) scores: 70, 50, 50, 30, 35, 65, 48, 52, 49, 51, 50, 53, 47, 51, 49, 52, 48, 60, 40, 48, 52, 50, 53, 47, 51, 49, 50, 56, 44, 58, 42, 50, 51, 49, 50.

117. Using SPSS, find the z scores for all the SAT scores in the following distribution: 500, 480, 520, 490, 510, 540, 460, 350, 650, 700, 300, 480, 500, 520, 500.

Point out to your students that the z scores they found that were identical, save for sign (such as -.20755 and .20755) were caused by the fact that they are based on raw score values that are exactly the same distance from the mean (as in the case just shown which has a mean of 500 and is based on raw scores of 480 and 520). This helps reinforce the z score definition as a value found by subtracting the raw score from the mean and then dividing by the standard deviation.

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CHAPTER 5 Z SCORES, T SCORES AND OTHER NORMAL CURVE TRANSFORMATIONS

Activity:Curing the Algebra Phobe Sooner or later in your teaching career, probably sooner, you will run across a student who although not really math disabled is very ill at ease when it comes to algebraic equations. In fact, in this day and age of electronic calculators and statistical computer programs, the student who fears arithmetic is actually easier to deal with than is the algebra-phobe. You can explain patiently to the former that the arithmetic involved in the course will all be done by electronics, but for the algebra-phobe a different tactic may be tried. For example when solving for the various unknowns in the z score equation, the algebra-phobe will look up at you in sheer terror. One little device that has been found useful is to give such a student the following diagram:

Using the Thumb to Cover the Unknown and Solve the Equation The idea here is to have the student cover the term to be found, and read out what's left uncovered as the answer. Tell your student, for example, that if the Z score is needed, cover the Z and read what's left uncovered, which

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will then be X-M over the SD. Now have the student stop and write out the equation, that is Z = X-M SD Similarly, if the SD is needed use the same procedure, that is cover the SD and read X-M over Z as the answer. Again, stop and write out this equation, SD = X-M Z If either the mean or the raw score are needed, cover the X – M and write out the equation that is now left uncovered, that is X - M = (Z)SD).

Then if it's the raw score, X, that is asked for, keep it where it is, to the left of the equals sign, and place the M to the right of the equals sign and change its sign to +, so that X = (Z)(SD) + M

Finally, if the M is needed, again write out X-M = (Z)(SD).Keep -M on the left of the equals sign and change its sign to plus. Then put the X on the right side and change its sign to -, such that M = (Z)(SD) - X Activity: Translating into T scores

Again, cover the term being asked for and write out the resulting equation.

Thus, if one wanted the z score, cover the z and read the equation as Z = T-50/10

.

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Or to get the T score, cover the T-50 and read the equation as T-50 = (Z)(10). Then move the 50 to the right of the equals sign and change it to a plus, or T = (Z)(10) +50

Activity: Using Table 5.1 - Stanines When discussing stanines, deciles, quartiles and normal-curve equivalents, the use of Table 5.1 should put even the fearful students at ease. It will also reinforce their awareness of both z scores and percentiles, and the relationship between the two. I've had students who finally had the magic moment of insight while working out transformations directly from the table.

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B: Multiple Choice Items 5-1. When converting a z score into a raw score, anytime the z score is negative in sign a. so too will be the raw score b. so too will be the mean c. so too will be the standard deviation d. the value of the raw score will be less than the mean 5-2. When converting a z score into a raw score, anytime the z score is positive in sign a. so too will be the raw score b. so too will be the mean c. the value of the raw score will be greater than the mean d. none of these, since the conversion can only be made when z is negative 5-3. A z score of -1.50 indicates that the raw score is a. one and one-half percentage points below the mean b. one and one-half standard deviation units below the mean c. one and one-half raw score units below the mean d. negative 5-4. A z score of +3.00, means that a. the raw score is exactly 3 standard deviation units above the mean b. the raw score is exactly 3 percentage points above the mean c. the raw score is exactly 3 raw score units above the mean d. none of these 5-5. On a normal distribution, percentiles may be converted into raw scores a. by first translating the percentile into a z score b. by first translating the mean into a standard score c. by first translating the percentile into its decile equivalent d. all of these 5-6. Any percentile which is greater than 50 will convert into a raw score a. which is greater than 50 b. which is to the left of the mean c. which is greater than the mean d. which is larger than the value of the percentile 5-7. A percentile rank of 95 indicates that a. 95% of the distribution is above it b. 47 and a half % of the distribution is above it c. 95% of the distribution is below it d. 47 and a half % of the distribution is below it 57

5-8. A percentile rank of 45 indicates that a. it is 5 percentage points below the mean b. it is 45 percentage points below the mean c. it is 95 percentage points below the mean d. it is above the mean 5-9. A percentile rank of 85 indicates that a. 15% of the distribution is below it b. 35% of the distribution is below it c. 35% of the distribution is between it and the mean d. it is below the mean 5-10. When converting percentiles into z scores, it is essential to know a. the value of the mean b. the value of the standard deviation c. the value of the raw score d. the percentage of cases falling between the given percentile and the mean 5-11. Whenever the value of the standard deviation is positive, then a. the z score must be positive b. the percentile must be greater than 50 c. the raw score must be above the mean d. none of these, since the standard deviation is always positive 5-12. Whenever the value of the z score is positive, then a. the percentile must be greater than 50 b. the raw score must be greater than the mean c. neither of these, since the z score is always positive d. both a and b, but not c 5-13. All z scores falling to the left of the mean a. must be given negative signs b. must result in negative percentiles c. must result in negative raw scores d. none of these 5-14. Whenever the percentage values in the z score table indicate an absolute tie a. use the higher of the two z scores b. use the lower of the two z scores c. use the z score which is closest to the mean d. the z score cannot be used

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5-15. If someone were to score at the 84th percentile, the resulting z score would be approximately a. .50 b. 1.00 c. .34 d. none of these, since a percentile never yields a z score 5-16. If someone were to score at the 16th percentile, the resulting z score would be approximately a. -.50 b. 1.00 c. -.34 d. none of these, since a percentile never yields a z score 5-17. On a normal distribution, the value of the standard deviation can be found from a knowledge of the a. mean, median and mode b. mean, z score and raw score c. z score, raw score and percentile d. curve's kurtosis 5-18. If the mean of a normal distribution were equal to 100, and a raw score of 99 was found to be at the 10th percentile, then a. the standard deviation must be extremely small b. the standard deviation must be extremely large c. the standard deviation must be less than the z score d. the standard deviation could not be calculated from this information 5-19. If the standard deviation of a normal distribution were equal to 15 and someone had a raw score of 122, which fell at the 90th percentile, then the mean a. must be greater than 122 b. must be less than 122 c. must be between 90 and 122 d. could not be found from the above information 5-20. Whenever the z score equation is used in order to calculate the mean, the mean must always be greater than the raw score a. when the raw score has a percentile value of less than 50 b. when the raw score has a percentile value of greater than 50 c. when the raw score is negative d. when the z score is positive

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5-21. Whenever the z score equation is used in order to calculate the mean, the mean must always be less than the raw score a. when the raw score has a percentile value of less than 50 b. when the raw score is positive c. when the z score is negative d. when the raw score has a percentile value of greater than 50

5-22. The percentile value for a given raw score must always be at least 50 or above whenever a. the mean is greater than the raw score b. the raw score is more than 1 percentage point away from the mean c. the equivalent z score is greater than 50 d. the equivalent z score is positive 5-23. When a raw score translates into a negative z score, the resulting percentile must be a. negative b. greater than 50 c. less than 50 d. none of these 5-24. When a raw score translates into a negative z score, then the mean of the distribution a. must be higher than the raw score b. must be lower than the raw score c. must fall above the 50th percentile d. must fall below the 50th percentile 5-25. In a normal distribution, the mean falls above the 50th percentile only when a. the z score is positive b. the raw score is negative c. the raw score is below the 50th percentile d. none of these 5-26. The T score can only be used a. when the mean of the raw score distribution equals 50 b. when the z score distribution is no longer normal c. when the mean of the raw score distribution is 100 d. when the distribution is normal

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5-27. The value of the mean of the T score distribution is always equal to a. the value of the raw score mean b. the value of the z score mean c. the value of the standard deviation d. 50 5-28. The range of the T score distribution always equals approximately a. 50 b. 60 c. 30 d. the same value as the range of the raw score distribution 5-29. A T score of 60 is equal to a z score of a. 0 b. +2 c. +1 d. 50 5-30. A T score of 30 is equal to a z score of a. -2 b. +2 c. +1 d. 0 5-31. When a raw score converts to a percentile of greater than 50, then the resulting T score must be a. negative b. equal to 10 c. equal to the z score d. greater than 50 5-32. A T score of 60 is exactly a. one standard deviation unit above the mean b. ten standard deviation units above the mean c. ten standard deviation units below the mean d. the same as a percentile rank of 60 5-33. The value of the standard deviation of the T distribution is equal to a. the standard deviation of z b. the standard deviation of the raw score distribution c. the standard deviation of the raw score distribution, divided by 10 d. 10

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5-34. If the mean of the raw score distribution equals 68 with a standard deviation of 3, then a person with a raw score of 65 would have a T score of a. 60 b. 50 c. 47 d. 40 5-35. If the mean of the raw score distribution equals 120 with a standard deviation of 5, then a raw score of 130 would have a T score of a. 55 b. 60 c. 70 d. 30 5-36. A raw score which is at the 50th percentile translates into a T score of a. 50 b. 10 c. 70 d. cannot tell from information given 5-37. When the standard deviation of the raw score distribution equals 10, then the mean of the raw score distribution must equal a. 50 b. 100 c. 15 d. cannot tell from information given 5-38. The T score distribution always assumes a a. mean of 50 and an SD of 10 b. mean of 100 and an SD of 15 c. mean of 0 and an SD of 1 d. none of these 5-39. The z score distribution always assumes a a. mean of 50 and an SD of 10 b. mean of 100 and an SD of 15 c. mean of 0 and an SD of 1 d. none of these 5-40. The raw score distribution always assumes a a. mean of 50 and an SD of 10 b. mean of 100 and an SD of 15 c. mean of 0 and an SD of 1 d. none of these 62

5-41. If the T score equals 51, then the equivalent raw score a. must be greater than 100 b. must be less than 100 c. must be greater than the raw score mean d. must be greater than the T score mean 5-42. A raw score which falls exactly at the 5th decile, must convert to a T score of a. 50 b. 100 c. 0 d. none of these 5-43. A raw score which falls exactly at the second quartile, must convert to a T score of a. 50 b. 100 c. 0 d. 10 5-44. On a T score distribution, 68% of the cases must fall between T scores of a. 50 and 60 b. 0 and 10 c. 40 and 50 d. 40 and 60 5-45. Any raw score which has a percentile rank of 99, has a T score of at least a. 10 b. 20 c. 80 d. 99 5-46. When T equals 50, then z must equal a. 50 b. 10 c. 0 d. cannot tell from the above information 5-47. When T equals 50, then the raw score must equal a. 50 b. 10 c. 0 d. cannot tell from the above information

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5-48. When the standard deviation of the raw score distribution is equal to less than 1.00, then a. the standard deviation of the T distribution becomes less than 10 b. the standard deviation of the z distribution becomes less than 10 c. the standard deviation of the z distribution becomes less than 1 d. none of these

5-49. When converting z to T, all positive z scores yield T scores a. of greater than 50 b. of greater than 10 c. of greater than 0 d. none of these, since z cannot be converted to T 5-50. The distribution of T scores is assumed to be a. Gaussian b. normal c. mesokurtic d. all of these

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C: True or False: For the following, indicate T (True) or F (False) 5-51. All z scores greater than 0 must coincide with percentiles which are greater than 50.

5-52. All T scores greater than 20 must coincide with percentiles which are greater than 50.

5-53. A person with a T score of greater than 50, must have a negative z score.

5-54. When T=50, then z=0.

5-55. The standard deviation of the T distribution must always be greater than 50.

5-56. When the value of the mean is less than the value of the raw score, then the raw score must convert to a T score of greater than 50.

5-57. The standard deviation of the T score distribution is always equal to 10.

5-58. If one knows the mean, the raw score and the z score, then the z score equation can be used to calculate the standard deviation.

5-59. If a distribution is normal, any percentile can be converted into its equivalent z score.

5-60. When the value of T equals 40, the z score must be positive.

5-61. When the z score is negative, so too is the T score.

5-62. For a given raw score to be converted into a z score, one must know both the mean and standard deviation of the raw score distribution.

5-63. When a T score is greater than 50, so too is its equivalent percentile.

5-64. Raw scores may never be converted into T scores unless the raw score distribution is normal.

5-65. The range of the T score distribution is approximately 10.

5-66. When a given raw score value is less than the mean of the raw score distribution, its equivalent z score must be positive.

5-67. On the T score distribution the standard deviation always equals 10.

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5-68. If the mean of an IQ distribution equals 100, with a standard deviation of 15, then an IQ score of 85 is equivalent to a T score of 35.

5-69. All T scores are positive in sign.

5-70. A z score of -1, indicates a point one standard deviation unit below the mean.

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D: For the following questions, calculate the values. Assume a normal distribution in all questions.

5-71. On a test with a mean of 200 and an SD of 50, what raw score would fall at the 86th percentile?

5-72. With a mean of 100 and an SD of 20, what raw score would fall at the 75th percentile?

5-73. The mean is 100, with an SD of 15. What raw score would fall at the 11th percentile?

5-74. The mean is 65, with an SD of 5.00. A raw score of 57 would fall at what percentile?

5-75. With a mean of 100 and an SD of 20, what raw score would fall at the 27th percentile?

5-76. The T score on a certain distribution is 28. The mean of the raw score distribution is 110, with an SD of 14.93. What is the value of the equivalent raw score?

5-77. The mean on a test is 75 with an SD of 10. What is the raw score for a percentile rank of 80?

5-78. With a mean of 105 and an SD of 6.30, what raw score falls at the 93rd percentile?

5-79. The mean is 38.70 with an SD of 6.31. What score falls at the 11th percentile?

5-80. The mean on a test is 90, with an SD of 20. What raw score would be equivalent to a percentile rank of 29?

5-81. On a distribution with a mean of 75 and an SD of 15, what raw score falls at the 95th percentile?

5-82. The mean height for a group of children is normally distributed, with a mean of 55" and an SD of 3.71". One child is at the 39th percentile of the height distribution. How tall is that child?

5-83. A certain raw score equals 85 and is at the 15th percentile. The standard deviation is 15.27. What is the mean of the distribution?

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5-84. A certain raw score equals 85 and is at the 85th percentile. The standard deviation is 15.27 What is the mean of the distribution?

5-85. A certain raw score equals 160 and is at the 53rd percentile. The standard deviation is 10.00. What is the mean of the distribution?

5-86. A certain raw score equals 71 and is at the 72nd percentile. The standard deviation is 3.00. What is the mean of the distribution?

5-87. A certain raw score equals 12 and has a percentile rank of 30. The standard deviation is 2.93. What is the mean of the distribution?

5-88. The mean equals 60. One individual scores at the 13th percentile and has a raw score of 50. What is the standard deviation?

5-89. The mean equals 55. One individual scores at the 39th percentile and has a raw score of 42. What is the standard deviation?

5-90. The mean equals 100. One individual scores at the 20th percentile and has a raw score of 85. What is the standard deviation?

5-91. The mean equals 100. One individual scores at the 58th percentile and has a raw score of 107. What is the standard deviation?

5-92. The mean equals 30. One individual scores at the 26th percentile and has a raw score of 26. What is the standard deviation?

For questions 93 through 97,find the T score.

5-93. z = 1.28

5-94. z = -.10

5-95. z = 2.25

5-96. z = 1.65

5-97. z = 0 68

5-98. With a mean of 150 and an SD of 17, find the T score for a raw score of 80.

5-99. With a mean of 68 and an SD of 3.07, find the T score for a raw score of 65.

5-100. With a mean of 15 and an SD of 2.38, find the T score for a raw score of 12.

5-101. With a mean of 68 and an SD of 2.95, find the T score for a raw score of 75.

5-102. The T score on a certain distribution is 28. The mean of the raw score distribution is 110 with an SD of 14.93. What is the value of the raw score?

5-103. The T score on a certain distribution is 57. The mean of the raw score distribution is 95 with an SD of 10.51. What is the value of the raw score?

5-104. The T score on a certain distribution is 75. The mean of the raw score distribution is 430 with an SD of 82. What is the value of the raw score?

5-105. The T score on a certain distribution is 20. The mean of the raw score distribution is 100 with an SD of 14.97. What is the value of the raw score?

5-106. With an SD of 7, a raw score of 55 is at the 23rd percentile. Find the mean of the raw score distribution.

5-107. With an SD of 35, a raw score of 145 is at the 53rd percentile, Find the mean of the raw score distribution.

5-108. With an SD of 15, a raw score of 115 is at the 84th percentile, Find the mean of the raw score distribution.

5-109. With an SD of 7, a raw score of 25 is at the 40th percentile, Find the mean of the raw score distribution.

5-110. With an SD of 10, a raw score of 60 is at the 58th percentile, Find the mean of the raw score distribution.

5-111. With an SD of 90, a raw score of 550 is at the 71st percentile, Find the mean of the raw score distribution.

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CHAPTER 6 PROBABILITY A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: The Lottery and Combinations When looking at the section (enclosed box) on combinations, permutations and statistical inference, students often challenge the claim that all winning lottery numbers are really equally likely. Isn't it more difficult to win with a consecutive string of numbers, such as 1, 2, 3, 4, 5 and 6 than with a random sequence? Point out to the class that all sequences are equally likely and that when one selects say 6 numbers out of 36, where the probability of winning is roughly one out of two million, that probability remains the same regardless of which six numbers are selected. Consecutive strings don't come up very often because there simply aren't many of them. The number groupings 1 through 6, 2 through 7, and so on up to 30 through 36, are very few (there are only 30 of them) when compared to the enormous number of non-consecutive groupings, and that's why they rarely appear as winning combinations. Rest assured however that any given combination of numbers is going to occur with the same frequency as any other combination. The trouble is that any combination of 6 numbers out of 36 is only going to win once in every 2 million selections. Despite class objections, this means that the combination that won last week's lottery is just as likely as any other combination to win next week, and the week after, and even the week after that.

Activity: Low Probabilities and the Impossible This might be used as a lead-in to the myth that extremely low probability events don't exist, as in statements that proclaim that the chances of certain things occurring are so low as to be impossible. Even though the probability of winning the lottery is extremely low, eventually someone does win.

Probability statements don't preclude the event (unless the probability is zero). In fact even very low probabilities insist that in the long run the event has to occur, very infrequently to be sure, but occur it eventually will. Ask your students for low probability events that may have occurred in their lives. Two stories might be used to get the ball rolling, both taken from newspaper accounts. At Fenway Park in Boston, two men were seated mid-way up in the bleachers, when they both noticed an old and dear friend sitting below them, just behind the bullpen (we'll call the friend Pete). Pete, as everyone who knew him was aware, was a recovering alcoholic who had almost lost his family, home and job because of his past drinking problem. As the friends gazed down, however, it was obvious that Pete at that moment had lost his resolve and had consumed an enormous quantity of beer. By the time the two concerned friends worked their way down to him, Pete was virtually comatose.

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After the game they were faced with the question of what to do with old Pete.

They decided to drive him home (to Providence, Rhode Island, which was far out of their way). When they arrived at Pete's home, they debated as to who was going to have the honor of ringing the doorbell and announcing to Pete's wife his current condition. One of them finally got out of the car, rang the bell, explained the situation to Pete's wife, only to be told that Pete was at that very instant comfortably seated in the living room, watching TV. The probability of having two persons look so much alike may be extremely low, but in Boston it happened. Incidentally, they took the sleeping stranger back to Boston and propped him up in front of the souvenir stand across from the ball park. Having a look-alike, although obviously rare, is certainly not impossible. Out of the 5 billion persons on the planet, there is almost certainly someone who comes very close to looking just like any one of you.

Maybe not close enough to fool your mother (mothers can differentiate between MZ twins), but close enough to fool most people. In fact, that's why they have those celebrity look-alike contests.

In another news account, a woman, a reading specialist from Connecticut, was in New York City on a shopping tour. As the day progressed she had to go to the bathroom, but could not find a public toilet. Finally in desperation, she went into a funeral home where a wake was in progress. She quickly signed the guest book, paid her respects to the deceased, and then asked the funeral director for the lady's room. Months later, now back at home, she received a letter from a New York attorney stating that the deceased had bequeathed money to anyone who showed up at his wake, and that her share was $18,000, check enclosed. This is not meant to encourage using toilets at funeral homes, just to stress that low probability events indeed do occur.

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B: Multiple Choice Items 6-1. Probability is defined as a. the number of total events over the number of chances b. the chances an event cannot occur over the chances it can c. the number of times a specific event will occur over thee total number of events d. the total number of events over the number of times a specific event will occur 6-2. Probability is equal to a. s/t b. t/s c. odds divided by chances d. none of these 6-3. A probability value of .25 means that an event will occur a. once out of 25 times b. 25 times out of 100 c. once out of 5 times d. 75 times out of 100 6-4. If an event can occur once out of 20 times, its probability value is a. .20 b. .80 c. .95 d. .05 6-5. If an event can occur five times out of 100, its probability value is a. .20 b. .80 c. .95 d. .05 6-6. Suppose you had flipped a coin 19 times and "heads" had come up each time.

The probability of flipping a "head" on the 20th flip would then be a. .50 b. .05 c. .95 d. .01

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6-7. Suppose you had flipped a fair coin 19 times and "tails" had come up each time. The probability of now tossing a "head" on the 20th flip would then be a. .50 b. .05 c. .95 d. .99 6-8. When events cannot influence or be influenced by other events, they are said to be a. low-probability events b. high-probability events c. chance events d. independent events 6-9. The so-called "Gambler's Fallacy" makes the mistake of not recognizing a. the low-probability events b. the high-probability events c. the independent-events situation d. that losses occur more frequently than wins 6-10. The so-called law of averages only applies a. under controlled conditions b. when the probabilities are conditional c. in the long run d. in the short run 6-11. If the odds against an event are 5 to 1, the probability of its occurrence is a. .17 b. .05 c. .01 d. .25 6-12. If the odds against an event are 19 to 1, the probability of its occurrence is a. .17 b. .05 c. .01 d. .25

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6-13. If the odds against an event are 3 to 1, the probability of its occurrence is a. .17 b. .05 c. .01 d. .25

6-14. Odds against are based on how often a given event a. will occur out of 100 times b. will occur out of any number of times c. will occur just once d. will not occur at all, to the number of times it will 6-15. Probability is based on how often a given event a. will occur out of 100 b. will occur out of 52 times c. will occur just once d. will not occur at all 6-16. When the frequency of an event is 25%, its probability is a. 3 to 1 b. .25 c. .75 d. none of these 6-17. When the frequency of an event is 75%, its probability is a. 3 to 1 b. .25 c. .67 d. none of these 6-18. When the frequency of an event is 95%, its probability is a. .95 b. .05 c. .50 d. Cannot be found from the information given 6-19. If an event has a probability value of .95, its frequency of occurrence is a. 5% b. 95% c. 19 to 1 d. 47.5%

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6-20. When tossing coins, the frequency of occurrence of "heads" turning up is a. .50 b. .50% c. 1 to 1 d. 50% 6-21. When tossing coins, the odds against a "head" turning up are a. .50 b. .50% c. 1 to 1 d. 2 to 1 6-22. Under the normal curve, the probability of a score occurring within one standard deviation (plus or minus) of the mean is a. .68 b. .50 c. 2 to 1 d. .68%

6-23. Under the normal curve, the probability of a score occurring above the mean is a. .68 b. .50 c. 2 to 1 d. .68% 6-24. Under the normal curve, the probability of a score occurring below the mean is a. .68 b. .50 c. 2 to 1 d. .68% 6-25. When an event must occur, its probability value is a. .99 b. 1.00 c. 0 d. 100% 6-26. When an event cannot occur, its probability value is a. .99 b. 1.00 c. 0 d. 100% 75

6-27. The reason z scores may be used to calculate probabilities is that a. frequencies are easily converted into probabilities b. all z scores are independent of each other c. all z scores are included within a standard deviation unit (plus or minus) of the mean d. all z scores are included within 2 standard deviation units (plus or minus) of the mean 6-28. The probability of obtaining a z score of +1.00 or higher, is a. 1.00 b. .16 c. .84 d. .34 6-29. The probability of obtaining a z score of +1.00 or lower, is a. 1.00 b. .16 c. .84 d. .34 6-30. The probability of obtaining a z score of 0 or above is a. 0 b. .16 c. .34 d. .50

6-31. Under the normal curve, the probability of any single score falling below the mean is a. 0 b. .16 c. .34 d. .50 6-32. Under the normal curve, the probability of any single score being above the 50th percentile is a. 0 b. .16 c. .34 d. .50

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6-33. Under the normal curve, the probability of any single score being between z scores of + 3.00, is a. 1.00 b. -1.00 c. .99 d. .50 6-34. No probability value may ever be greater than a. 1.00 b. .99 c. 0 d. there is no upper limit to the size of a given probability value 6-35. No probability value may ever be less than a. 1.00 b. .99 c. 0 d. there is no upper limit to the size of a given probability value 6-36. Under the normal curve, the probability of any score falling below the 50th percentile, is a. 0 b. .16 c. -.50 d. .50 6-37. The use of z scores for obtaining probabilities is only possible when a. the events are independent b. the probabilities are conditional c. the distribution curve is symmetrical d. the distribution curve is normal 6-38. When combining probabilities, to find the probability of one event OR another event, the separate probabilities should be a. multiplied b. divided c. subtracted d. added 6-39. When combining probabilities, to find the probability of one event AND another event, the separate probabilities should be a. multiplied b. divided c. subtracted d. added 77

6-40. When combining probabilities, to find the probability of one event AND another event, the combined probability will be a. higher than either of the separate probabilities b. lower than either of the separate probabilities c. exactly the same as the separate probabilities d. probability values should never be combined

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C. True or False: For the following, indicate T (True) or F (False) 6-41. All probability values lie within the range of +1 to -1.00.

6-42. No probability value can ever be greater than 1.00.

6-43. When an event occurs at a frequency of 95%, its probability value is .05.

6-44. An event which occurs at a high frequency also has a high probability of occurring.

6-45. When the odds against an event occurring are 9 to 1, the probability that the event will occur is .10.

6-46. A probability value of 0 indicates that the event can never occur 6-47. When a specific event has a probability value of .05, then its frequency of occurrence is 95%.

6-48. When an event must occur, its probability is 100.

6-49. On a normal distribution 68% of the cases must fall above the mean 6-50. The ratio, specific event divided by total events, defines probability.

6-51. Probability is always stated in terms of how often an event will occur.

6-52. When events are included between two negative z scores the probability value for those events must be less than .50.

6-53. The probability that any given event may be included between two positive z scores must be greater than .50.

6-54. The probability of any event occurring above the mean is at least .68.

6-55. The "Gambler's Fallacy" overlooks the fact that probability values may never be greater than 100.

6-56. If a coin is flipped 10 times, and by chance, heads turn up 10 times, on the next flip the probability of again obtaining a head is .50.

6-57. The probability of any event falling within one standard deviation of the mean (plus or minus) is .68.

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6-58. A negative z score must yield a positive probability value.

6-59. Under the normal curve, the exclusion of the most extreme scores refers only to the highest scores in the distribution.

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D: For the following questions, calculate the values. Assume a normal distribution in all questions.

6-60. The mean is 55 with an SD of 15. What is the probability of selecting at random a score which falls between 60 and 65?

6-61. The mean weight for a group of students is 130 pounds, with an SD of 8 pounds. What is the probability of selecting a single student who weighs 125 pounds or more?

6-62. The mean IQ in a certain sorority is 120 with an SD of 5. What is the probability of selecting at random a student whose IQ is between 130 and 135?

6-63. A certain air-force missile has an average range of 1500 miles with a standard deviation of 62 miles. What is the probability that any single shot will travel 1400 miles or less?

6-64. With a mean of 40 and an SD of 20.45, what is the probability of selecting at random a score of 45 or higher?

6-65. With a mean of 95 and an SD of 12, what is the probability of selecting at random a score of between 89 and 101?

6-66. With a mean of 55 and an SD of 6, what is the probability of selecting at random someone scoring between 48 and 50?

6-67. At a certain race track, the horses run the mile in an average time of 135 seconds, with an SD of 13 seconds. What is the probability of selecting at random a horse that could run the mile in 130 seconds or less?

6-68. With a mean of 70 and an SD of 10, what is the probability of selecting a score of between 75 and 90?

6-69. With a mean of 60 and an SD of 4, what is the probability of selecting at random a score between 56 and 58?

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6-70. The average life of a certain light bulb is 200 hours with an SD of 27 hours. What is the probability of selecting at random a bulb lasting 150 hours or less?

6-71. At a certain college, the mean college-board score in math is 400 with an SD of 97. What is the probability of selecting at random a student who scored between 370 and 450?

6-72. The mean weight of adult U.S. females in the mid 1990s was 153 pounds with an SD of 30 pounds. What is the approximate probability of selecting a woman who weighed exactly a. 155 pounds? b. 150 pounds c. 145 pounds d. 95 pounds 6-73. The mean WAIS-III IQ among adult Americans is 100 with an SD of 15. What is the approximate probability of selecting an adult whose IQ was exactly a. 120 b. 110 c. 100 d. 90 6-74. If the probability of a certain event is .50 and the probability of a second event is .50, what is the probability of having the first event occur AND also the second event occur? a. .25 b. .50 c. .75 d. 1.00 6-75. If the probability of a certain event is .50 and the probability of a second event is .50, what is the probability of having the first event occur OR the second event occur? a. .25 b. .50 c. .75 d. 1.00

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6-76. In a certain house of correction, 64% percent of the current population of inmates had recidivated. What is the probability of selecting an inmate at random who had recidivated? a. .36 b. .64 c. .50 d. .46 6-77. Using data from the previous problem, what is the probability of selecting an inmate who had NOT recidivated? a. .36 b. .64 c. .50 d. .46.

6-78. At a certain prison it was found that 71% of the inmates had had a first arrest at the age of 15 or younger. It was also found that 70% of the population were identified as having had a drug problem. What is the probability of selecting at random an inmate who had been arrested at 15 or younger and had also been identified as having a drug problem? a. .14 b. .50 c .95 d. .46 6-79. At a certain casino on a certain day, 12% of the participants left at the end of the evening as winners (had more money than when they entered). What is the probability of selecting at random a participant who had had a winning evening? a .12 b. .24 c. .48 d. .88 6-80. Using data from the previous problem, what is the probability of selecting at random a participant who had had a losing evening? a. .12 b. .24 c. .48 d. .88

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CHAPTER 7 STATISTICS AND PARAMETERS A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Ask Me and I'll Tell You (even if I don't know what you're talking about).

A real problem facing the data-collecting pollsters is that people don't want to appear uninformed, and will often give opinions on issues of which they are totally ignorant. One study found that over a third of the respondents offered opinions, and some very strong opinions, on the non-existent "Public Affairs Act". It might be instructive for your students to use these questions, and others that they create, to poll their dorms or classes. Students who have been involved in this sort of polling report back, sometimes years later, that they learned one of life's important lessons doing just that. We, as teachers, are not trying to create cynics, but we do want our students to be concerned enough not to accept at face value the voluminous polling data they are daily exposed to. As shown in the text even subtle changes in the questions being asked often produce dramatically different results.

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B: Multiple Choice Items 7-1. The techniques of inferential statistics are designed to yield a. eternal truth b. data descriptions c. probability estimates d. all of these 7-2. Predicting the characteristics of an entire group, after having measured a small group, is the major goal of a. probability theory b. descriptive statistics c. measurement theory d. inferential statistics 7-3. An entire group of persons, things or events having at least one trait in common defines the a. population b. sample c. statistic d. measurement scale 7-4. The size of a given population may be a. finite b. infinite c. determined on the basis of the total number of observations sharing a given trait d. all of these 7-5. Any value which is based on having measured the entire population is called a. a parameter b. a statistic c. a finite measure d. none of these 7-6. Populations are to parameters as samples are to a. populations b. statistics c. infinite measures d. finite measures 7-7. Any value which is based on having measured a sample is called a. a parameter b. a statistic c. an infinite measure d. biased 85

7-8. When the population mean differs from the sample mean, the result is called a. bias b. statistical regression c. measurement transformation d. sampling error 7-9. When a sample mean is greater than the true, population mean, the resulting difference is called a. bias b. statistical regression c. measurement transformation d. sampling error 7-10. When sample means are consistently lower than the true, population mean, the result is called a. biased b. statistical regression c. measurement transformation d. sampling error 7-11. If the names of all the students at Omega University were placed in a fishbowl, and the researcher selected out 10% of the names, the resulting sample would be a. biased b. random c. stratified d. far too small for meaningful analysis 7-12. In order to make accurate population estimates, the sample must be a. very large b. representative of the population c. manipulated by the experimenter d. free of any possible sampling error

7-13. If the names of all the students at good old C.V.(Commuter Varsity) were dropped into a fishbowl, they would constitute a. the population of C.V. students b. a sample off C.V. students c. a random sample of C.V. students d. all of these, depending on the size of the enrollment at C.V.

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7-14. Random sampling occurs when a. the sample is selected from a single population b. a random choice determines the first subject to be selected c. the sample is selected from an infinite population d. all members of the population have an equal chance of being selected 7-15. It is assumed that random sample a. must contain bias b. is too much a matter of chance to be dependable c. is only valid when used for description purposes d. will probably be representative of the population 7-16. If Omega University had a total student enrollment of 600, and 5999 of these students were selected for testing, the group being test would constitute a. sample b. a population c. a parameter d. a statistic 7-17. Omega University has a total student enrollment of 6000, and all 6000 students were tested for IQ. The mean IQ for the group was 115. This value of 115 constitutes a a. statistic b. population c. sample d. parameter 7-18. If 10 Omega University students were selected out of the total population of 6000, and median IQ for this small group equaled 118, then this value of 118 would constitute a a. statistic b. population c. sample d. parameter 7-19. If the mean IQ of all Omega University students was equal to 115, and a small sample of students was selected from the population whose mean IQ was 120, then this difference of 5 points would equal a. parameter b. the statistic c. the sampling error d. the deviation quartile

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7-20. If every 10th student entering the college library on a Tuesday afternoon were selected for testing, the group thus selected would constitute a. a sample b. a random sample c. a quota sample d. a statistic 7-21. When a group is selected from a population on the basis of sharing the same percentages of trait characteristics as are known to exist in the population, then the group so selected would constitute a. a random sample b. a stratified sample c. a biased sample d. a sampling distribution 7-22. When a group is truly selected randomly from a population, the probability of its mean value being higher than the parameter mean is a. 0.00 b. .50 c. 1.00 d. .33 7-23. Sampling error is assumed to result a. when any sample is selected b. only when a non-representative sample is selected c. only when a biased sample is selected d. only when the researcher makes a mistake 7-24. Whenever the sampling error is consistently in one direction, the result is a. a negative sampling error b. a positive sampling error c. a population exclusion d. bias 7-25. The probability of obtaining a random sample mean which is lower than the population mean is a. 1.00 b. 0 c. .50 d. none of these, since a sample mean can never be lower than a population mean

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7-26. The normal, expected difference between a statistic and a parameter is called a. sampling error b. bias c. controlled randomization d. none of these, since statistics and parameters may never differ 7-27. The reason the Literary Digest's 1936 election poll was so inaccurate, was that a. the sample was too small b. it contained sampling error c. the sample was biased d. the wrong statistical tests were employed for extrapolating to the population 7-28. Any sample which contains fewer than 50% of the total population a. must be biased b. must be non-random c. must be inaccurate d. none of these 7-29. The sample selected by the Literacy Digest in 1936 was a. far too small to be accurate b. overly represented by affluent voters c. overly represented by Democrats d. truly random 7-30. The problem with the Gallup poll of 1948 (when Dewey was predicted to beat Truman) was that a. the polling ended too soon b. the "undecideds" were split 50/50 c. many of the "undecideds" had, in fact, already made up their minds d. all of these 7-31. The sum total of all registered voters in Canada represents a. the population of potential Canadian voters b. a parameter c. statistic d. sampling error 7-32. A biased sample can never be a. selected from a single population b. selected when the parameters are known c. selected when the entire population is available d. representative of the population 89

7-33. By definition, one can never obtain a statistic unless a. the entire population has been measured b. a random sample has been measured c. a quota sample has been measured d. none of these, since a statistic refers to a measure of any type of sample 7-34. When graphing the sampling distribution of means, the ordinate represents a. sample mean b. a population mean c. a statistical mean d. the frequency of occurrence 7-35. When graphing the sampling distribution of means, the various sample means are represented on a. the abscissa b. the ordinate c. the Y axis d. none of these 7-36. Each point on the abscissa of the distribution of sample means represents a. a parameter b. a statistic c. the sampling error d. the population error 7-37. The mean of the entire distribution of sample means represents a. a parameter b. a statistic c. the sampling error d. none of these 7-38. The sampling distribution of means assumes normality only when a. all samples are randomly selected b. all samples have been selected from a single population c. the samples represent groups of sufficient size d. all of these 7-39. The Central Limit Theorem states that the sampling distribution of means is a. only available by inference b. only available when sample sizes are less than 10 c. only available when the very high and very low means have been eliminated d. approximately normal in shape

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7-40. The range of values for the sampling distribution of means a. is larger than the range of the underlying individual scores b. is identical to the range of the underlying individual scores c. is smaller than the range of the underlying individual scores d. is always equal to 30 or higher 7-41. When the sampling distribution of means is randomly selected from an entire population of individual scores, the highest and lowest individual scores a. become lost through averaging b. must be arbitrarily eliminated before the sampling distribution becomes available c. assume the characteristics of parameters d. produce a bimodal distribution 7-42. When the standard deviation of the entire distribution of random sample means had been calculated, the resulting value is called a. the deviation score b. the parameter mean c. the standard error of the mean d. sampling error 7-43. The probability estimates provided by inferential statistics are only necessary when a. the population values are unknown b. the parameters are given c. the population values are provided d. none of these 7-44. When comparing the size of the standard error of the mean with the size of the standard deviation of the underlying distribution of individual scores, a. the standard error of the mean is always larger b. the standard error of the mean is always smaller c. the standard error of the mean is sometimes larger and sometimes smaller, depending on the sample size d. none of these 7-45. The fact that the distribution of random sample means is assumed to approximate normality is expressed in a. the Gaussian Law b. Pascal's triangle c. the Central Limit theorem d. none of these, since the premise is false

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7-46. In order to calculate the z test, one must know a. the sample's true standard deviation b. the amount of sampling error present c. the parameter mean, the parameter standard deviation and the sample mean d. the exact amount of bias contained in the sample 7-47. The z test may be used to determine a. if the sample is representative of a known population b. if a sample contains bias c. the exact amount of sampling error contained in the parameter mean d. the amount of sampling error contained in the population's standard deviation 7-48. The z test assumes that a. the sample mean is equal to the population mean b. the standard deviation of the sample scores is equal to the standard deviation of the distribution of sample means is normal c. the distribution of sample means is normal d. none of these, since the z test is "assumption free" 7-49. The standard error of the mean must equal a. the standard deviation of the sample's distribution of individual scores b. the standard deviation of the entire population's distribution of individual scores c. the "average" of the sample's distribution of individual scores and the population's distribution of individual scores d. the standard deviation of the entire sampling distribution of means

7-50. When all possible samples have been randomly selected from a single population, the standard deviation of the resulting distribution of means is called a. the "true" standard deviation of the sample's individual scores b. the "true" standard deviation of the population's individual scores c. the standard error of the mean d. sample variance plus error variance 7-51. One measure of the overall sampling error in the entire distribution of sample means is the a. standard error of the mean b. population mean c. population median d. sample's true amount of bias

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C: True or False: For the following, indicate T (True) or F (False) 7-52. A population may never share more than one common trait.

7-53. A statistic always refers to a sample measure.

7-54. Statistic is to sample as parameter is to population.

7-55. True population may never be infinite.

7-56. Bias is the difference between the sample measure and the population measure.

7-57. With true random sampling, every observation in the population has an equal chance of being selected.

7-58. When sampling error is constant and in one direction, the sampling must be biased.

7-59. The major error in inferential statistics is to commit the "sampling error".

7-60. Each and every time a sample is selected, sampling error must be assumed.

7-61. The Central Limit theorem states the exact probability value of estimating the true mean.

7-62. The Central Limit theorem states that the sampling distribution of means is normal in shape.

7-63. Stratified sampling refers to the actual number of different samples being selected from a single population.

7-64. When samples are selected randomly, they are assumed to be representative of the population.

7-65. The z test assumes that the sampling distribution of means is normal.

7-66. For the z test, the null hypothesis states that the sample and parameter means are equal.

7-67. The z test assumes that the sample's standard deviation equals the standard error of the mean.

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7-68. The z test assumes knowledge of the two population parameters, the mean and the standard deviation.

7-69. The Central Limit theorem only applies when the distribution of individual sample scores is normal.

7-70. The null hypothesis for the z test, since it is based on chance, must always be accepted.

7-71. The z test draws a comparison between the sample means and the known population mean.

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D: For the following questions, calculate the values.

7-72. With a known population mean of 100, and a known standard error of the mean of 7.5, what is the probability of selecting at random a sample whose mean is equal to 110 or greater?

7-73. With a known population mean of 1500, and a known standard error of the mean of 42.50, what is the probability of selecting at random a sample whose mean is 1450 or less?

7-74. With a known population mean of 10 and a known standard error of the mean of 1.47, what is the probability of selecting at random a sample whose mean is 8.50 or less?

7-75. With a known population mean of 68 and a known standard error of the mean of 2.58, what is the probability of selecting at random a sample whose mean is 70 or greater?

7-76. With a known population mean of 50 and a known standard error of the mean of 4.00, what is the probability of selecting at random a sample whose mean is 52 or greater?

7-77. With a known population mean of 75 and a known standard error of the mean of 5.00, what is the probability of selecting at random a sample whose mean is 70 or less?

7-78. The standard deviation of the population of individual scores is 22.50. A given random sample of 100 scores produces a mean of 210. Calculate the standard error of the mean.

7-79. The standard deviation of the population of individual scores is 15. A random sample of 10 scores yields a mean of 100. Calculate the standard error of the mean.

7-80. With a normal distribution, if the standard deviation in the population of individual verbal SAT scores is 100, then the Range should approximate_________ 7-100. A random sample of 45 subjects yields a mean score of 500. Calculate the standard error of the mean.

7-81. The standard deviation of the population of adult female height scores is 3 inches.

A random sample of 50 women yields a mean height of 64 inches.

Calculate the standard error of the mean.

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7-82. The standard deviation of the population of individual adult male weight scores is 25 pounds. A random sample of adult males produces a mean of 180 pounds. a. Calculate the standard error of the mean. b. Test the hypothesis, via the z test, that this sample could still be representative of a population whose mean is known to be 170 pounds.

7-83. The standard deviation of the population of individual scores is 15. A random sample of 40 subjects yields a mean of 105. a. Calculate the standard error of the mean. b. Test the hypothesis, via the z test, that this sample could be representative of a population whose mean is known to be 100.

7-84. The standard deviation of the population of men's collar sizes is 1.02 inches. A random sample of 12 men was selected and yielded a mean collar size of 15.50 inches. a. Calculate the standard error of the mean. b. Use the z test to establish whether this sample could represent the population of men's collar sizes which is known to have a mean of 15 inches.

7-85. The standard deviation of the population of individual Digit Span scores is 3.00.

(The Digit Span test is a sub-test of the Wechsler IQ test). A random sample of 20 subjects yielded a mean Digit Span score of 7.25. a. Calculate the standard error of the mean. b. Test the hypothesis, using the z test, that the sample could be representative of a population whose mean is known to be 10.00.

7-86. The standard deviation of the population of individually measured adult male spine lengths is 1.02 inches. A random sample of 25 men is selected and the mean spine length was recorded at 30 inches. a. Calculate the standard error of the mean. b. Test the hypothesis that this sample could be representative of the general population of adult males whose mean spine length is known to be 28 inches.

7-87.The standard deviation of the population of adult female spine lengths is .92 inches.

A random sample of 40 women was selected and the mean spine length was found to be 26 inches. a. Calculate the standard error of the mean. b. Use the z test to determine whether this sample could be representative of a the general population of women whose mean spine length is known to be 24 inches.

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7-88. The standard deviation of the population of elementary school mathachievement scores is 7.78. A random sample of 20 elementary school children was selected and yielded a mean achievement score of 80.04. a. Calculate the standard error of the mean. b. Use the z test to establish whether this sample could be representative of the general population of elementary school children whose mean math achievement score is known to be 78.00.

7-89. A researcher in the area of IQ wishes to measure a sample of children using the WISC III. The standard deviation on the WISC-III is 15. How large a sample should be selected if the researcher wishes to be .95 accurate to within 5 IQ points?

7-90. A researcher in the area of IQ wishes to measure a sample of children using the WISC III. The standard deviation on the WISC-III is 15. How large a sample should be selected if the researcher wishes to be .95 accurate to within 3 IQ points?

7-91. A researcher in the area of IQ wishes to measure a sample of children using the WISC III. The standard deviation on the WISC-III is 15. How large a sample should be selected if the researcher wishes to be .99 accurate to within 5 IQ points?

7-92. A researcher in the area of IQ wishes to measure a sample of children using the WISC III. The standard deviation on the WISC-III is 15. How large a sample should be selected if the researcher wishes to be .99 accurate to within 3 IQ points?

7-93.The standard deviation of the population of adult female spine lengths is .92 inches.

A random sample of 40 women was selected and the mean spine length was found to be 26 inches. a. Calculate the standard error of the mean. b. Use the z test to determine whether this sample could be representative of a the general population of women whose mean spine length is known to be 24 inches.

7-94. The parameter mean score on the AST (Anti-Social Tendencies) test is 50 with an SD of 10. A group of 25 inmates from a local house of correction (all of whom were doing time for B&Es, breaking and entering) were given the test during orientation and their mean score on the test was 46. a. Calculate the standatd error of the mean. b. Use the z test to determine whether this sample is representative of the population at large.

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CHAPTER 8 PARAMETER ESTIMATES A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Interval Estimates and Precision Stress the point that for interval estimation, there has to be a constant trade-off between accuracy and precision. To set up a confidence interval that is going to be accurate 99% of the time, then the interval will be substantially wider than if the 95% level is chosen. Using, as an example, sample scores of 13, 11, 9, 7, 7, 7, 5, 3, 1, the estimated standard error of the mean will be 1.248. At the .95 level, the confidence interval will run between 9.872 and 4.128, whereas at .99 it expands 11.187 and 2.813.

Also, the MORE VARIABILITY in the original data set, the wider the confidence interval and the less precise the resulting estimation. As an extreme example, the May 1993 issue of the British journal "Nature" reported on a statistical study conducted by J.R. Gott which purported to show that the longevity of things (including humankind)can be estimated with a high degree of accuracy simply by looking at the various histories.

Intelligent human beings, who Gott alleges first appeared on earth 200,000 years ago, are predicted to remain on the planet for a maximum of 7.8 million years on the high side or 5,128 years on the low side. That's an interval of over 7 MILLION YEARS. He also predicts that the manned space program will last somewhere between 10 months and 1250 years, and that the New York Times will continue publication for somewhere between 3.6 years and 142 years (New York Times, June 1, 1993). These interval estimates may be 95% accurate, but they are hardly precise.

Activity: Interval Estimates, Hypothesis Testing and the Alpha Level: Point out that with interval estimates we don't have to test the null hypothesis. In short when we establish that a certain predicted range of values will capture the true parameter mean 95% of the time or 99% of the time, we don't have to make the statistical decision to accept or reject null. On the other hand, if we test our sample mean against an hypothesized, or known, population mean, then the null hypothesis does have to be tested and a statistical decision has to be made. Thus, we can't commit the Type 1 error with an interval estimate, simply because with this technique we are never rejecting Ho (in fact, never testing Ho).

As an example, take the following ordered set of sampled IQ scores: 145,113,105,104,103,102,101,101,100,100,100,100,99,99,98,97,96,95,87,55.

The mean is 100.000 and the estimated standard error of the mean is 3.441, and the tabled value of t at .01 for 19 degrees of freedom is 2.861.

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First, have the class create a .99 confidence interval for predicting the population mean.

CI (.99) = (+ 2.861)(3.441) + 100 = +9.845 + 100 = 109.845 = -9.845 + 100 = 90.155 Thus, we are estimating that the true parameter mean falls somewhere between 109.845 on the high side and 90.155 on the low side, and our confidence level for this estimation has a probability level of .99.

Now, using the same data, have them TEST THE HYPOTHESIS that this sample could be representative of a population whose mean is 110 (a value just above the confidence limits just shown). t = 100-110/3.441 t = -2.906 At the .01 level, we then test the Null Hypothesis that the sample mean of 100 does in fact equal the assumed population mean of 110, and with 19 degrees of freedom we have to at least reach a t ratio of + 2.861. In this case we more than reach it since our t ratio is greater than the tabled value and we reject the null hypothesis. Since our decision was to reject, we conclude that this sample probably does not represent the assumed population.

Also, since we rejected null, there is now the possibility of committing the Type 1 error, which for this example had been set at an alpha of .01 (one chance in a hundred that we made the wrong decision). Thus, although both examples point to the same conclusion, in one case we estimated a parameter, and in the other case we tested the null hypothesis against an assumed parameter.

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B: Multiple Choice 8-1. Estimating the value of the population mean on the basis of the sample mean alone produces a(n) a. parameter b. point estimate of a parameter c. estimated statistic d. none of these, since the population mean cannot be estimated 8-2. The fact that the point estimate of the population mean may be inaccurate (even with random sampling) is due largely to a. bias b. alpha error c. sampling error d. none of these, since the population mean cannot be estimated 8-3. Whenever a sample is selected and measured, one must assume that the sample mean a. will probably differ from the population mean b. will probably be identical with the population mean c. will over-estimate the population mean d. will under-estimate the population mean 8-4. A point estimate of a population value is referred to as a. a biased estimator b. a true parameter c. a true variability measure d. an hypothesized parameter 8-5. According to the text, the highest value that should be assigned to the alpha level is a. .95 b. .99 c. .10 d. .05

8-6. Other things being equal, an alpha level of .05 should lead to a rejection of the null hypothesis a. more often than when alpha is set at .01 b. more often than when alpha is set at .10 c. less often than when alpha is set at .01 d. none of these, since when alpha is set at .05, the null hypothesis cannot be rejected

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8-7. If it is determined that there is no difference between the population mean being represented by the sample and the mean of the population at large, then a. the alpha level must have failed b. the alpha level must have been rejected c. the alpha level must have been committed d. there was a failure to reject the null hypothesis 8-8. When a given difference is determined to be "significant", then a. the alpha level must have failed b. the alpha level must have been rejected c. the null hypothesis must have been rejected d. there was a failure to reject the null hypothesis 8-9. When the null hypothesis is rejected, then we know for certain that a. the mean of the sample equals the mean of the population b. the mean of the sample differs from the mean of the population c. the mean of the sample is a biased estimator d. none of these, since the statistical decision is a probability statement and not a statement of certainty 8-10. When we fail to reject null hypothesis, then we know for certain that a. the mean of the sample equals the mean of the population b. the mean of the sample differs from the mean of the population c. the mean of the sample is a biased estimator d. none of these, since the statistical decision is a probability statement and not a statement of certainty 8-11. When an area (between high and low values) of the sampling distribution of means have been bracketed on the basis of a reasonable expectation of containing (estimating) the population mean, then a. the central limit theorem has been proven b. the point estimate has been proven c. the alpha level has been calculated d. the confidence interval has been calculated 8-12. The true population mean is expected to fall outside a .99 confidence interval a. 1% of the time b. 99% of the time c. 100% of the time d. none of these, since the true mean may never fall outside the confidence interval

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8-13. Increasing the confidence-interval level from .95 to .99 a. increases the likelihood of including the population mean within its limits b. decreases the likelihood of including the population mean within its limits c. increases the number of degrees of freedom d. none of these, since a confidence interval may never include the true population mean 8-14. The single-sample t ratio is used to compare a. two population means b. a sample mean with an assumed or known population mean c. two sample means d. a single sample's mean with that same sample's standard deviation

8-15. With a single-sample t ratio, the null hypothesis states that a. the sample mean could readily be part of the same distribution whose overall mean is already known. b. the sample mean in all likelihood could not be part of the same distribution whose overall mean is already known c. the sample mean and the population mean are significantly different values d. none of these, since the null hypothesis always assumes that at least two sample means have been obtained 8-16. A single-sample t ratio presents the difference between two means in units of a. the estimated standard error of the mean b. the estimated population standard deviation of the underlying distribution of individual scores c. the estimated standard error of difference d. none of these, since the single-sample t ratio cannot compare two means 8-17. A single-sample t ratio is only considered to be significant when a. the population values are known b. the population values are assumed c. its value falls within the confidence interval d. the null hypothesis has been rejected 8-18. Other things being equal, the higher the value of the t ratio a. the greater the likelihood of rejecting Ho b. the less the likelihood of rejecting Ho c. the more certain that the samples represent a single population d. the greater the likelihood of sampling error

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8-19. Rejecting Ho when in fact we should have failed to reject it, causes the a. standard error b. sampling error c. omega error d. Type 1 error 8-20. For the t ratio, the larger the sample size a. the larger the alpha level b. the larger the beta level c. the larger the number of groups being compared d. the larger the number of degrees of freedom 8-21. The Type 1 error states the probability of being wrong whenever a. There is a failure to reject Ho b. Ho is rejected c. Ho is tested d. the sample means are assumed to be equal 8-22. For the t ratio, the larger the difference between the sample mean and the population mean (other things being equal), the greater the likelihood of a. rejecting Ho b. failing to reject Ho c. sampling error d. committing the standard error 8-23. When alpha is set at .05 and Ho is rejected, then the probability is a. .95 that there was a failure to reject Ho b. .95 that Ho should not have been rejected c. .05 that Ho should have been rejected c. .95 that Ho should have been rejected 8-24. Whenever Ho is rejected, then a. it is possible to commit the Type 1 error b. it is impossible to commit the Type 1 error c. the only possible error is sampling error d. none of these 8-25. For the t test, other things being equal, the smaller the value of the alpha error a. the less likely it becomes to reject Ho b. the more likely it becomes to reject Ho c. the less likely it becomes to fail to reject Ho c. the fewer the degrees of freedom

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8-26. The relationship between the exact value of alpha and the number of degrees of freedom, is that a. the greater the alpha level, the greater the degrees of freedom b. the smaller the alpha level, the greater the degrees of freedom c. degrees of freedom and alpha level take on the same value d. none of these, since the alpha level and degrees of freedom are unrelated 8-27. Whenever the calculated value of t is equal to or greater than the tabled value of t, for a given number of degrees of freedom, then a. There is a failure to reject Ho b. Ho is rejected c. Ho is no longer relevant d. the alpha level becomes zero 8-28. Whenever the calculated value of t is less than the tabled value of t (for a given number of degrees of freedom), then a. There is a failure to reject Ho b. Ho is rejected c. Ho is no longer relevant d. the alpha level increases

8-29. The relationship between the alpha level and the confidence level is such that a. the lower the alpha, the higher the confidence level b. the lower the alpha, the lower the confidence level c. they are not related until the confidence level reaches .95 d. none of these, since they are not related at all 8-30. In order to reach a confidence level of .95, then alpha must be set at a. .05 b. .95 c. .99 d. none of these 8-31. If a researcher were to conduct 100 experiments, and in each case the null hypothesis were rejected at an alpha of .05, then a. chances are that 5% of the decisions were incorrect b. chances are that 95% of the decisions were incorrect c. chances are that less than 5% of the decisions were correct d. none of these, since null can never be rejected 100 times in a row

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8-32. The estimated standard error of the mean can be calculated on the basis of the information a. provided by a single, individual score b. provided in a single sample c. provided by comparing two sample means d. provided by comparing a pair of parameters 8-33. In order to calculate the estimated standard error of the mean, one need only know a. the sample's standard deviation and the size of the sample b. the sample mean c. the parameter mean d. the standard deviation of the distribution of individual scores 8-34. The estimated standard error of the mean is a. an estimate of the sampling error b. a parameter c. a statistic d. equal to the range of individual scores, divided by 6 8-35. The estimated standard error of the mean is equal to a. the standard deviation of the sample b. the standard deviation of the sample, divided by the square-root of the size of the sample, less one c. the standard deviation of the population multiplied by the size of the population d. the difference between the mean of the sample and the mean of the population

8-36. The estimated standard error of the mean takes on a negative value only when a. most of the samples are below the mean b. populations of less than an N of 30 are involved c. populations of more than an N of 30 are involved d. none of these 8-37. The estimated standard error of the mean is always used to predict a. sample size b. bias c. central tendency d. the variability of the distribution of sample means

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8-38. When estimating the population mean on the basis of the information contained in a single sample, one must at least know a. the estimated standard error of the mean and the mean of the sample b. the z score for each raw score c. the standard deviation of the underlying distribution of individual scores d. the interdecile range 8-39. Multiplying the estimated standard error of the mean by 6 yields an approximation of a. the range of the entire distribution of individual scores b. the range of the entire sampling distribution of means c. the standard deviation of the sample d. the standard deviation of the population 8-40. The estimated standard error of the mean becomes a better predictor whenever the distribution of sample means a. is skewed b. is bimodal c. is platykurtic d. none of these 8-41. The estimated standard error of the mean is to the standard deviation of the entire distribution of all possible sample means as a. a statistic is to a parameter b. a parameter is to a statistic c. a parameter is to an estimate d. a deviation score is to a raw score 8-42. The term "confidence" expresses the probability that one's estimate is a. accurate b. not accurate c. inferred from a population d. based on a sample mean 8-43. The higher the probability value of the confidence interval a. the broader becomes the range of predicted mean values b. the more narrow becomes the range of predicted mean values c. the less the sampling error d. the greater the sampling error

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8-44. When the parameter mean is predicted to lie within a certain range of mean values, the technique employed is called a. point estimation b. error estimation c. interval estimation d. inferential validity 8-45. When the parameter mean is predicted as a single value, the technique employed is called a. point estimation b. error estimation c. interval estimation d. inferential validity 8-46. When the confidence interval has a probability value of .99, then a. the resulting prediction is never wrong b. the resulting prediction is wrong 99% of the time c. 99% of the resulting predictions are assumed to lie within the interval d. none of these, since the confidence interval has nothing to do with accuracy 8-47. For the t ratio, other things being equal, having 1 degree of freedom rather than 12 degrees of freedom makes it a. more likely to reject Ho b. less likely to reject Ho c. impossible to reject Ho d. none of these 8-48. The Type 1 error may never be committed when a. Ho is rejected b. There is a failure to reject Ho c. the direction of Ha is specified d. none of these, since there is always some possibility of committing the Type 1 error 8-49. The true parameter mean may be predicted by using a. either hypothesis testing or known variance values b. either hypothesis testing or interval estimation c. either interval estimation or its opposite, confidence intervals d. none of these

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8-50. In order to predict whether a certain set of random sample scores could possibly be part of a given population, the researcher could use a. the z score equivalents of each sample score b. the percentile equivalents of each sample score c. the one-sample t test d. the interdecile range

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C: True or False: For the following, indicate T (True) or F (False) 8-51. The true mean of the population must always be a known value before inferential techniques may be employed.

8-52. The standard error of the mean may never be negative.

8-53. A confidence interval of .99 predicts a narrower range of mean values than does an interval of .95.

8-54. A point estimate predicts a single parameter-mean value.

8-55. The techniques of inferential statistics are based on a probability model, and may sometimes lead to inaccurate predictions 8-56. The estimated standard error of the mean can only be calculated when the entire population of individual scores are available.

8-57. The estimated standard error of the mean may be calculated on the basis of the information contained in a single random sample.

8-58. The alpha level specifies the probability of being correct when rejecting null.

8-59. Other things being equal, the fewer the degrees of freedom, the easier it is to obtain a significant t ratio.

8-60. With very small sample sizes, the estimated standard deviation of the population yields a slightly higher value than does the true standard deviation of the sample.

8-61. Because of sampling error, the point estimate of the population mean may produce an inaccurate prediction.

8-62. The true standard deviation of the sample may be used directly to estimate the point value of the population mean.

8-63. The actual mean of the set of sample scores may be used directly to estimate the point value of the population mean.

8-64. When subjects are selected randomly, then the larger the sample size, the higher the likelihood of the sample mean approaching the population mean.

8-65. The true standard deviation of the entire distribution of random sample means is defined as the standard error of the mean.

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8-66. The concept of the standard error of the mean indicates that the actual population mean can never be a true parameter.

8-67. The concept of the standard error of the mean implies that the estimated population standard deviation must yield a value which over-estimates the true mean.

8-68. Regardless of sample size, the degrees of freedom assigned to the single-sample t ratio must remain constant.

8-69. The only t ratios which can possibly lead to a reject of the null hypothesis are those which are positive in sign.

8-70. Negative t ratios may sometimes lead to significant results.

8-71. With a sample size of 30, the single-sample t ratio has 1 degree of freedom.

8-72. Parameter estimates are considered to be hypotheses.

8-73. A significant difference is one that is probably not due to chance.

8-74. When there is a failure to reject Ho, one must conclude that the difference between the sample mean and the population mean is probably not due to chance.

8-75. The alpha level is typically set at either .95 or .99.

8-76. The Type 1 error is committed when here is a failure to reject Ho when in fact it should have been rejected.

8-77. Interval estimates, unlike point estimates, are used to predict a single parameter value.

8-78. Other things being equal, increasing the sample size, decreases the width of the confidence interval.

8-79. Other things being equal, the smaller the value of the estimated standard error of the mean, the narrower the confidence interval becomes.

8-80. Other things being equal, a .99 confidence interval produces a wider interval than when the confidence level is set at .95.

8-81. With infinite degrees of freedom, the t distribution approaches normality.

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8-82. With extremely large sample sizes, the difference in values between the estimated population standard deviation and the true standard deviation of the sample become minimal.

8-83. The smaller the standard deviation of the sample, for a given N, the larger is the estimated standard error of the mean.

8-84. For a given value for the sample's standard deviation, increasing the sample size increases the value of the standard error of the mean.

8-85. Because of sampling error, the point estimate of the population mean can never provide the "best" estimate of a single parameter value.

8-86. Sampling error always produces bias.

8-87. Random sampling helps to reduce bias.

8-88. True random sampling can be guaranteed to eliminate sampling error.

8-89. The single-sample t ratio may be used to compare the sample mean with an already known population mean.

8-90. The single-sample t ratio may be used when all the parameter values are known rather than assumed.

8-91.Reducing the alpha level from .05 to .01 increases the level of confidence.

8-92.Sampling error can be eliminated by using true random sampling.

8-93. Point estimates can never be based on sample values.

8-94. The type 1 error cannot be committed when alpha is set at .05.

8-95. When sample scores are compared with a population value that is known to be true, we are able to use the one-sample t ratio.

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D: For the following questions, calculate the values 8-96. For the following set of sample scores, find the estimated standard error of the mean: 25, 21, 17, 16, 15, 14, 11, 8, 7.

8-97. For the following set of sample scores, find the estimated standard error of the mean: 15, 14, 14, 13, 12, 12, 12, 11, 10, 9, 9.

8-98. For the following set of sample scores, find the estimated standard error of the mean: 16, 10, 8, 6, 5, 3, 1, 1.

8-99. For the following set of sample scores, find the estimated standard error of the mean: 60, 55, 53, 52, 50, 50, 47, 47, 44, 40.

8-100. For the following set of sample scores, find the estimated standard error of the mean: 20, 14, 16, 18, 9, 16, 18, 16, 15, 14.

8-101. For the following set of sample scores, find the estimated standard error of the mean: 23, 20, 21, 24, 18, 22, 24, 22, 20, 20.

8-102. For the following set of sample scores, find the estimated standard error of the mean: 12, 11, 9, 4, 8, 2.

8-103. For the following set of sample scores, find the estimated standard error of the mean: 4, 3, 2, 2.

8-104. For the following set of sample scores, find the estimated standard error of the mean: 8, 6, 7, 7, 3, 1, 8.

8-105. For the following set of sample scores, find the estimated standard error of the mean: 2, 1, 2, 3, 8, 5, 1.

8-106. For the following set of sample scores, estimate the population mean within a .95 confidence interval: 3, 6, 5, 3, 3, 7, 7, 5, 6.

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8-107. For the following set of sample scores, estimate the population mean within a .95 confidence interval: 15, 18, 12, 17, 19.

8-108. For the following set of sample scores, estimate the population mean within a .99 confidence interval: 10, 10, 12, 11, 10.

8-109. For the following set of sample scores, estimate the population mean within a .99 confidence interval: 50, 45, 45, 40, 40, 40, 35, 35, 30.

For questions 110 through 115, use the following: A researcher is interested in evaluating a certain brand of radial auto tire. Twenty tires are randomly selected from retail outlets throughout the country, and each is placed on a special machine which rotates the tires at a constant speed (equivalent to 55 miles per hour) against the friction equivalent of a 4000 pound auto being driven on a smooth highway. Each tire is run until there is no tread left. The number of miles (in thousands) were as follows: 40, 30, 32, 35, 39, 35, 31, 36, 37, 35, 34, 35, 37, 34, 36, 38, 35, 36, 35, 36.

8-110. Find the mean mileage for this sample.

8-111. Find the estimated standard error of the mean.

8-112. Estimate the population mean within a confidence interval of .95.

8-113. Estimate the population mean within a confidence interval of .99.

8-114. Setting alpha at .05, test the hypothesis that this sample of tires could represent a population whose mean was 36.50

8-115. Setting alpha at .01, test the hypothesis that this sample of tires could represent a population whose mean was 36.50 8-116. A random sample of 30 infants (age 12 months) was selected and given a Motoric Development test. The mean score for the sample was at the 13.20 month level. The estimated standard error of the mean was 1.25. Estimate whether the mean test age of the sample could be representative of a population whose mean test age is 12 months.

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8-117. A random sample of 61 fifth-grade students was selected and given the StanfordBinet IQ test. The mean score for the sample was 97, with an estimated standard error of the mean of 2.43. Estimate whether the sample mean could be representative of a population whose mean is 100.

8-118. A random sample of 20 fourth-grade students was selected and given the math sub-test of the Peabody Individual Achievement Test (PIAT). The sample mean was found to be 38.07 with an estimated standard error of the mean of 2.11. The population mean was known to be 42.50. Test whether the sample could be representative of the population.

8-119. A random sample of 41 children (all 5 years of age) was selected and given the Reading test of the WRAT (Wide Range Achievement Test). The sample mean was 20.57 and the estimated standard error of the mean was 1.98. Test whether the sample could be representative of a population whose mean was 23.52.

8-120. A random sample of 10 adult subjects was selected and tested on the Vocabulary sub-test of the Wechsler Adult Intelligence Scale (WAIS). Their scores were as follows: 12, 11, 10, 10, 10, 7, 6, 6, 5, 5. Test whether this sample could be representative of a population whose mean is known to be 10.00.

SPSS Computer Problem The MMPI2 was given to a random sample of 15 men, all of whom had been convicted of auto theft. Their scores on the ASP scale (Anti-Social Practices) were recorded. The population mean on this test is 50.00. The sample scores were as follows: 52,55,48,60,52,49,49,53,56,62,53,56,50,52,53.

8-121. What is the mean of the sample?

8-122. Find the single-sample t ratio.

8-123. Is this sample representative of the population at large?

A researcher selects a random sample of 15 young adult females, all of whom were engaged to be married within the next three months. The women were given the MCS (Marriage Commitment Scale), a test with a female population mean of 25.

Their scores were: 23,26,27,24,28,19,22,24,27,30,26,28,24,26,29.

8-124. What is the mean of the sample?

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8-125. Find the single-sample t ratio.

8-126. Is this sample representative of the population at large?

The average sentence (in months) for homicide is 227 (Gaines & Miller, 2003). A sample of inmates from a certain prison, all of whom having been convicted of homicide, was selected and their sentences calculated. The sentences follow: 200, 225, 197. 256, 300, 158, 220, 226, 230, 215, 237, 223, and 243.

8-127. What is the mean of the sample?

8-128. Find the single sample t ratio.

8-129. Is this sample representative of the population?

Gaines, L.K. & Miller, L.M. (2003). Criminal justice in action.

Belmont, CA: Wadsworth-Thomson.

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CHAPTER 9 THE FUNDAMENTALS OF RESEARCH METHODOLOGY A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: The Hawthorne Effect and John Henry Keep the student focus on the fact that good experimental research demands that the groups being compared are originally as equivalent as it's humanly possible to make them. The IV is manipulated and if the groups are now found to differ, we are allowed to assume that the IV must have caused this difference. However, as the chapter points out, other non-systematic events may have impinged on the groups during the experimental process. For example, again as stressed in the chapter, the Hawthorne Effect could have artificially increased the motivation level of the group. Maybe people out-do themselves when they receive the flattery and attention of seemingly important researchers. Or consider the researcher in the field of education who has come up with an exciting and innovative teaching technique that is so unique as to create what is called the "novelty effect". The novelty effect is another motivation enhancer that is due simply to persons being in a situation where they are doing something new and different. But as we all know from personal experience, eventually the novelty can wear off. And this can happen to the control group as well as the experimental group. Later when this new technique is tried out in the classroom, the motivation begins to erode and what worked in the lab is not working over the long haul. Finally, tell your students about something called the "John Henry" effect. John Henry was that folk-song miner who wanted to hold back the machine age by proving himself superior to the steam drill. Using only a hammer, he finally did beat the steam drill but then dropped dead from the effort. Sometimes even the control group will get so caught up in the competitive spirit that it will, over the short run, perform at an almost super-human level, a level that could never be maintained over the course of an entire school term. Or sometimes the teacher in the control group will feel personally threatened by possibly being supplanted by a new approach, say a computer-assisted social science unit. The teacher in the control group, using the old drill method, may, like John Henry, work him/herself and the class to the bone trying to out-perform the electronic wizardry being offered to the experimental group. The group then out-does itself and performs at a level way beyond what it normally would or thought it could. The effect of the treatment might then be masked over, even though the experimental group is also doing extremely well and in reality is profiting from the new instruction.

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Activity: Sensitization and Causal Inference Point out to your students that it is all too easy to become sensitized to certain events that happen to be consistent with underlying beliefs and attitudes and then begin to attribute causal powers to what are really only random connections. For example, many persons assume that the moon causes people to act in irrational ways and even commit murder, rape, robberies and suicide. Studies have shown, however, that the actual correlations between phases of the moon and anti-social behavior are in fact zero. The misapprehension may be due to the fact that people remember when someone does something bizarre during the full moon, but don't seem to recall all the fullmoon nights when nothing happens. Studies which have been done in this area, comparing numbers of cases in hospital emergency rooms, airline and train disasters, auto crashes, even the number of penalties at hockey games have found nothing. It appears that people are more sensitized to spot bizarre events when the moon is full, but simply forget or dismiss them when the moon is not full. In fact, the only consistently significant finding to come out of this research is that the belief in a moon effect correlates with a belief in spirit worlds, reincarnation, ESP, the Loch Ness monster, the abominable snow man and the fact that Elvis is still alive. We, as researchers, must be especially careful not to confuse expectations with root causes.

Activity: A Research Example Tell your students that one long-running controversy in the field of Education revolves around the issue of whether to adopt the pass-fail grading system in colleges and universities. Proponents of pass-fail claimed that students will actually learn more, that they will be more likely to explore different course areas, and that they will feel less anxiety when taking a course without the pressure of regular letter grades. The opponents, on the other hand, argued that students will be less motivated and therefore learn less if they are graded on the less precise pass-fail basis. A large number of studies have been conducted at a wide variety of educational institutions, and the data are fairly consistent on the following point: when students who are taking a course for regular grades are compared with students who are taking the same course on a pass-fail basis, the graded students achieve higher grades. The comparison is made possible by not telling the instructor which students are taking the course on the pass-fail basis and having the instructor assign letter grades to all students. The registrar then converts these to pass-fail for those students who had previously elected this option. This provision is included to protect the pass -fail students from possible instructor bias (since many instructors might penalize a student known to be in the pass-fail category, as though those students are not serious about the course). The data clearly show that there is a difference in grades between the two groups of students in the same class, those electing to take the course for pass-fail and those taking the course for regular letter grades. The 117

interpretations of this difference, however, vary widely. Some claim that this difference proves that pass-fail students simply don't work as hard or take the course as seriously as students taking the course for letter grades. Others claim that this difference proves that only the less competent students elect the pass-fail option. Others claim that this difference proves that students only take their most difficult (for them) courses under the pass-fail option and therefore are exploring areas they might otherwise attempt to avoid.

Others claim that this difference proves that students use the pass-fail option only in courses where the teacher is so personally uninspiring as to need the threat of letter grades in order to goad the students into studying.

What, in fact, do the data really prove? Nothing, other than that a difference in grades does indeed exist between the two groups. All the previously mentioned explanations are really only hypotheses, that is, guesses about the possible reasons for this established difference. One or even several of these hypotheses might eventually be supported, but at present none of the studies really proves any of the hypotheses. Yet in the hands of a statistical charlatan, the data may seem to an unsophisticated audience to prove whatever the statistician says they prove. So, again, when reading research studies, be alert. Think out the procedures carefully, and, above all, don't just read the conclusions.

Activity: Reviewing the Literature It can be most illuminating and personally rewarding to fledgling researchers to be given the opportunity to discover errors in published studies. Go over the following three studies in class, and then allow enough class time for the students to point out the errors and pitfalls in each. Tell them to really be critical and once they adopt that frame of mind, the class will literally reek of harsh criticism. After each presentation, make sure that key errors are not overlooked. Incidentally, these three studies are all old enough that the authors are not going to feel as though they are being picked on.

1) Rogerian Therapy Some time ago, a researcher attempted a study aimed at discovering the possible efficacy of the Rogerian, client-centered therapy technique among children with reading problems. A sample of 37 first and second grade children, all of whom had been diagnosed as poor readers on the basis of a standardized reading test (the Gates Primary Reading Test) as well as teacher evaluations, was selected (Axline, 1947). These children were then placed in a special class that was taught by a teacher trained in Rogerian, nondirective techniques. The atmosphere in this classroom was described as therapeutic, the children being encouraged to express their feelings openly in the presence of a warm, permissive, and accepting teacher. The Stanford-Binet IQs of the children ranged from 80 to 148. At the end of the school term, the children were again 118

tested on the Gates Primary Reading Test, and, in general, their reading-age scores improved, in one instance by a phenomenal 17 months. The researcher interpreted these gains as resulting from the nondirective teaching technique, which was of course the assumed IV. Key Errors: The lack of a separate control group leaves this study seriously flawed. We cannot tell whether the gains were really the effect of the teaching technique or of the several confounding variables: teacher personality, the novelty effect of being placed in a special class, the growth changes that normally take place among children during a school term and so on.

The students will find others. Finally, although your students won't be aware of this, in this particular study no statistical tests of significance were conducted, leaving the reader unable to determine whether the reading gains were in fact simply a result of chance, or whether the high IQ children profited more (or less) from the special class than did the low-IQ children.

2) Frustration-Aggression Perhaps one of American psychology's most cherished theories of aggression was launched many years ago by those famous, then-at-Yale psychologists, Neal Miller and John Dollard. They put forth the now-famous frustration-aggression hypothesis, which in its briefest form stated that frustration (in the form of goal blockage) always leads to aggression . They also believed that the aggression produced by frustration could sometimes be displaced on to minority groups, that is, could by exhibited in the form of racial prejudice. To test this hypothesis Neal Miller went to a government work camp that was situated deep in the woods, yet still close enough to civilization to allow the workers to get to a nearby town about once a week. Needless to say, the men eagerly looked forward to these visits! On one occasion, the men were especially anxious. The town's only movie theater had been holding a series of "bank night" drawings, and the previous week one of the workers at the camp had won a large sum of money. On the appointed night for the trip to town, the men were all brought together and given an attitude test toward Japanese and Mexicans.

They were told that as soon as they finished filling out the questionnaires, the bus would be ready to take them to town. The men cheerfully filled out the questionnaires, and then waited for the bus and waited and waited some more.

The bus never came. Instead of going to town they were told they had to stay in camp and take a series of difficult intelligence tests. No question about it, Miller knew how to manipulate the goal blockage IV. After a lot of complaining and moaning, the men finally settled down, took the IQ tests, and then, to add insult to injury, were told they had to again take the Japanese-Mexican attitude test. Their scores on the attitude posttest declined precipitously, indicating an increase in racial prejudice. Miller interpreted this increase as resulting from the action of the IV - that is, that not going to town produced the outgroup aggression (Miller & Bugelski, 1948).

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Key Errors: The problem with this study is that one cannot really be sure that the increase in prejudice resulted from not being allowed to take the bus into town. This was a one-group, repeated-measures experimental design that used the same group as its own control. In fact, a separate control group is needed here, a group that was not told anything about going to town but was still forced to take the attitude test, the intelligence tests, and the attitude test a second time. Without this control, the researcher can't be sure whether the attitude change was the result of not going to town or of the boredom generated from being forced to sit still for such a long period of time, or from the anger generated by having to take the same attitude test a second time.

3) College Teaching Techniques In a study of college teaching techniques, three different Intro-to-Psych classes were taught by three different methods (Faw, 1949). One class was taught in the traditional lecture format, the second in what was called a "student-centered" fashion and the third class alternated between the two techniques at each meeting. The results showed that students in Groups 2 and 3 indicated higher interest and enjoyment in the course than did student in the traditional class. They also felt that the social-emotional value of the course was greater. On an objective test of the course's content, however, there were no significant differences among the three groups (although, interestingly, the students in the traditional class felt that they had gained more information and knowledge).

Key Errors: The problem with this study is twofold. First, this was a quasiexperiment and there very well could have been systematic differences among the students in the three classes at the beginning of the term. Perhaps students who elect afternoon courses are different from those who opt for morning classes. More serious, however, was that the instructor playing the different teaching roles was also the researcher. It could very will be that this researcher's own convictions allowed him to carry out one of the roles more convincingly and with greater personal motivation than was the case with the other role. Also, the students must have been fully aware of who was in the experimental group and who was in the control group. This was, in short, definitely not a double-blind study, and because of this, the possibility of extraneous motivational variables may not have been fully controlled. The potential for confounding variables is far too great to make this a definitive test of teaching methods.

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Activity: The Student Researcher You as the primary researcher are interested in the possible effects of the drug Ritalin on the behavior of ADHD children. A group of hyperactive children is randomly selected and taken for treatment to a behavioralpsychology clinic. The group is then randomly broken down into two groups, group A and group B. The two groups are then measured on a hyperactivity scale.

Group A is given daily doses of Ritalin, while Group B is given a placebo.

Both groups receive identical daily behavioral treatment at the clinic. Then six weeks later the children are reevaluated regarding their hyperactivity. At the end of this six week period, the pre and post scores for all the children are compared.

1. What kind of research is this?

2. Identify the IV(s) and DV(s) 3. Were all the possible confounding variables controlled? If not, what other variables might the researcher have at least attempted to control?

4. Was the possibility of outside events contaminating the IV controlled?

5. Discuss the study's internal and external validity.

6. What type of statistical analysis would you propose?

7. If the differences are significant, what kinds of conclusions could you legitimately draw?

Ans: 1. This is experimental research, since the IV was manipulated by the researcher. It is a repeated-measures design with a separate and independent control group.

2. The IV is the drug, either Ritalin or a placebo.

The DV is the score on the hyperactivity scale. The assumption can be made that this is made up of interval data.

3. Not all the the possible confounding variables were controlled, although the study is fairly clean as it stands. However, two variables, age and gender, stand out as possible confounding variables. Younger children may display more hyperactivity than older children, may fight more or may be generally more restless, and of course, boys may be more hyperactive than girls.

Also, whether some or all the children were already on a drug, or had been previously on a drug is not mentioned. This could be important since how the children were first rated would be influenced by any medication they might already be on. Other variables not mentioned are race and socioeconomic status.

4. One possible outside event contaminating the IV is whether the study was double blind. If either the children or the researcher who was doing the ratings was aware of which group was receiving the Ritalin, it could influence the results. While the description states that both groups received identical daily behavior treatments, if the treatments were not provided by the same person, contamination could occur. Also, if the same 121

person rated the children on both the pre and post tests, then the halo effect could occur, with the second rating possibly influenced by the first rating. This is also a possible factor even if the second rating was done by a different person, but a person who was aware of the results of the first set of ratings.

5. Internal validity was partly answered above, since it depends on whether the pure effects of the IV can be established. Were the pre and post ratings done by the same person, by different but equally trained observers, were the outside variables truly controlled? The external validity is based on whether these results can be generalized to real-life situations and persons. Perhaps these children all came from the same geographical, socioeconomic and racial backgrounds and therefore may not represent children in general.

6. Since the groups were independently selected, the comparison of the change scores for each subject in each group can be handled via the independent t test. This would be an adequate analysis, as long as the distribution of change scores was fairly normal and neither set of scores approached the upper or lower limits of the scale (ceiling or cellar effects).

7. If the t ratio is significant, and if the direction of the difference favors the Ritalin group, then it may be assumed that Ritalin was the cause of the decrease in hyperactivity.

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B: Multiple Choice Items 9-1. The statement that "one can prove anything with statistics" is only true when a. correlational research has been used b. post-facto research has been used c. the data have been cross-referenced d. the reader is totally naive regarding statistical procedures 9-2. The most common error in reading and interpreting research studies is in a. assuming that the data have been incorrectly reported b. assuming that the correct statistical tests have been used c. assuming that the causal factor has been isolated d. all of these 9-3. In any antecedent-consequent relationship, the independent variable a. precedes the dependent variable b. follows the dependent variable c. occurs simultaneously with the dependent variable d. none of these 9-4. In any antecedent-consequent relationship, the dependent variable a. precedes the independent variable b. follows the independent variable c. occurs simultaneously with the independent variable d. none of these 9-5. In correlational research, the independent variable is usually placed a. on the abscissa b. on the ordinate c. on each data point d. none of these, since correlational research can never be graphed 9-6. In correlational research, whichever variable one predicts from is called a. the criterion variable b. the independent variable c. the dependent variable d. the control variable 9-7. When it is assumed that information about the X variable may be used to predict the Y variable, then the X variable is a. the independent variable b. the dependent variable c. the control variable d. the criterion variable 123

9-8. In any input-output relationship, the dependent variable is considered to be a. the input b. the output c. the relationship itself between the two measured variables d. all of these, depending on the type of research 9-9. An example of a subject-assigned independent variable would be a. IQ b. race c. gender d. all of these 9-10. An example of a manipulated independent variable would be a. IQ b. income level c. amount of education d. none of these 9-11. When the researcher has actively altered the environmental conditions, then the independent variable has been a. assigned b. manipulated c. confounded d. strengthened 9-12. In a study on the effects of TV violence on viewer aggressiveness, the dependent variable would be the a. control group b. TV violence c. viewer aggressiveness d. experimental group 9-13. In a study on whether increasing illumination increases worker productivity, the independent variable would be the a. illumination levels b. worker productivity rates c. causal relationship d. experimental group 9-14. If a researcher wished to assess whether or not a certain drug will increase IQ scores, the independent variable would be the a. drug levels b. IQ scores c. IQ test d. subjects whose IQ's were changed 124

9-15. One should not infer a direct cause-and-effect relationship when the research method is a. experimental b. post-facto c. double-blind d. repeated-measures 9-16. The two basic types of research are a. post-facto and correlational b. post-facto and cross-sectional c. post-facto and experimental d. post-facto and longitudinal 9-17. When subjects are measured, then followed through the years and measured again, the research technique is called a. longitudinal b. cross-sectional c. matched group d. controlled manipulation 9-18. When doing human-research studies on the long-term effects of growth and development, it is best to use a. longitudinal research b. cross-sectional research c. after-only research d. anecdotal research 9-19. In experimental research, the independent variable is a. the response made by the subject b. the group receiving the placebo c. actively manipulated by the experimenter d. the control group 9-20. In experimental research on humans, the independent variable is a. some form of stimulus presented to the subjects b. under the full, active control of the experimenter c. the potential causal half of the cause-and-effect relationship d. all of these

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9-21. When neither the subjects nor the experimenter are aware of which group is experimental and which is control, then the study is said to be a. contaminated b. confounded c. concealed d. double-blind 9-22. In experimental research on humans, the dependent variable is a. manipulated by the experimenter b. the causal half of the cause-and-effect relationship c. some form of stimulus d. a measure of the subject's response

9-23. In a study on whether college graduates earn more money then non-college graduates, the research method is almost certainly a. post-facto b. experimental c. matched-subjects d. before-after 9-24. The best way to distinguish between experimental and post-facto research is to discover whether a. the independent variable has been confounded b. the independent variable has been manipulated or assigned c. the dependent variable has been manipulated or assigned d. the appropriate statistical test has been used 9-25. In the study, cited in the text, of the relationship between the amount of time a teacher spends smiling and the achievement level of the students, it was shown that a. the less time a teacher smiles, the better the students do on an achievement test b. the more time a teacher smiles, the better the students do on an achievement test c. the time a teacher spends smiling is independent of student achievement d. a smiling teacher causes a student to get better grades In the following questions (26 through 30), you are to identify the alternative which is NOT appropriately grouped with the others 9-26. a. after-only b. repeated-measures c. post-facto d. matched-subjects 126

9-27. a. independent variable b. assigned variable c. manipulated variable d. dependent variable 9-28. a. response measure b. independent variable c. dependent variable d. out-put measure 9-29. a. correlation b. association c. prediction d. experimental method 9-30. a. correlation b. cause-and-effect relationship c. experimental method d. manipulated independent variable 9-31. The assigning of numbers to measurements according to certain rules is called a. prediction b. point estimation c. matrix information d. measurement 9-32. When numbers are assigned on the basis of discrete, mutually exclusive categories, the resulting scale is called a. nominal b. ordinal c. interval d. ratio 9-33. When data are in nominal form, the only information provided is a. the frequency of occurrence within categories b. greater than or less than, but not how much greater or less c. the value of absolute zero d. nominal data provide no information to the researcher

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9-34. When the measurement scale provides information regarding greater than or less than, but not how much greater or less, it is in the form of a. nominal data b. ordinal data c. interval data d. ratio data 9-35. An example of a nominal scale would be a. the number of pieces of mail going to Post Office A versus Post Office B b. the recorded heights of a group of first-grade children c. the order of finish for the horses at the Kentucky Derby d. all of these 9-36. Whenever we discover how many persons, things, or events have X or Y in common, we are dealing with the a. nominal scale b. ordinal scale c. interval scale d. the mere number of persons, things, or events having X or Y in common cannot be scaled 9-37. Whenever measured observations are rank ordered, the data form is a. nominal b. ordinal c. interval d. rank-ordered observations cannot be measured 9-38. When the only mathematical rule involved is that of equality versus nonequality, the measurement scale must be a. nominal b. ordinal c. interval d. ratio

9-39. Whenever we are given information regarding equality versus non-equality, greater than or less than, but not how much greater or less, the data form is a. nominal b. ordinal c. interval d. the concept of "greater than" can never be scaled

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9-40. Whenever we are presented with "nose-counting" data, such as how many persons are registered as republicans versus those registered as democrats, the data form is a. nominal b. ordinal c. interval d. ratio 9-41. If the number of jail inmates are categorized on the basis of whether or not they exhibit overt aggression during a free-time period, the data form is a. nominal b. ordinal c. interval d. data that are merely categorized fit no specific scale type 9-42. When measures provide information of not just greater than or less than, but also how much greater or less, the data form is a. nominal b. ordinal c. interval d. a and b, but not c 9-43. When frequency data are categorized into a nominal scale, the categories must a. overlap slightly b. overlap completely c. be completely independent d. frequency data cannot form a nominal scale 9-44. The concept of "mutually exclusive" categories is central to a. nominal scaling b. ordinal scaling c. interval scaling d. both a and c, but not b 9-45. If we were to know not only the order of finish of the cars at the Indianapolis "500", but also the actual elapsed times for each car, the data form would be a. at least nominal b. at least ordinal c. at least interval d. none of these

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9-46. If the population of registered voters in a given district were to be rank-ordered on the basis of socio-economic status. the data form would be a. at least nominal b. at least ordinal c. at least interval d. socio-economic status cannot be measured 9-47. The difference between interval and ratio scales is that ratio scales a. form independent categories b. form ordered sequences c. have absolute zeroes d. there is no difference 9-48. Most social scientists treat IQ data as though they are a. nominal b. ordinal c. interval d. ratio 9-49. Whenever the distance between successive scale points are assumed to be equal, the scale of measurement is at least a. nominal b. ordinal c. interval d. ratio 9-50. With interval data, we can say everything except a. X differs from Y b. X is greater than Y c. X is two units greater than Y d. X is twice Y 9-51. The reason that it is incorrect to say that person A's IQ is twice person B's is that IQ is not scaled as a. nominal data b. ordinal data c. interval data d. ratio data

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9-52. Of all the measurement scales covered in this chapter, the one containing the most information is the a. nominal scale b. ordinal scale c. interval scale d. ratio scale 9-53. Of all the measurement scales covered in this chapter, the one containing the least information is the a. nominal scale b. ordinal scale c. interval scale d. ratio scale 9-54. The issue of which scale of measurement has been used largely determines a. which statistical tests can be used b. whether the statistics are descriptive or inferential c. whether the statistics are inferential or predictive d. whether the data have been "rigged" or "faked" 9-55. The concept of generalizing from a few observations to an entire group is central to the area of a. descriptive statistics b. nominal scaling c. ratio scaling d. inferential statistics 9-56. Of the following, the measures which comes closest to ratio scaling are a. the numbers of people using fluoride versus non-fluoride toothpaste b. the final standings of the American League East's baseball teams c. the area codes used for direct telephone dialing d. the salaries of a group of factory workers 9-57. If it is known that X is greater than Y, and that is all that is known, then the measurement scale must be a. nominal b. ordinal c. interval d. a descriptive measure of this type can never be scaled 9-58. The measured heights of a group of basketball players form a(n) a. nominal scale b. ordinal scale c. interval scale d. ratio scale 131

9-59. Statistical tests designed to handle interval data, may also be used on a. nominal data b. ordinal data c. ratio data d. a test designed for interval data cannot handle data of any other type 9-60. Nominal data can always be converted into a. ordinal data b. interval data c. ration data d. nominal data cannot be converted into any other scale since it depends only on equality versus non-equality 9-61. Whenever a researcher randomly selects equivalent groups, and then exposes them to different stimulus conditions, and then measures them to see if significant difference can be observed, the total procedure is an example of a. experimental research b. post-facto research c. cross-sectional research d. none of these 9-62. The basic goal of good post-facto research should always be a. to establish a causal factor b. to test the hypothesis of difference c. to test the hypothesis of association d. to make better-than-chance predictions 9-63. When a post-facto relationship is found between two variables A and B, it is always possible, though not certain, that a. A caused B b. B caused A c. a third variable, X, has caused A plus B d. all of these 9-64. A study which compares the growth in the number of PTA mothers to the number of small business failures would be an example of a. experimental research b. post-facto research c. quasi-experimental d. repeated-measures

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9-65. When using the post-facto method, one may test the hypothesis of a. difference b. association c. both of these d. neither of these 9-66. When using the experimental method, the final goal is to test the hypothesis of a. difference b. association c. both of these d. neither of these Questions 67 through 72 will be bases on the following study: A researcher wishes to test whether the drug, magnesium pemoline, has any influence on intelligence. Two groups of subjects are randomly selected from the population of Omega University students. Group A is given the magnesium pemoline and then tested on the Stanford-Binet IQ test. Group B is given a placebo and, then, also tested on the Stanford-Binet. Both groups are tested under identical environmental conditions.

9-67. The study described above is an example of a. experimental research b. post-facto research c. correlational research d. longitudinal research 9-68. The design of this study is an example of a. the between-subjects method b. the before-after method c. the matched-subjects method d. the repeated measures method 9-69. The independent variable in this study is (are) the a. Stanford-Binet test b. magnesium pemoline versus the placebo c. control group d. IQ scores 9-70. The group which received the placebo is called a. the experimental group b. the control group c. the matched group d. the model group 133

9-71. The dependent variable in this study is (are) the a. magnesium pemoline b. testing conditions c. IQ scores d. there was no dependent variable in this study 9-72. If the IQ scores in Group A had been significantly higher than those in Group B, we may conclude that a. the magnesium pemoline probably caused an increase in IQ scores b. the magnesium pemoline probably caused a decrease in IQ scores c. cause and effect statements cannot be inferred from this study d. none of these 9-73. When subjects are measured on some trait they already possess, and are then categorized on the basis of that trait, the independent variable has been a. assigned b. manipulated c. confounded d. actively altered by the experimenter 9-74. When it is established that a change in the independent variable actually produces a change in the dependent variable, then the research must be a. post-facto b. cross-sectional c. experimental d. none of these 9-75. In experimental research, when the same group is used as both an experimental and control group, the research design is a. matched-subjects b. before-after c. between-subjects d. none of these 9-76. With the matched-subjects design, the subjects are equated on some variable which is assumed to have a possible influence on the a. independent variable b. dependent variable c. both of these d. neither of these

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9-77. The whole point of using one of the various experimental designs is to a. create equivalent groups of subjects b. randomize the alpha error c. prevent any sampling error d. prevent any possibility of alpha error 9-78. When a t test is applied to experimental research data, the groups are being compared with regard to their performances on the a. independent variable b. dependent variable c. confounding variable d. controlled variable Questions 9-79 through 9-84 will be based on the following: A researcher wishes to test the hypothesis that intelligence is a sex-linked characteristic. A random sample of 1000 women is selected and given IQ tests.

Similarly, a random sample of 1000 men is selected and also given IQ tests.

The mean IQ's of the two groups are compared and the difference is found to be significantly in favor of the females.

9-79. This study proves that a. a non-chance difference exists between male and female IQ's b. being female causes one to have a higher IQ c. being male causes one to have a lower IQ d. none of these, since the research is strictly post-facto 9-80. The research technique used in this study is an example of a. experimental research b. post-facto research c. longitudinal research d. b and c, but not a 9-81. In this study, the variable being manipulated by the researcher is a. sex b. intelligence c. the IQ tests d. none of these

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9-82. This study proves that a. a causal relationship exists between sex and intelligence b. a causal relationship exists between intelligence and sex c. sex and intelligence interact to cause an effect above and beyond either of the variables taken separately d. this study does not address the question of causality, since it is post-facto research 9-83. The results of this study suggest that a. sex may be used to predict IQ b. IQ is independent of sex c. sex is the causal factor in producing one's IQ d. the IQ difference should not be generalized to the population from which the sample was selected 9-84. When the experimental method has been used, and a significant difference is established between the means of two sample groups, then it is highly probable that a. the sample groups could not have been equivalent to start with b. the sample groups now represent different populations c. the sample groups still represent a single population d. the population parameters are known 9-85. When the achievement levels of two groups of college seniors are being evaluated on the basis of whether of not the students had previously attended a public secondary school, then the research is a. experimental b. quasi-experimental c. repeated-measures d. post-facto 9-86. When subjects change their behavior simply as a result of the experimenter paying attention to them, this is called a. the halo effect b. the Hawthorne effect c. the Rosenthal effect d. the Zeigarnik effect 9-87. When a person is viewed as being positive in one area, and then is assumed to be positive in other areas, this is called a. the double-blind error b. the nominal fallacy c. the beta error d. the halo effect

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9-88. The repeated-measures experimental design may lead to ambiguous results unless a. the subjects are matched on some relevant variable b. the dependent variable is measured only after the introduction of the independent variable c. the alpha error is increased to at least .50 d. there is a separate control group 9-89. The potential for the Hawthorne effect is most pronounced when the researcher is using a. the between-subjects design b. the matched-subjects design c. the one-group repeated-measures design d. an assigned independent variable 9-90. The best way to prevent the Hawthorne effect is to use a. post-factor research technique b. objective observers c. a separate control group d. none of these, since preventing the Hawthorne effect is itself a major research error 9-91. When the researcher is unable to examine the pure effects of the independent variable, then the independent variable is said to be a. manipulated b. confounded c. held constant in the group d. none of these 9-92. In experimental research, whenever extraneous influences cause changes on the dependent variable, then a. the independent variable has not been properly manipulated b. the dependent variable has been confounded c. the dependent variable has not been adequately measured d. the independent variable had been confounded 9-93. The independent variable is said to have been confounded whenever a. the response differences could have been caused by other factors b. the response differences are significant c. null is rejected d. the Type 1 error is committed

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9-94. In experimental research, cause-and-effect statements should not be made unless a. the sample groups are significantly different before the independent variable is introduced b. the sample groups are significantly different after the independent variable is introduced c. the sample groups represent a common population after the independent variable is introduced d. the null hypothesis is accepted Questions 9-95 through 9-101 will be based on the following: A researcher was interested in discovering whether attitude changes can be more effectively brought about when persons are made to feel part of the decision making process. A group of women was selected and randomly assigned to one of two groups, one group being placed in a large lecture hall, and the other in a small discussion-group setting. In the lecture setting, a speaker exhorted the women to use less expensive cuts of meat, whereas in the small-group setting a different group leader, although raising the same issues as the lecture had, also encouraged the subjects to participate in the discussion and offer suggestions, Several weeks later both groups of women were checked at home to determine whether they were indeed using the meat cuts which had been urged. Only 3% of the lecture group members had the meat cuts, whereas 32% of the discussion group members had the meat cuts. The difference was clearly significant.

9-95. The research method involved was a. post-facto b. longitudinal c. experimental d. designed to eliminate any possibility of alpha error 9-96. The design of the study was a. between-subjects b. repeated-measures c. matched-subjects d. none of these 9-97. Whether the women were subjected to the lecture method or the group discussion method, defined the a. control group b. assigned variable c. dependent variable d. independent variable

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9-98. The independent variable was a. an assigned-subject variable b. manipulated c. held constant d. accounted for by the difference in group variability 9-99. Whether or not the women later had the meat cuts defined a. the assigned independent variable b. the manipulated independent variable c. the error scores d. the dependent variable 9-100. The fact that the two groups were led by different persons created the possibility of incurring a. the Type 2 error b. the halo effect c. the Hawthorne effect d. confounding variables 9-101. The fact that the two groups differed significantly regarding whether or not they had the meat cuts, means that a. Ho was accepted b. Ho was never tested c. Ho was rejected d. the dependent variable was not affected 9-102. In evaluating the results of experimental research, secondary variance is considered to be a. an essential component of the design b. a result of the action of the independent variable c. the result of the influence of any uncontrolled, or confounding, variables d. the error term in the F ratio 9-103. When secondary variance is allowed to occur in experimental research, a. the design is assumed to have internal validity b. the design is assumed to have external validity c. the design is assumed to be tightly controlled d. the design is assumed to be "flawed" 9-104. When the design of an experiment simulates real-life conditions, it is said to be high in a. internal validity b. external validity c. internal reliability d. all of these 139

9-105. When the design of an experiment is such that the influence of the independent variable can be unambiguously interpreted, the experiment is said to be high in a. internal validity b. external validity c. external reliability d. all of these 9-106. According to the text, the major factor in achieving internal validity is in the correct use of a. statistical techniques of analysis b. representative sampling methods c. inductive logic d. an adequate control group 9-107. A study which produces large amounts of secondary variance will probably be found to be extremely low in a. sampling error b. Type 2 error c. random error d. internal validity 9-108. If a researcher is interested in whether a new training technique affects reading achievement scores, and the group chosen for the study is already high in reading-achievement levels, the lack of any further significant increase may be due to the a. researcher having chosen the .05 rather than the .01 level of statistical significance b. halo effect c. Hawthorne effect d. ceiling effect 9-109. When a significant difference is found between the DV scores of the experimental and control groups, it always means that this difference is a. important and profound b. due to chance c. due to confounding variables d. none of these

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9-110. For the quasi-experimental design, the following are all true, except a. the independent variable is manipulated. b. the independent variable identifies different treatment conditions. c. the subjects are randomly assigned to the sample groups. d. the independent variable is always an antecedent variable.

9-111. When the researcher is doing an experiment in which intact groups are being compared, the experimental design is considered to be a. post facto. b. correlational. c. path analytical. d. quasi experimental.

9-112. When doing experimental research, if the sample groups are not formed on the basis of the subjects' having been randomly assigned, then, of the following, which design could be used? a. post facto. b. correlational. c. path analytical. d. quasi experimental.

9-113. One of the fundamental requirements of all true experiments is that the different groups be a. equivalent at the outset. b. tested under at least three different conditions. c. matched on several covariates. d. none of these, since in the true experiment different groups can never be used.

9-114. The safest way to avoid the cause-and-effect trap is to a. be sure to obtain statistical significance. b. operationally define all terms. c. use the methodology of the true experiment. d. always use a post-facto analysis.

9-115. When a subject variable is used as an IV, it is a. manipulated. b. assigned after the fact. c. an obvious causal variable. d. all of these.

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9-116. In a matched-subjects design a. the groups of subjects are equated on the basis of overall averages. b. the subjects in the groups are matched person-for-person. c. the matching of all subjects is done statistically, after the fact. d. the matching is based completely on the IV manipulation.

9-117. In order to assess whether or not the IV had any impact, it is necessary to always compare the groups on the basis of the a. dependent variable. b. covariate. c. matching variable. d. none of these, since groups are not typically compared.

9-118. Any time a variable has been controlled, it has presumed to be a. held constant. b. administered only to the control group. c. administered only to the experimental group. d. all of these, depending on the experimental design.

9-119. When the results of a rat study conducted in the laboratory are extrapolated to humans in the work place, the study becomes weak in a. internal validity. b. external validity. c. reliability. d. statistical power.

9-120. The problem of pre-test sensitization is most acute in the a. after-only design. b. one-group before-after design. c. matched-subjects design. d. cross-sectional design.

9-121. Pre-test sensitization has the potential for a. confounding the IV. b. confounding the DV. c. increasing test reliability. d. creating constants out of variables.

Questions 9-122 through 9-124 are based on the following simulation: A researcher attempted to discover if the amount of learned material retained by first-grade children was a function of the mode by which the material was presented. Two first-grade classes are selected, Miss Emily McKenna's class and 142

Miss Clair Keating’s class. The study began at 9 AM on a Monday morning. By a flip of a coin, Mrs. Vivian's class was chosen to receive a visual presentation, whereas in Miss McKenna's class an auditory presentation of the same material was used. One hour later, the two groups were compared regarding the amount of material that was retained.

9-122. The research procedure in this case illustrates the a. quasi-experimental method. b. post-facto method. c. repeated-measures method. d. true experiment.

9-123. If the children had instead been randomly assigned to the two groups, then the above case would have illustrated the a. quasi-experimental method. b. post-facto method. c. repeated-measures method. d. true experiment.

9-124. The independent variable(s) in this study is (are) the a. the retention scores. b. the age of the subjects. c. the time of testing. d. the mode of presentation, audio vs. visual.

9-125. A study comparing an inexpensive generic drug sold over the counter with the prescription drug Singulair, found no significant difference in the treatment of hay fever (Mucha et al, 2006). The researchers randomly divided a group of male and female adult hay fever sufferers into two groups, one group receiving daily doses of Singulair while the other group was given the generic drug. After several weeks the groups were compared and no difference was found regarding their hay fever symptoms. This illustrates the a. quasi-experimental method. b. post-facto method. c. cross-sectional method. d. true experiment.

9-126. In this study the independent variable was a. Which drug the groups were given b. The incidence of hay fever c. The age of the subjects d. The gender of the subjects.

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9-127. In this study the dependent variable was a. Which drug the groups were given b. The incidence of hay fever c. The age of the subjects d. The gender of the subjects.

9-128. What statistical decision did the researchers make? a. Accept the Null Hypothesis b. Reject the Null hypothesis c. Keep the Null hypothesis form being involved in the decision. d. None of these 9-129. Comparing the experimental method with the quasi-experimental method a. The experimental method has a higher degree of internal validity. b. The experimental method has a higher degree of external validity c. The experimental method is the only one in which the IV is manipulated. d. All of these.

9-130. When an experimenter assigns a grouping variable, that variable is the a.dependent variable b.independent variable c.control variable d.all of these, depending on the method being used.

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C: True or False: For the following, indicate T (True) of F (False) 9-125. Experimental methodology demands the use of a manipulated independent variable.

9-126. Whenever two separate sample groups are being compared, the research must be experimental.

9-127. Whenever the independent variable had been assigned, the research must be post-facto.

9-128. The main purpose of post-facto research is to establish equivalent groups subjects.

9-129. It is virtually impossible to manipulate such variables as age, sex, race or income.

9-130. Experimental research only allows for the testing of the hypothesis of association.

9-131. In experimental research, the dependent variable always defines the differences in the treatment conditions undergone by the subjects.

9-132. When groups of subjects are equal to begin with, but are then treated differently by the experimenter, the research technique is post-facto.

9-133. One method which is used in order to create equivalent groups of subjects is the matched-subjects design.

9-134. The between-subjects, repeated-measures or matched-subjects designs may be used when conducting experimental research.

9-135. Post-facto research techniques may only be used to test the hypothesis of difference.

9-136. Whenever the hypothesis of difference is tested, the research method must have been experimental.

9-137. In order to establish that one group of subjects is significantly taller than another group, the experimental method must be used.

9-138. A study designed to test the effect of a prison education program on recidivism, would have recidivism as the independent variable.

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9-139. In the antecedent-consequent relationship, the dependent variable is always the antecedent variable.

9-140. When an independent variable is actively manipulated, the research must be postfacto.

9-141. The Hawthorne effect occurs whenever a group of subjects changes as a direct result of the flattery and attention received from the experimenter.

9-142. The Hawthorne effect is most likely to occur when the research design is matched-subjects.

9-143. An experiment which is high in internal validity is usually relatively free of the effects of confounding variables.

9-144. Primary variance is assumed to result from the action of the independent variable.

9-145. The technique of counterbalancing assures that external validity will be minimized.

9-146. The frequency of occurrence within mutually exclusive categories provides nominal data.

9-147. The concept of "greater than or less than" is a crucial component of nominal data.

9-148. With interval data, there must be an absolute zero point.

9-149. Ordinal data provide information regarding greater than of less than, but not how much greater of less.

9-150. Interval data require that there be an equal distance between successive scale points.

9-151. The mathematical rule of "equality versus non-equality" is central to nominal data.

9-152. Nominal data always imply equal distances between successive scale points.

9-153. With ordinal data, assumptions can be made regarding how much greater X is than Y.

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9-154. If IQ data are interval, then one can safely say that an IQ of 140 is twice as high as an IQ of 70.

9-155. Measurement is the assigning of numbers to observations according to certain rules.

9-156. Statements such as "twice as large" or "half as large" can only be made when the data are ordinal.

9-156. When a significant difference is found between the scores of the experimental and control groups, there is no possibility that this result could have occurred on the basis of chance.

9-157. When testing the null hypothesis, chance can never be completely ruled out.

9-158. In any study, experimental or post-facto, the independent variable must always be under the researcher’s full control.

9-158. A study on the effects of race on the length of prison sentences found no significant difference regarding the length of the sentences as a function of race (Spohn & Holleran, 2000). The independent variable in this study was race.

9-159. A study on the effects of race on the length of prison sentences found no significant difference regarding the length of the sentences as a function of race (Spohn & Holleran, 2000). The dependent variable in this study was race.

9-160. A study on the effects of race on the length of prison sentences found no significant difference regarding the length of the sentences as a function of race (Spohn & Holleran, 2000). The statistical decision in this study was to accept the null hypothesis.

9-161. In the classic Kansas City preventive control study (1973) it was found that applying different police patrol strategies had absolutely no impact on crime rates. The independent variable in this study was based on the crime rates.

9-162. In the classic Kansas City preventive control study (1973) it was found that applying different police patrol strategies had absolutely no impact on crime rates. The dependent variable in this study was based on the crime rates.

9-163. In the classic Kansas City preventive control study (1973) it was found that applying different police patrol strategies had absolutely no impact on crime rates. The statistical decision in this case was to accept the null hypothesis.

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9-164. In studies of prison riots, it has been found that prison riots increase as security measures are heightened (Useem & Kimball,1989). In these studies the independent variable is the heightened security.

9-165. In studies of prison riots, it has been found that prison riots increase as security measures are heightened (Useem & Kimball,1989). In these studies the dependent variable was based on the riots.

9-166. In studies of prison riots, it has been found that prison riots increase as security measures are heightened (Useem & Kimball,1989). These were all post-facto studies…

References Axline, V.M. (1947). Nondirective therapy for poor readers. Journal of Consulting Psychology, 1, 61-69.

Faw, V.E. (1949). A psychotherapeutic model of teaching psychology. American Psychologist, 4, 104-109.

Kelling, G.L., Pate, T., Dieckman, D., Brown, C.E. (1973). The Kansas City Preventive Control Experiment. Washington, DC: The Police Foundation.

Miller, N.E., & Bugelski, R. (1948). Minor studies of aggression. Journal of Psychology, 25, 437-442.

Mucha, S.M., DeTineo, M., Naclerio, R.M. & Baroday, F.M. (2006), Comparison of montelukast and pseudoephedrine in the treatment of rhinitis. Archives of Otolaryngol Head Neck Surgery, 132, 164-172.

Spohn, C. & Holleran, D. (2000). The imprisonment penalty paid by young, unemployed black male offenders. Criminology, 35, 281-297.

Useem, B., & Kimball, P. (1989). State of siege: U.S. prison riots. New York, Oxford Press.

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CHAPTER 10 THE HYPOTHESIS OF DIFFERENCE A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Student’s t as a RATIO For the two-sample independent t, stress the fact that the t ratio is indeed a RATIO - the difference between the two means in units of the estimated standard error of difference. Too often the students will observe the sheer magnitude of the difference between the two means and assume the difference is either significant or not depending on the size of that mean difference. They must be reminded that the variability within each of the sample distributions (which will make up the standard error of difference) is just as important as the difference between the means. For a given mean difference, the less variability within the sampling distributions then the higher the t ratio and the greater the likelihood of rejecting Ho. For example, the following two sets of sample scores reveals a mean difference of 4 points.

Sample #1 Sample #2 26 22 22 18 22 18 20 16 20 16 20 16 20 16 18 14 16 12 14 10 Mean 1 = 19.800 Mean 2 = 15.800 SD1 = 3.156 SD2 = 3.156 s1 = 3.327 s2 = 3.327 SE of mean 1 = 1.052 SE of mean 2 = 1.052 SE of difference = 1.488 t = 2.688 With t at .05 and 18 df, the table value of t is 2.102 - leading to a reject of Ho at P < .05. Thus, with relatively little within variability, the mean difference of 4 points is significant.

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Activity: Increasing the Variability Within Now, however, by stretching the ranges of both distributions and increasing the two standard deviations (the within variability), the same mean difference may lead to an accept of Ho. Sample #1 Sample #2 32 28 24 20 22 18 20 16 20 16 20 16 20 16 18 14 14 10 8 4 Mean 1 = 19.800 Mean 2 = 15.800 SD 1 = 5.896 SD 2 = 5.896 s1 = 6.215 s2 = 6.215 SE of mean 1 = 1.965 SE of mean 2 = 1.965 SE of difference = 2.779 t = 1.439 Accept Ho; the difference of 4 points between the means is NOT SIGNIFICANT. Activity: Getting Creative with the Standard Deviation.

During the calculation of the one or two sample t test, if a student inadvertently calculates the standard deviation of the sample scores incorrectly, a very interesting phenomenon occurs. This is aimed at the student who misreads the SD equation and assumes that ΣX² is the same as (ΣX)².

Don't assume that this will never happen in your class. If you teach the course long enough, and have large enough enrollments the probability is approximately 1.00 that at least one of your students will confuse the two summation Xs. Thus on the following distribution, the standard deviation so calculated turns out to be the enormous value of 27.30.

X X² 12 144 10 100 12 144 14 196 12 144 10 100 8 64 X = 78 X² = 892 150

M =ΣX/N = 78/7 = 11.14 Instead of using ΣX², or 892, the student used (ΣX)², 78² , or 6084.

Thus, SD became equal to √(6084/7)² - 11.14 =√ 869.14-124.10 = √ 745.04 = 27.30.

This obviously creates a standard deviation that is far too large. At this point the student might have remembered the logic check that the SD cannot be greater than R/2, or in this case 6/2 = 3, and without question, 27.30 is a tad larger than 3. Missing this thought, however, the student went plunging on, and next used this erroneous SD to find the estimated standard error of the mean.

SEm = SD/√N-1 = 27.30/ √6 = 27.30/2.45 = 11.14.

What has been found, therefore, is not the standard error of the mean, but the plain, old arithmetic MEAN of the sampling distribution, a value that was already known as soon as 78 was divided by 7. Point out strongly that this is a rather circuitous route to something as easily found as the arithmetic mean.

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B: Multiple Choice Items 10-1. The distribution of differences is composed of the differences between a. raw scores and the mean b. the highest and lowest scores in a distribution c. the standard deviation and the standard error of the mean d. pairs of randomly selected sample means 10-2. When graphing the distribution of differences, the abscissa is used to represent a. a sample mean b. the difference between the mean of the individual raw scores and the sample mean c. the difference between two sample means d. the frequency of occurrence 10-3. When graphing the distribution of differences, the ordinate is used to represent a. a sample mean b. the difference between the mean of the individual raw scores and the sample mean c. the difference between two sample means d. frequency of occurrence 10-4. When the difference between pairs of randomly selected means are obtained, the resulting distribution is called a. the distribution of means b. the distribution of differences c. the distribution of individual sample scores d. the ogive distribution 10-5. When all samples are drawn from a single population, the mean of the distribution of differences should approximate a. zero b. +1.0 c. -1.0 d. the mean of the distribution of means 10-6. If the mean of the distribution of differences were to equal -6.00, this would indicate that a. the standard deviation of the distribution is a function of the range b. the samples probably represent different populations c. the samples were probably selected from a single population d. none of these, since the mean can never be negative

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10-7. Whenever the mean of the distribution of differences equals zero, then one can be fairly safe in concluding that a. the samples probably represent a single population b. the samples probably represent different populations c. there was absolutely no sampling error d. none of these, since the mean can never equal zero 10-8. The estimated standard deviation of the distribution of differences is used to predict the a. standard error of the mean b. standard error of difference c. standard error of the estimate d. actual standard score 10-9. If all the possible pairs of sample means were obtained from a single population, the mean of the distribution of differences would be a. a statistic b. a parameter c. identical with the mean of the distribution of means d. equal to one-sixth of the range 10-10. The estimated standard error of difference may be obtained on the basis of the information contained in a. a single sample b. a pair of random samples c. the sampling error d. a single z score 10-11. The estimated standard error of difference is an estimate of a. centrality b. variability c. normality d. none of these, since the estimated standard error of difference is not a statistic.

10-12. The Central Limit theorem states that the distribution of differences is a. always based on a pair of samples b. always based on a single sample c. a parameter d. approximately normal in shape

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10-13. When selecting a pair of random-sample means from a single population, the probability of the first mean being higher than the second mean is a. 1.00 b. 0 c. .50 d. either a or b, but never c 10-14. The distribution of differences is a. a sampling distribution b. a distribution of parameters c. always leptokurtic in shape d. always platykurtic in shape 10-15. Regarding the distribution of differences, the assumption of normality is based on a. Pascal's triangle b. Gossett's law c. the Central Limit theorem d. the Gaussian law 10-16. When pairs of samples are selected from two separate populations, the mean of the distribution of differences is assumed to be a. some value other than zero b. zero c. a constant d. equal to the mean of the distribution of means

10-17. In order to use the estimated standard error of difference for making probability estimates, both samples must a. have equal means b. have unequal means c. be free of sampling error d. have similar amounts of variability 10-18. In order to ensure a sampling error of zero, each sample must a. be randomly selected b. be equal in size c. be selected from the same population d. none of these, since one can never assume a sampling error of zero

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10-19. Whenever the distribution of differences is normally shaped, and all its assumptions are met, then a. the samples must represent different populations b. the samples must represent a single population c. more than two populations must be involved d. none of these, since when its assumptions are met, the distribution of differences is always assumed to be normally shaped 10-20. In order to calculate the estimated standard error of difference, all that need be known is (are) a. the means of the two samples b. the sizes of the two samples c. the standard deviation of the two samples d. the standard errors of the mean for each of the two samples 10-21. The independent t test may be used to evaluate the difference between a. two sample means b. two or more sample means c. the mean and the standard deviation d. the mean and the variance 10-22. The independent t ratio may be used to evaluate whether a. two samples represent a single population b. two samples represent different populations c. both of these d. neither of these 10-23. For the independent t test, the null hypothesis states that the samples a. represent a single population b. represent different populations c. are of infinite size d. each have mean scores of zero 10-24. For the independent t test, degrees of freedom are based on a. the population sizes b. the sample sizes c. the number of samples being compared d. none of these 10-25. When the degrees of freedom reach infinity, the t distribution a. becomes identical with the distribution of raw scores b. gets larger c. approximates normality d. cannot be obtained 155

10-26. For the independent t test, the null hypothesis is always a. a function of sample size b. a function of variability c. a function of centrality d. the same as the "chance" hypothesis 10-27. For the independent t test, the hypothesis of "no difference" is called the a. alternative hypothesis b. null hypothesis c. central limit hypothesis d. hypothesis of association 10-28. When a difference is said to be "significant", it means that the difference probably a. does not exist in the population b. does exist in the population c. is no longer related to the population d. none of these 10-29. When a difference is considered "significant", it indicates that the difference is probably a. meaningful and profound b. due to chance and chance alone c. not due to chance alone d. none of these 10-30. Like the T score, the t ratio is a measure of a. individual performance b. how a single sample performed c. central tendency d. none of these, since the T score involves a measure of individual performance and the t ratio involves group comparisons.

10-31. The difference between the T score and the t ratio is that a. the T score always results in a higher value than does the t ratio b. the T score refers to several sample means, whereas the t ratio only refers to two sample means c. the T score is used as a measure of individual performance, whereas the t ratio evaluates the possibility of a performance difference between two independent groups d. none of these, since T and t are the same

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10-32. For the t ratio, when the null hypothesis is rejected, then a. the difference is assumed to be significant b. the difference is assumed to equal zero c. the difference is assumed to be positive d. none of these 10-33. For the independent t test, the null hypothesis is always stated in terms of a. the sample means b. the population means c. both of these d. neither of these 10-34. The independent t ratio specifies how far the difference between the sample means deviated from a. an assumed mean of 50, in units of standard deviations b. an assumed mean of 100, in units of estimated standard errors of difference c. an assumed mean of zero, in units of estimated standard errors of difference d. the standard deviation of the population 10-35. The t test for two independent samples may only be used when a. the selection of subjects for the first group does not depend in any way on the selection of subjects for the second group b. the subjects used in the two samples are matched on some relevant variable c. the same subjects are used twice d. the degrees of freedom are independent of the sample sizes 10-36. When the alternative hypothesis is accepted, then a. the sample means are assumed to be equal b. the population means are assumed to be equal c. chance is rejected d. a significant difference has been established 10-37. Whenever the null hypothesis is rejected, then a. the alternative hypothesis must also be rejected b. the alternative hypothesis must be accepted c. the alternative hypothesis must be tested separately d. none of these, since the null hypothesis and the alternative hypothesis are not related to each other

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10-38. The statement that the difference between the two population means is zero, represents a. the Type 1 error b. the sampling error c. the alternative hypothesis d. the null hypothesis 10-39. The alternative hypothesis for the independent t test is also called a. the hypothesis of association b. the hypothesis of difference c. the null hypothesis d. none of these 10-40. When the null hypothesis is rejected, then a. the "chance" explanation is completely ruled out b. the "chance" explanation, though not ruled out, becomes improbable c. the "chance" explanation is fully accepted d. the "chance" explanation becomes irrelevant 10-41. The alpha level sets the probability for the a. Type 1 error b. Type 2 error c. Type 3 error d. none of these 10-42. According to the text, the value beyond which alpha should not be set, is a. .01 b. .05 c. .50 d. 1.00 10-43. The t distribution assumes normality only when a. there are infinite degrees of freedom b. there is a large degree of bias c. the null hypothesis has been rejected d. none of these, since the t distribution can never be normal 10-44. With infinite sample sizes, t ratios of +1.96 a. include only the middle-most 68% of all t values b. include only the middle-most 50% of all t values c. include only the middle-most 5% of all t values d. exclude only the most extreme 5% of all t values

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10-45. With one degree of freedom, a significant t ratio will occur a. 1% of the time b. 99% of the time c. 68% of the time d. none of these, since degrees of freedom reflect sample sizes not frequency of occurrence.

10-46. If an independent t ratio is to be calculated for a two-group design, 20 subjects in each group, then the degrees of freedom equal a. 1 b. .05 c. 40 d. 38 10-47. The power of a statistical test, such as the independent t test, can be increased by a. increasing the sample size b. increasing the alpha error c. using a one-tail test d. all of these 10-48. Anytime the probability of beta is reduced, then a. power increases b. power decreases c. the likelihood of rejecting null is minimized d. the likelihood of obtaining a significant difference is minimized 10-49. Power can be defined as a. One plus beta b. Two plus beta c. One minus beta d. One minus alpha 10-50. Every degree of freedom value specifies a. a different t distribution b. the same t distribution c. that the t distribution must be assumed to be normal d. the number of sample groups being compared 10-51. If the tabled value for t, for a certain number of degrees of freedom, were 3.17, and the calculated value of t were 3.17, then a. There is a failure to reject Ho b. Ho is rejected c. alpha becomes equal to zero d. the t distribution must be normal 159

10-52. To do a one-tail t test, as opposed to a two-tail, a. the direction of the difference must always be specified beforehand b. the direction of the difference need only be specified after null has been rejected c. the direction of the difference must only be specified when null is accepted d. the direction of the difference becomes irrelevant 10-53. The sign of the t ratio only becomes important to the statistical decision when a. null is accepted b. the degrees of freedom are infinite c. alpha is set at .05 or less d. calculating a one-tail t test 10-54. The t distributions deviate from normality when a. the null hypothesis is being tested b. the alpha is less than .05 c. sample sizes become smaller d. none of these, since the t distribution never deviates from normality 10-55. Whenever the alternative hypothesis is written as "mean 1 is greater than mean 2", then a. a one-tail t test is allowed b. only a two-tail t test is allowed c. an independent t test is not allowed d. sample sizes must be greater than 30 10-56. The only time a difference between two sample means may be called "significant" is when a. they are at least 10 points apart b. they differ by at least .05 points c. the first mean is larger than the second mean d. the null hypothesis has been rejected 10-57. A powerful test is one that is a. less likely to commit the Type 2 error b. more likely to commit the Type 2 error c. more likely to accept the null hypothesis d. less likely to detect a population difference

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10-58. The higher the likelihood of rejecting null when null really should be rejected, then a. the higher becomes the value of beta b. the less powerful the statistical test c. the more powerful the statistical test d. none of these, since rejecting the null hypothesis is not related to either power or beta.

10-59. The effect size is determined by how much difference there is between the a. two types of error, Type 1 and Type 2 b. population means in units of population standard deviations c. population mean and the sample variance d. sample mean and the population variance 10-60. The larger the effect size, the greater the likelihood a. of ignoring the null hypothesis b. of rejecting the null hypothesis c. of committing the Type 2 error d. that the population means are not significantly different

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C: True or False: For the following, indicate T (True) or F (False) 10-61. When the two sample means are equal, the t ratio will always lead to an accept of the null hypothesis.

10-62. Student’s t test for two independent samples assumes that the variances within each of the two groups are NOT significantly different from each other.

10-63. The assumption of homogeneity of variance can be tested using the Levine Test.

10-64. When the Levene test results in a significant value, .05 or less, then the variances can be assumed to be equal.

10-65. If the Levene test resulted in a significance level greater than .05, then homogeneity of variance can be assumed.

10-66. The mean of the distribution of differences becomes zero only when all sample pairs represent different populations.

10-67. When the direction of the difference between two sample means is not specified, we must use the two-tail t.

10-68. The larger the sample sizes, the greater the degrees of freedom.

10-69. With an infinite number of degrees of freedom the t distribution assumes normality.

10-70. When the null hypothesis is rejected, then the alternative hypothesis must be accepted.

10-71. The estimated standard error of difference estimates the value of the population mean.

10-72. The estimated standard error of difference may never assume a negative value.

10-73. The estimated standard error of difference is an estimate of the variability occurring within the entire distribution of differences.

10-74. The t ratio may only be calculated when the data are in ordinal form.

10-75. When all the possible sample pairs have been selected from a single population, the mean of the distribution of differences is expected to approximate zero.

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10-76. When there is a failure to reject Ho then one may assume that the two sample groups represent a common population.

10-77. The central limit theorem suggests that the entire distribution of differences between pairs of successively drawn sample means will approximate normality.

10-78. The distribution of differences is a distribution of sample measures.

10-79. The true standard error of difference is equal to the standard deviation of the entire sampling distribution of differences when all the sample pairs in the population have been measured.

10-80. The greater the number of degrees of freedom for the t ratio, the less the likelihood of rejecting the null hypothesis.

10-81. As was true for the z test, the t ratio assumes that all the parameter values are known.

10-82. For a given t value, the likelihood of rejecting null increases when a two-tail, rather than a one-tail, t value is applied.

10-83. For the t ratio, degrees of freedom are based on the number of samples rather than the size of the samples.

10-84. When all the possible pairs of random sample means have been selected from the population, the mean of the sampling distribution of differences is a parameter.

10-85. The t ratio assumes that the two samples being compared are similar with respect to variability.

10-86. The t ratio, since it is a parametric test, makes no assumptions regarding the shape of the population distribution from which the samples have been selected.

10-87. The t ratio should only be calculated when the data are in at least interval form.

10-88. The estimated standard error of difference is an interval estimate of the true mean of the distribution of underlying individual scores.

10-89. The larger the effect size, the greater the likelihood of detecting population difference with inferential statistical tests.

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10-90. Increasing the sample sizes, increases the probability of incurring the Type 2 error.

10-91. The larger the effect size, the greater the likelihood of rejecting the null hypothesis.

10-92. When an inferential statistical test such as the t test is showing significant results, then the effect size must be large.

10-93. When there is a significant difference in the population, then a powerful statistical test is likely to detect that difference.

10-94. The population effect size is estimated in the numerator of the t ratio.

10-95. The Levene test may only be calculated with nominal data.

10-96. When homogeneity of variance is assumed, then the Levene test will NOT be significant, 10-97. The Levene test assesses the degree of skewness in both distributions.

10-98. Student’s t should only be used when the homogeneity of variance assumption is met.

10-99. Student’s t can only be used with nominal data.

10-100. Student’s t can only be used with ordinal data.

10-101. Student’s t demands at least interval data.

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D. For the following questions, calculate the values.

Items 102 - 107 will be based on the following: A researcher is interested in whether social assertiveness might be influenced by training procedures. Two groups of college sophomores are randomly selected. Group A is assigned to the "training sessions", whereas the subjects in Group B are allowed to get on with their lives in any way they choose. Both groups are then given the "Social Assertiveness" test, (high scores indicating greater assertiveness), and the results were as follows: Group A Group B 14 12 11 10 10 10 10 8 8 7 6 3 8 8 9 6 10 5 11 4 10-102.Calculate the means for each group

10-103. Calculate the standard deviation for each goup.

10-104

Calculate the SEms for each group.

10-105

Calculate the standard error of difference.

10-106

Calculate the independent t ratio

10-107

Is the t ratio significant?

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For questions 108 through 108, use the following: A researcher wonders if students will spend more time studying for a quiz if they are told that a high mark on that quiz will excuse them from writing a term paper. Two groups of college students were randomly selected from a large lecture course in art history. The students in Group 1 were simply told that they would have their first art-history quiz two days later. In Group 2, they were told the same thing, except that the promise of "no term paper" for high quiz performance was added. Just before taking the quiz, all students in both groups reported (in hours) the time they had spent preparing. In Group 1, the times were 6, 4, 9, 3, 2, 5, 7, 4, 7, 6. In Group 2 the times were 12, 5, 13, 5, 5, 7, 9, 6, 9, 6.

10-108. Calculate the means for each group

10-109. What is the meaning of the negative t ratio?

10-110. As a one-tail t, since the direction of the difference might have been predicted, do you reject or accept the null hypothesis? If rejected, what is the significance level?

10-111. A researcher selects a pair of random samples. In Group 1 the scores are 12, 6, 9, 10, 8, 7. In Group 2, the scores are, 10, 4, 7, 8, 7, 6. Find the standard error of difference.

10-112. A researcher selects a pair of random samples. In Group 1 the scores are 20, 14, 16, 18, 16, 15, 14. In Group 2 the scores are 23, 20, 21, 24, 18, 22, 24, 22, 20, 20. Find the standard error of difference.

10-113. A researcher selects a pair of random samples. In Group 1 the scores are 9, 12, 7, 8, 6. In Group 2 the scores are 6, 5, 7, 6, 5. Find the t ratio.

10-114. A researcher selects a pair of random samples. In Group 1 the scores are 25, 25, 24, 24, 23, 233, 22, 21, 20. In Group 2 the scores are 22, 20, 20, 18, 15, 15, 12, 12, 10. a. Find the t ratio b. Do you reject Ho? c. Do the two groups represent a single population?

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10-115. A researcher selects a pair of random samples. In Group 1 the scores are 21, 19, 177, 13, 11, 9, 7.In Group 2 the scores are 20, 18, 15, 11, 11, 8,4. a. Find the t ratio b. As a two-tail t, should Ho be rejected?

If rejected, state the probability of the alpha error. c. Do the two groups represent a single population?

10-116. A researcher selects a pair of random samples. In Group 1 the scores are 14, 10, 10, 9, 8, 6, 4. In Group 2 the scores are 12, 10, 6, 5, 4, 2, 2. a. Find the t ratio b. As a two-tail t, should Ho be rejected?

If rejected, state the probability of the alpha error. c. Do the two groups represent a single population?

10-117. A researcher selects a pair of random samples. In Group 1 the scores are 3, 7, 6, 5. In Group 2 the scores are 4, 6, 2, 2. a. Find the t ratio b. As a two-tail t, should Ho be rejected?

If rejected, state the probability of the alpha error. c. Do the two groups represent a single population?

10-118. A researcher selects a pair of random samples. In Group 1 the scores are 10, 4, 12, 5, 4, 6, 8, 7. In Group 2 the scores are 6, 4, 9, 3, 2, 5, 7, 4. a. Find the t ratio b. As a two-tail t, should Ho be rejected?

If rejected, state the probability of the alpha error. c. Do the two groups represent a single population?

10-119. A researcher selects a pair of random samples and predicts beforehand that Group 2 will score higher than Group 1. The scores in Group 1 are 11, 10, 8, 3, 7, 1. The scores in Group 2 are 12, 11, 8, 4, 9, 2. a. Find the t ratio b. As a one-tail t, should Ho be rejected?

If rejected, state the probability of the alpha error. c. Do the two groups represent a single population?

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10-120. A researcher selects a pair of random samples and predicts beforehand that Group 1 will score higher than Group 2. The scores in Group 1 are 6, 7, 10, 9, 8. The scores in Group 2 are 3, 4, 9, 7, 5. a. Find the t ratio b. As a one-tail t, should Ho be rejected?

If rejected, state the probability of the alpha error. c. Do the two groups represent a single population?

10-121. The appropriate value for assessing effect size for the two sample t ratio is a statistic called a. a b. b c. c d. d

A group of 20 violent drug abusers, all of whom had committed homicide while under the influence of a psychostimulant, was selected from the population of a certain prison. Half the men had been cocaine abusers while the other half had been on methamphetamines.

All were abstinent at the time of testing. The men were tested on the CAQ (Clinical Analysis Questionnaire) and their Factor Q1 scores were compared (Factor Q1 includes Radicalism, Hypochondriasis and Boredom & Withdrawal). Their scores, converted to T scores, were as follows: Cocaine Group: 65,58,65,62,60,61,70,71,68,75.

Methamphetamine Group: 60,53,52,55,54,59,62,60,60,56.

10-122. Find the means for each group.

10-123. Do the two groups differ significantly?

10-124. Identify the independent variable.

10-125. Identify the dependent variable.

10-126. Was this experimental or post-facto research?

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Since psychopaths account for a quarter of all prisoners in the United States (and for as much as 50% of all violent crime) researchers have been examining the personality traits that may play a role in this condition. Tow groups of inmates, half diagnosed by Psychiatrists as psychopathic and the other not so diagnosed were given the IRI (Interpersonal Reactivity Index as a measure of empathy). The scores, converted to T scores were as follows: Psychopaths: 35,40,28,44,45,48,42,38,40,44.

Non-Pychopaths: 52,60,50,48,55,50,54,55,44,40.

10-127. Find the means for each group.

10-128. Do the two groups differ significantly?

10-129. Identify the independent variable.

10-130. Identify the dependent variable.

10-131. Was this experimental or post-facto research?

A researcher was interested in studying the possible deterrent effect of sentencing. A sample of 12 recidivists was compared with another sample of 12 who had stayed clean.

Both groups had been convicted of distributing drugs and both groups had served time.

For each subject the length of the previous sentence in months was found.

Non-Recidivists Recidivists 36 60 48 72 54 59 32 35 66 56 45 55 37 42 46 57 35 60 42 64 28 97 24 28

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10-132. Find the mean sentence for each group.

10-133. Do the two groups differ significantly?

10-134. Identify the independent variable.

10-135. Identify the dependent variable.

10-136. Was this experimental or post-facto research?

10-137. Did the results answer the researcher’s question?

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CHAPTER 11 THE HYPOTHESIS OF ASSOCIATION: CORRELATION A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Correlation and the Cause-and-Effect Trap Explain to your students that many correlation studies commit what the scientists call the post-hoc fallacy, which is written in Latin as "post hoc, ergo propter hoc" (translated as "because it came after this, therefore it was caused by this"). It's like the rooster taking credit for the sunrise. Many examples of this fallacy will come readily to mind. Before asking your class for examples, get the ball rolling by providing some of your own. Remind the students of some common examples from everyday life, such as someone complaining that it always rains right after the car is washed, as though washing the car caused it to rain. Or the traffic is always especially heavy when you have an important meeting to go to, as though all the other drivers know when the meeting has been planned and gang up in a contrived conspiracy to force you to be late. Or perhaps while watching a sporting event on TV you have to leave the room for a few minutes. When you return you discover that your favorite team has finally scored. You angrily hypothesize that the team deliberately held back until you weren't there to enjoy the moment. Some published research findings have jumped to conclusions very similar to these rather blatant examples. Ask your class for other examples, especially from their own memory banks. One student remembered his very first day in school when as a quaking five-year old, he simply forgot to bring home his sweater. To cover this transgression, he actually blamed another student for having taken it. The next day he did not want to go back to school, complaining that "every time I go to school, they steal my sweater".

Warn your students to be extremely careful when reading research studies that are based on correlational evidence. Just because the study appears in a journal and includes lots of statistical notations (which always tend to lend an air of authority), it doesn't mean that the conclusions are necessarily valid. Remind the class that they still have to look at the findings through the clear lens of common sense. Let the reader beware who does not at least to some extent adopt a critical posture. And this warning should not be ignored in situations where the data have been statistically analyzed but should be especially heeded when the data have been statistically analyzed.

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Activity: The Spearman rs or rho Every year J.D. Powers and Associates report the Customer Satisfaction Index on the current year's vehicles. For one year, circa 2010, the rankings for the highest rated cars were as follows: 1. Lexus, 2. Infiniti, 3. Saturn, 4. Mercedes-Benz, 5. Audi, 6. Cadillac, 7. Toyota, 8. Acura, 9. Jaguar, 10. Lincoln, 11. Honda, 12. Oldsmobile, 13. Buick, 14. Saab, 15. Volvo, 16. Subaru. Present this list, first in alphabetical order, and ask the class to guess as to how they ranked.

Alphabetized they are: Acura, Audi, Buick, Cadillac, Honda, Infiniti, Jaguar, Lexus, Lincoln, Mercedes-Benz, Oldsmobile, Saab, Saturn, Subaru, Toyota and Volvo. Then pair the median rank (ask the students to do this), with the J.D. Powers rank and run a Spearman correlation between the two. Some enterprising students have even gone out and found the base sticker price for these vehicles, and run correlations between the Powers ranking and the cost of the vehicle. All of this tends to result in lively discussions, especially when you tell them that the following sixteen cars ranked at the bottom of the list, and although Powers does not release the actual rankings of the lowest rated cars, they were (in alphabetical order): BMW, Chevrolet, Chrysler, Dodge, Eagle, Ford, Geo, Hyundai, Mazda, Mercury, Mitsubishi, Nissan, Plymouth, Pontiac, Suzuki and Volkswagen.

Activity: Hard Data?

We as educators and social researchers tend to think of the social sciences as dealing with observations that don't lend themselves to precise quantification, and sometimes they don't. But we too often have an inferiority complex when it comes to precision, and we buy into stories like the one concerning the market researcher who was trying to gauge the size of a TV audience. The data arrives, and the researcher proudly shouts out that last night the XYZ show was seen by 10.0284398753% of the viewing audience, grandly using lots of decimal places to enhance the perception of precision. Then we find out how the data were gathered. Phone calls went out to what was alleged to be a representative sample, and the respondents were asked what TV show they were watching, or if the TV was not on what would they watch in this time slot if the TV had been on. In many cases the respondents then asked what's on during that time period, and after being told, finally make a selection. And that's how the researcher got that good hard data that warranted at least ten decimal places. Yeah right! But it's not always that bad even though we often think of ourselves as the measurement practitioners who are always plagued with "soft" data. We tend to think of the physicist as having a much easier time because of data that are far more amenable to precise quantification. However, measurements can't be any more accurate than the level of precision involved in their construction.

We might think that something as physical as measuring the length and width of a dollar bill would be easy and highly reliable. But what measurement units should we use? Inches are undoubtedly close enough, but suppose we demanded 172

more and more precision, and decided to use centimeters or even millimeters. Or take the bio-medical researchers who have attempted to evaluate something as minute as a virus that actually measures only 1 attometer. An attometer is so small that it would take 1 quadrillion of them to cover the period at the end of this sentence. In fact, if one were to enlarge a series of attometers, each one to the size of that period, and then line them up end to end they would stretch out 1 billion kilometers, or approximately the distance between the earth and Saturn. If we think measuring IQs is difficult, getting reliable measures using attometers might not be as easy as a stroll in the park on a June afternoon.

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B: Multiple Choice Items 11-1. A maximum linear correlation is expressed as a. 100% b. 99 c. +10.00 d. +1.00 11-2. The more a correlation deviates from zero, a. the more it is certain to be positive in sign b. the more it is certain to be negative in sign c. the more it is certain that the measured variables are independent of each other d. the better its predictive accuracy 11-3. When two sets of measures have been found to be totally independent of each other, then the a. correlation between them must be negative b. correlation between them must be positive c. correlation between them must be zero d. no attempt to calculate the correlation should be made 11-4. Of the following correlation coefficients, which expresses the weakest association? a. .10 b. 0 c. -.50 d. -1.00 11-5. Of the following correlation coefficients, which expresses the strongest association? a. .10 b. 0 c. -.50 d. -1.00 11-6. When two variables, A and B, are strongly correlated, it means that a. A has caused B b. B has caused A c. some third variable has caused both A and B to occur together d. none of these

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11-7. The correlation coefficient, in and of itself, a. never proves causation b. never rules out causation c. allows for better-than-chance predictions d. all of these 11-8. The most common error in interpreting correlational research studies is in a. assuming that the data have been incorrectly reported b. assuming that the appropriate statistical analyses have been used c. assuming that a causal factor has been isolated d. all of these 11-9. Each data point on a scatter plot represents a. the frequency of occurrence b. a pair of scores c. a score on one measurement d. none of these 11-10. A scatter plot on which the array of points goes from upper left to lower right, indicates a. a positive correlation b. a negative correlation c. a zero correlation d. all of these, depending on the strength of the correlation 11-11. The major reason for using the correlational techniques is to a. make cause-and-effect statements b. make better-than-chance predictions c. identify a specific percentile d. establish a significant difference 11-12. On a scatter plot, the ordinate is used to represent a. the scores on the Y measure b. the scores on the X measure c. the frequency of occurrence d. the product moment 11-13. When the array of points on a scatter plot slopes from lower left to upper right, then a. the correlation is positive b. the correlation is negative c. there is no correlation d. no predictions can be made

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11-14. The originator of the product-moment correlation technique was a. Karl Gauss b. Blaise Pascal c. Karl Pearson d. Alexandra McKenna 11-15. When variable A correlates highly with B, then a. A has caused B b. B has caused A c. variable B correlates highly with A d. none of these 11-16. The correlation between parental income and the amount of financial aid awarded by the college to students should most probably be a. positive b. negative c. zero d. none of these, since income is a nominal measure 11-17. Among elementary school children, the correlation between age and height would most probably be a. positive b. negative c. zero d. non-linear 11-18. A "significant" correlation is one which a. is especially meaningful b. does not deviate from zero c. indicates a valid difference d. can be generalized to the population 11-19. In order to establish the significance of a correlation, one must know the value of the correlation coefficient and also a. the number of paired scores b. whether it relates to other measures c. Whether it specifies the direction of the association d. the sign of the correlation 11-20. Negative correlations a. may never be significant b. may never be used for prediction c. may never generalize to the population d. none of these 176

11-21. When a relationship is inverse it means that the correlation must be a. non-significant b. non-linear c. significant d. negative 11-22. For the Pearson r, when Ho is rejected, it means that a. the correlation is positive b. the correlation is negative c. the correlation is significant d. none of these 11-23. The Pearson r may be used to test the hypothesis of a. difference b. association c. central limit d. all of these 11-24. When a correlation is not significant, it means that a. There is a failure to reject Ho b. the correlation can't be generalized to the population c. the correlation is negative d. a and b, but not c 11-25. When a correlation is discovered between two variables, A and B, it is possible, though not certain that a. A has caused B b. B has caused A c. X has caused A plus B d. all of these 11-26. When the knowledge of one event allows for the prediction of another event, then the two events are a. significantly different b. causally related c. correlated d. due to chance 11-27. When high scores on the first variable associate with high scores on the second, and low scores on the first associate with low scores on the second, then the correlation is a. non-existent b. positive c. negative d. zero 177

For questions 28 through 33 use the following.

A researcher selects a random sample of college students and measures them on both the number of hours spent watching TV and grade-point average.

The results were as follows: Subject # Hours of TV GPA X Y 1 50 1.90 2 20 2.20 3 19 2.40 4 10 3.00 5 7 2.90 6 15 2.50 7 5 3.50 8 40 1.95 9 30 2.00 10 0 3.90 11-28. The correlation between the two variables is generally a. positive b. negative c. zero d. positive, negative and sometimes even zero 11-29. If these data were to be placed on a scatter plot, the general slope of the array of data points would be a. straight across b. upper left to lower right c. lower left to upper right d. U shaped 11-30. Each point on the scatter plot portraying these data would indicate a. performance on GPA only b. hours of TV viewing only c. both performance on GPA and hours of TV viewing d. none of these 11-31. If the correlation between TV viewing and GPA were significant, then the only valid conclusion one can draw from these data, is that a. TV viewing lowers GPA b. TV viewing increases GPA c. TV viewing is independent of GPA d. A knowledge of TV viewing time allows for the prediction of GPA

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11-32. If the researcher uses these data to predict GPA, then the scatter plot's abscissa would represent a. TV viewing time b. GPA c. the correlation d. frequency of occurrence 11-33. If the researcher uses these data to predict GPA, then the scatter plot's ordinate would represent a. TV viewing time b. GPA c. the correlation d. frequency of occurrence 11-34. The more any correlation value deviates from zero a. the more likely it is to be significant b. the stronger the association between the two variables c. the greater its predictive efficiency d. all of these 11-35. In general, the array of points on a scatter plot tends to form the shape of a. a circle b. a rectangle c. a square d. an oval 11-36. The reason for the typical shape assumed by the array of points in a scatter plot, is due to a. the central tendency expected on the two distributions b. the wide variability expected on the two distributions c. the significant difference between the means of the two distributions d. the significant difference between the variabilities of the two distributions 11-37. For the Pearson r, the null hypothesis states that a. there is a significant correlation in the population b. there is no correlation in the sample c. there is no correlation in the population d. the sample no longer represents the population

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11-38. For the Pearson r, the alternative hypothesis states that a. there is a significant correlation in the population b. there is no correlation in the sample c. there is no correlation in the population d. the sample no longer represents the population 11-39. Regardless of the degrees of freedom, any Pearson r of .90 or higher must be a. significant b. non-significant c. more than significant d. none of these, since without the degrees of freedom significance cannot be determined 11-40. In doing a Pearson r on 20 pairs of scores, degrees of freedom would equal a. 20 b. 18 c. 40 d. 38 11-41. In doing a Pearson r with 10 degrees of freedom, there must have been a. 10 separate scores b. 22 separate scores c. 10 pairs of scores d. 12 pairs of scores 11-42. For a given number of degrees of freedom, when the obtained r is equal to the critical tabled value of r, then a. There is a failure to reject Ho b. Ho is rejected c. Ho is re-written as Ha d. Ho cannot be evaluated 11-43. When Ho is rejected for the Pearson r, then a. the correlation is assumed not to be significant b. the correlation is assumed to be significant c. alpha error is eliminated d. none of these, since when Ho is rejected no statement of significance is possible

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11-44. With 5 degrees of freedom, the critical tabled value of the Pearson r is .75 for an .05 alpha level. If one were to obtain a Pearson r value of -.76, then a. There is a failure to reject Ho b. Ho would be rejected c. the alpha level would become negative d. none of these, since a negative Pearson r cannot be evaluated 11-45. Any Pearson r which is significant at an alpha of .01, must also a. be significant at .001 b. be significant at .05 c. be significant at any alpha level d. be positive 11-46. The Pearson r may only be used when the data form is at least a. nominal b. ordinal c. interval d. ratio 11-47. For the Pearson r, as the sample size increases, so also will a. the degrees of freedom b. the alpha level c. the number of measures taken on each subject d. none of these, since sample size is not relevant to the Pearson r 11-48. The proportion of the variability in the Y distribution which is accounted for by the variability in the X distribution, is determined by a. the Spearman r b. the null hypothesis c. the coefficient of contingency d. the coefficient of determination 11-49. A Pearson r of .70 yields a coefficient-of-determination accuracy estimate of a. 70% b. 35% c. 49% d. .05% 11-50. A Pearson r value of -.20 a. can never be significant b. must always be significant c. is significant only at p of .01 d. none of these, since the number of pairs of scores was not given 181

11-51. The Pearson r assumes that a. the samples are randomly selected b. the association between the variables is linear c. the measures do not depart from normality d. all of these 11-52. Homoscedasticity refers to the a. central tendency within one of the two distributions b. variability within the two distributions c. number of degrees of freedom d. all of these 11-53. When the underlying distribution of interval scores is violently skewed, the appropriate correlational technique should be a. the Pearson r b. the t test c. the Spearman r d. none of these, since correlations cannot be computed for skewed distributions 11-54. When both sets of measures are in ordinal form, the appropriate correlational technique should be a. the Pearson r b. the t test c. the Spearman r d. the coefficient of determination 11-55. When one set of measures is in interval form, and the other in ordinal, then the appropriate correlational technique should be a. the Pearson r b. the t test c. the Spearman r d. the coefficient of determination 11-56. To calculate the degrees of freedom for the Spearman r, one must a. subtract 2 from the number scores b. subtract 2 from the number of paired scores c. add the number of scores to the constant 2 d. none of these 11-57. For the Spearman r, the null hypothesis states that a. there is no correlation in the sample b. there is no correlation in the population c. the correlation is significant d. the alpha level is zero 182

11-58. For the Spearman r, in order to establish a significant correlation in the population a. There is a failure to reject Ho b. Ho must be rejected c. Ho must be equal to alpha d. none of these, since the Spearman r cannot be used to infer population characteristics 11-59. For the Pearson r, rejecting Ho with an alpha of .05, means that a. the correlation is not significant b. there is no possibility that Ho should have been accepted c. the probability is .95 that Ho should have been accepted d. the probability is .05 that Ho should have been accepted 11-60. The value of the Pearson r indicates a. the strength of the linear relationship between two sets of interval measures b. the strength of the non-linear relationship between two sets of interval measures c. the strength of the linear relationship between two sets of nominal measures d. all of these 11-61. In order to rank-order the following IQ scores, 120, 110, 110, 105, the appropriate ranks would be a. 1, 2, 3, 4 b. 1, 2, 4, 5 c. 1, 1.5, 1.5, 2, 3 d. 1, 2.5, 2.5, 4 11-62. Whenever interval data are converted into ordinal data, a. the tied interval scores must be averaged, and assigned the same rank value b. the tied interval scores must be eliminated from the analysis c. the tied interval scores must be assigned different rank values d. none of these, since interval data should not be converted into ordinal 11-63. Whenever ordinal data are converted into interval data, a. the tied ordinal scores must be averaged, and assigned the same interval value b. the tied ordinal scores must be eliminated from the analysis c. the tied ordinal scores must be assigned different interval values d. none of these, since ordinal data should not be converted into interval

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11-64. Whenever a correlation coefficient deviates significantly from zero, then a. it may or may not be used for predictions, depending on its size b. it may be used to establish totally accurate predictions c. it may not be used for predictions d. it may be used for establishing better-than-chance predictions, regardless of its size 11-65. A significant correlation of -.40 a. cannot be used for prediction b. predicts with an accuracy of 40% c. predicts more accurately than a significant correlation of +.35 d. predicts with less accuracy than a positive correlation of any size 11-66. The Spearman r may be appropriately used whenever a. both sets of measure are in ordinal form b. one set of measures is in interval form, and the other in ordinal form c. both sets of measures are in interval form, but the distributions are badly skewed d. all of these 11-67. For the Pearson r, a non-significant correlation a. should not be used for predictions b. cannot be generalized to the population c. is simply due to chance d. all of these 11-68. In order to make better-than-chance predictions, the Pearson r must be a. positive b. greater than .70 c. greater than .90 d. significant 11-69. A significant Pearson r a. may be generalized to any population b. may only be generalized to the population from which the sample measures were selected c. may only be generalized to finite populations d. may never be generalized to a population

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C: True or False: For the following, indicate T (True) or F (False) 11-70. Whenever measures are in interval form, regardless of the shape of the distribution, the Pearson r may be used.

11-71. When portraying a correlation on a scatter plot, the array of data points usually forms an elliptical shape.

11-72. If the correlation between X and Y is zero, and the correlation between X and Z is zero, then the correlation between Y and Z should be zero.

11-73. The Spearman r is to ordinal data as the Pearson r is to interval data.

11-74. Significant correlations always predict with an accuracy which is betterthan-chance.

11-75. The higher the value of the Pearson r, the lower the value of the coefficient of determination.

11-76. Whenever a correlation is significant, then a cause-and-effect relationship must be present.

11-77. On a scatter plot, the ordinate should always displays the measures from the Y distribution.

11-78. A Pearson r value of .50 or higher must be significant, regardless of the sample size.

11-79. For the Pearson r, the sample size, not the value of r, determines the number of degrees of freedom.

11-80. For the Spearman r, the degrees of freedom are established in the same way as for the Pearson r.

11-81. A Pearson r of .80 means that 80% of the information about Y being contained in X. 11-82. A significant Pearson r (of any value) means that some information about is being contained in X. 11-83. All positive correlations predict with a higher level of accuracy than do all negative correlations.

11-84. For any correlation to be considered significant, the null hypothesis must first be rejected.

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11-85. The Pearson r has the potential for yielding its highest numerical value when both sampling distributions, X and Y, represent the entire range of normally distributed values.

11-86. When attempting to assess the correlation between two distributions, X and Y, if either of the two variables fail to contain the high or low end of its distribution, the resulting coefficient will tend to over-estimate the true correlation in the population.

11-87. The restricted range phenomenon suggests that the value of the correlation between two variables will diminish as the distributions of scores being compared become more homogeneous.

11-88. For a given sample size, the Pearson r is more likely to show significance than is the Spearman r.

11-89.The Pearson r is a non-parametric test of correlation.

11-90.The Spearman r is a parametric test of correlation.

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D: For the following questions, calculate the values.

Questions 91 - 93 will be based on the following: A group of 10 subjects was given a personality test. Among the traits measured on the test were Orderliness and Social Conformity. The scores were as follows: Subject # Orderliness Social Conformity 1.

9 11 2.

6 8 3.

12 13 4.

7 6 5.

10 10 6.

4 6 7.

9 11 8.

10 9 9.

9 9 10.

2 2 11-91. Find the linear correlation between the two traits.

11-92. Was the correlation significant?

11-93. What conclusions might you draw?

Questions 94-96 are based on the following. A group of 10 college sophomores was randomly selected and given a personality test that measured both orderliness and rebelliousness. The scores were as follows: Subject # Orderliness 1.

9 2.

6 3.

12 4.

7 5.

10 6.

4 7.

9 8.

10 9.

9 10.

2

Rebelliousness 6 8 2 6 10 11 11 9 9 13

11-94. Find the linear correlation between the two traits.

11-95. Was the correlation significant?

11-96. What conclusions might you draw?

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Questions 97-102 will be based on the following: A group of 10 college seniors was randomly selected and given a WAIS-III IQ test. The IQs were recorded and then compared with each student's rank in class. The data are as follows: Student# IQ Rank In Class 1.

120 1 2.

125 2 3.

122 3 4.

110 4 5.

95 5 6.

105 6 7.

100 7 8.

110 8 9.

100 9 10.

94 10 11-97. The IQ scores represent which scale of measurement?

11-98. The class ranks represent which scale of measurement?

11-99. Which correlation technique must therefore be applied?

11-100. What is the correlation between IQ and Rank in Class?

11-101. Is the correlation significant?

11-102. What conclusions might you draw?

11-103. For the following, calculate the Pearson r.

X Y 10 12 8 10 6 8 4 6 2 4

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11-104. For the following, calculate the Pearson r.

X Y 10 2 8 4 6 6 4 8 2 10

11-105. For the following scores, a. Find the Pearson r. b. If Ho is rejected state the level of significance.

X 12 2 5 9 11 10 4 1

Y 10 3 7 5 9 8 6 3

11-106.For the following ranks, a. Calculate the Spearman r b. Should Ho be rejected or accepted? If rejected, state the significance level.

R1 2.5 2.5 1.0 5.5 7.0 8.0 4.0 5.5

R2 1 2 5 7 8 6 4 3

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11-107. For the following ranks, calculate the Spearman rs R1 R2 4.0 5 8.5 4 6.0 3 1.0 1 2.5 2 2.5 6 6.0 8 8.5 7 6.0 9 10.0 10

11-108. Should Ho be rejected or accepted?

11-109. If rejected, state the significance level.

A researcher wishes to test the hypothesis that there is a correlation between IQ and math ability. A random sample of 8 subjects was selected. Each subject was given an IQ test, and then rank-ordered in terms of mathematical ability.

IQ Score Math Rank 130 1 92 5 115 4 115 3 82 6 82 7 60 8 120 2 11-110. Calculate the appropriate correlation.

11-111. Should Ho be rejected or accepted? If rejected, state the level of significance.

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A researcher is interested in testing the hypothesis that income is negatively related to family size. A random sample of 10 families was selected from the population of a large mid-western city. All families were assessed on both their per-family income and their numbers of children.

Family A B C D E F G H I J

Per-family income $190,000 $30,000 $19,000 $40,000 $23,000 $40,000 $25,000 $40,000 $28,000 $30,000

Number of children 2 6 7 3 6 4 5 3 1 5

Since the income distribution was badly skewed, it was decided not to use the Pearson r.

11-112. Calculate the appropriate correlation

11-113. Should Ho be rejected or accepted? If Ho is rejected, state the level of significance.

For questions 114 through 119 use the following: A researcher wishes to find out whether there is a dependable relationship between the severity of earthquakes and temperature for the 7 days immediately preceding the quake. A sample of 10 measured earthquakes was randomly selected from those recorded since 1965. For each area both the Richter value (higher value meaning more severe quakes) and the mean temperature for the 7 days preceding the quakes were recorded. The data follow: Area Richter Scale Value Mean Temperature A 8.0 65 B 7.0 67 C 4.9 70 D 6.5 66 E 5.0 65 F 5.5 60 G 5.0 63 H 4.5 64 I 5.0 65 J 2.0 65 191

Neither distribution appears to be skewed.

11-114. Calculate the correlation.

11-115. How many degrees of freedom should be assigned?

11-116. Should Ho be rejected or accepted?

11-117. If rejected, state the level of significance.

11-118. On the basis of your statistical decision, which error, Type 1 or Type 2 might you have committed?

11-119. On the basis of the above data, can the severity of earthquakes be predicted on the basis of previous temperature?

11-120 Calculate the t ratio, and make a statistical decision at the .05 level for the following combinations of r and N. a. r = .25 and N = 31 b. r = .25 and N = 62 c. r = .60 and N = 27 d. r = .70 and N = 15 11-121. Calculate the t ratio and make a statistical decision at the .05 level for the following combinations of r and N. a. r = .18 and N = 122 b. r = .92 and N = 5 c. r = .45 and N = 12 d. r = .38 and N = 28 11-122. A researcher has computed two significant Pearson r values from independent random samples. Sample A, 60 subjects, produced an r of .98 and Sample B, 50 subjects, produced an r of .71. Do the two sample correlations represent a common population correlation?

11-123. A researcher has computed two significant Pearson r values from independent random samples. Sample A, 163 subjects, produced an r of .40 and Sample B, 158 subjects, produced an r of .20. Do the two sample correlations represent a common population correlation?

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11-124. A researcher has computed two significant Pearson r values from independent random samples. Sample A, 171 subjects, produced an r of .75 and Sample B, 175 subjects, produced an r of .60. Do the two sample correlations represent a common population correlation?

A researcher is interested in whether a correlation exists between Alcohol scores and Drug Abuse scores on the MAQ (Maryland Acquisition Questionnaire). A group of inmates, all of whom had at least two previous incarcerations, were given the MAQ and their Drug and Alcohol scores were compared.

Subj# Drug Alcohol 1 34 40 2 75 60 3 48 53 4 55 47 5 40 50 6 42 48 7 56 52 8 44 54 9 54 44 10 52 56 11 48 51 12 50 41 13 47 38 14 53 55 15 60 40 16 40 60 17 48 58 18 52 60 19 50 48 20 53 47 11-125 Find the Mean for each set of scores

11-126. Find the Pearson r between Drug and Alcohol scores

11-127. Is it significant?

11-128. What kind of research is this experimental or post-facto?

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A study was done to determine if a correlation exists between age of first arrest and the number of years of education an inmate had. A random sample was selected from the male population of a certain house of correction and their first-arrest ages were tallied along with their years of education.

Subj# Age first arrest Education Years 1 15 8 2 17 9 3 32 12 4 25 11 5 42 16 6 18 11 7 20 10 8 19 9 9 20 11 10 16 10 11-129 Find the Mean for each set of scores.

11-130. Find the Pearson r between Drug and Alcohol scores.

11-131. Is it significant?

11-132. What kind of research is this experimental or post-facto?

A researcher is using a standardized test of AP (Anger Potential) and selects 16 subjects to take the test which has a population mean of 20 and an SD of 6.00. The test scores were then converted into both z scores and T scores (See Chapter 5). The scores are as follows: Subj# X (raw scores) z T 1 5 -2.50 25.00 2 14 -1.00 40.00 3 14 -1.00 40.30 4 17 -0.50 45.00 5 17 -0.50 45.00 6 17 -0.50 45.00 7 20 0.00 50.00 8 20 0.00 50.00 9 20 0.00 50.00 10 20 0.00 50.00 11 23 0.50 55.00 12 23 0.50 55.00 13 23 0.50 55.00 14 26 1.00 60.00 15 26 1.00 60.00 16 35 2.50 75.00 194

11-133. Find the Pearson r correlation matrix for all three variables.

11-134. Please explain your result.

The New York Times conducted a survey aimed at determining the rankings of all the U.S. presidents in terms of leadership qualities, administrative abilities and possible failures. Those polled included historians, political scientists and public opinion. The following are the results of the poll. In the first column is the historical order of the presidents and the second column contains the overall ranking. Determine if there is a correlation between the order of the presidency and the president’s ranking. That is, are the earlier presidents (Washington, Adams Jefferson etc.) consistently getting higher rankings than the more recent presidents (Clinton and Bush).

Historical Order Ranking 1.

Washington 2 2.

Adams 13 3.

Jefferson 4 4.

Madison 15 5.

Monroe 21 6.

JQ Adams 16 7.

Jackson 14 8.

Van Buren 40 9.

WH Harrison 39 10.

Tyler 31 11.

Polk 9 12.

Taylor 28 13.

Fillmore 33 14.

Pierce 41 15.

Buchanan 42 16.

Lincoln 1 17.

A Johnson 24 18.

Grant 18 19.

Hayes 27 20.

Garfield 34 21.

Arthur 22 22.

Cleveland 19 23.

B Harrison 30 24.

Cleveland 19 25.

McKinley 17 26.

T Roosevelt 5 27.

Taft 29 28.

Wilson 10 29.

Harding 35 30.

Coolidge 26 31 .

Hoover 36 Data Continue on Next Page 195

32.

FD Roosevelt 3 33.

Truman 7 34.

Eisenhower 6 35.

Kennedy 11 36.

L Johnson 12 37.

Nixon 38 38.

Ford 25 39.

Carter 32 40.

Reagan 8 41.

Bush 20 42.

Clinton 23 43.

Bush 37 11-135. Find the appropriate correlation.

11-136. Is it significant?

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CHAPTER 12 ANALYSIS OF VARIANCE A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Comparing t and F Spend some time showing students the relationship between the F ratio and SOMETHING THEY ARE ALREADY FAMILIAR WITH, the t ratio. In the t ratio we had the difference between the two means (a value based on how much distance there was between the two groups) in the numerator, and the estimated standard error of difference (a measure of how much variability was occurring within the two groups) in the denominator. Point out that with the F ratio the MS between groups (comparable to the distance between the two means in the t test) is in the numerator and the MS within groups (comparable to the estimated standard error of difference) is in the denominator. In both cases, then we are evaluating how far the group means are spread out from each other as a function of how much internal variability is occurring within the sample groups. With the F ratio itself, the resulting value tells us directly what that relationship is. For example, an F ratio of 9.00 tells us specifically that the variation between the groups is 9 times larger than the variability within the groups. The same information can be found in the t ratio when the value of t is squared. Thus a t ratio of 3.00 is telling us that the variability between groups is nine times greater than the variability within the groups. Thus if we had two groups of 10 persons each, the degrees of freedom would be 18, and to reject at .05 we would need a t value of 2.101. Analyzing these same data with the F ratio, we would have 1 and 18 degrees of freedom, and at .05, we would need an F ratio of 4.41, which is the square of 2.101.

Activity: F, t and the Direction of the Difference While discussing this comparison of t and F, it might be wise to point out that unlike the t ratio, there is no one-tail version for the F ratio, meaning that the F ratio cannot be used to predict the direction of a given difference. Nor can the F ratio itself identify which of the various sample groups is deviating significantly from chance, although by following up a significant F ratio with a test such as Tukey's HSD this information can become available.

Activity: ANOVA and Independence For the ANOVAS presented in this chapter, the students should be constantly reminded that these F ratios can only be used when the experimental design is after-only (between groups), and that the sample groups must be independent of 197

each other. You might use this example. A researcher selects four independent random samples from the population of seventh graders in a large metropolitan school district. The hypothesis to be tested is that time of day influences the retention of learned material. Each of the groups is presented with the same material on the history of the stamp tax in colonial America just prior to the Revolutionary War. Group A is given the lesson between 9:00 and 9:30 AM, Group B between 11:00 and 11:30 AM, Group C between 1:00 and 1:30 PM, and Group D between 3:00 and 3:30 PM. All groups are tested immediately following the lesson, and the retention scores (on the next page) were: A 12 10 10 9

B C D 10 9 6 7 6 5 7 5 5 4 2 4

The students may then do a one-way ANOVA on these data to assess the possibility of significant differences among the four groups, and if differences are established, they may be attributed to the independent variable (time at which the lesson is presented).

Solving the above data yields F = 5.402 Reject Ho. Significant at p < .05 and Tukey's HSD = 4.276 Since the four means are A = 10.25, B = 7.00, C = 5.50, and D = 5.00, then the only differences that are significant are those between Group A and C, and between Group A and D. Activity: The ANOVA and Post-Facto Research The ANOVA can also be applied to post-facto research, as long as the subjects are independent of each other. For example, a researcher suspects that percentage of body fat among ten-year old boys is related to metabolic rate. A group of subjects is randomly selected from a population of ten-year old boys, and each subject is assigned to one of three groups on the basis of metabolic rate (the subject variable in this study). All subjects are then measured for body-fat percentages. The data, in the form of percentage of body fat, are as follows: GROUP A GROUP B GROUP C (LOW METABOLISM) (NORMAL METABOLISM) (HIGH METABOLISM) 24 22 20 20 18 16 20 22 16 18 20 15 12 20 14 198

The ANOVA can now tell us whether boys who are known to differ with respect to metabolic rate also differ with respect to body fat. Since the IV in this case is a subject variable and not manipulated, no cause-and-effect statements should be made, regardless of the size of the F ratio. If the F ratio is significant, it could be that some other variable might be producing both a low metabolic rate and an increase in body fat.

For the above data, F = 2.478; Accept Ho, difference is not significant. Tukey's is therefore not appropriate.

The Levene Test If your students are using SPSS, they are certain to ask why the output file provides two F ratios and two sets of degrees of freedom (and even different p values) for the SAME data set. Point out to them that SPSS produces results for data sets that do satisfy the homogeneity of variance requirement as well as those that don’t. Explain to your students that the Fisher’s F that they are currently studying should only be used when the equal variance assumption can be met and should not be used if the equal variance assumption has been violated. As a blatant example, explain that Fisher’s F ratio cannot handle a situation where the scores in Group 1 are leptokurtic, the scores in Group 2 are mesokurtic and the scores in Group 3 are platykurtic. That is, for Fisher’s F the variances among the groups should be similar, not necessarily identical, but at least similar.

And this is not just a matter of opinion. There is a test for the equal variance assumption, called the Levene test, and it is provided by SPSS for every F ratio (just as it was shown previously for the t ratio). Point out that equal variance can only be assumed when the significance level for the Levene is GREATER than .05. Only then should one report the F ratio next to “equal variance assumed”. If the Levene value is .05 or less, i.e. is a significant value, then equal variance CANNOT be assumed, and the F ratio next to “equal variance not assumed” should then be used. Also, have them note that each of those F ratios may accompany a different number of degrees of freedom, usually fewer degrees of freedom when equal variance cannot be assumed.

For those students who want more information on this subject, ask them to read Special Unit 2 in the back of the book. There they will find a complete discussion of the problem of unequal variance with equations for both t, F and their accompanying degrees of freedom.

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B: Multiple Choice Items 12-1. The F ratio was developed by a. William Sealy Gossett b. Sir Ronald Fisher c. Karl Pearson d. Sir Bryant McKenna 12-2. The F ratio may be used to assess differences between (among) a. two sample means b. three sample means c. four sample means d. all of these 12-3. Analysis of experimental data by the F ratio means that the independent variable could have been manipulated a. at only two levels b. at two or more levels c. at only three levels d. none of these, since F cannot handle experimental data 12-4. Doing successive t tests within the context of a single experiment a. is theoretically the same as doing an F ratio b. is theoretically the same as doing a Pearson r c. is theoretically the same as doing a Spearman r d. may inflate the alpha error 12-5. With a two-group experimental design, the independent variable has been manipulated at a. one level b. two levels c. three levels d. four levels 12-6. Comparing all means from a four-group design would require the calculation of a. four t tests b. six t tests c. one t test d. fifteen t tests 12-7. If one had to calculate 15 successive t tests, it would mean a. testing Ho 5 times b. testing Ho once c. testing Ho 15 times d. none of these, since t never tests Ho 200

12-8. With the t test, as the number of decisions to reject Ho increases, so too a. does the number of degrees of freedom b. does the sampling error c. does alpha d. all of these 12-9. With the F ratio, the overall hypothesis among 4 group means may be tested by making a. 4 statistical decisions b. 3 statistical decisions c. 2 statistical decisions d. 1 statistical decision 12-10. For the F ratio, if alpha is set at .05, it remains at .05 if comparisons are made between (among) a. 2 group means b. 10 group means c. 45 group means d. all of these 12-11. The alternative hypothesis for the F ratio is that the sample means a. represent a single population b. represent completely different populations c. do not all represent a single population d. none of these 12-12. When the distribution of raw scores within a sample forms a leptokurtic shape, then a. the variance within the sample is small b. the variance within the sample is large c. the variance within the sample is negative d. the variance within the sample equals the standard deviation 12-13. When the distribution of scores within a sample forms a platykurtic shape, then a. the variance within the sample is small b. the variance within the sample is large c. the variance within the sample is negative d. the variance within the sample equals the standard deviation 12-14. The only time the variance becomes equal to the standard deviation is when a. the variance is small b. the variance is large c. the variance is eliminated d. the variance is equal to 1.00 or zero 201

12-15. The SS, sum of squares, is equal to a. the sum of the squared scores b. the sum of the scores, squared c. the sum of the squared deviations of all the scores from the mean d. the sum of the squared deviations of all the scores from the mean, divided by the degrees of freedom 12-16. When the SS, sum of squares, is large, so too is the a. standard deviation b. variance c. both of these d. neither of these 12-17. The smallest possible value for the SS, sum of squares, is a. -5.00 b. 0 c. .01 d. 1.00 12-18. The concept of "total variability" is based on a. how far the scores in all groups deviate from the total mean b. how far the sample means deviate from each other c. how far the scores in a single group deviate for their own group mean d. how far each sample mean deviates from the total mean 12-19. The concept of "between variability" is based on a. how far the scores in all groups deviate from the total mean b. how far the scores in a single group deviate from their own group mean c. how far each sample mean deviates from the total mean d. all of these, depending on the between degrees of freedom 12-20. The concept of "within variability" is based on a. how far the scores in all groups deviate form the total mean b. how far the scores deviate from their own sample means c. how far each sample mean deviates from the total mean d. none of these 12-21. When the variance within a sample group is equal to 16, then the standard deviation for that sample is equal to a. 16 b. 8 c. 1.00 d. none of these

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12-22. The sums of squares components relate such that SS total is equal to a. SS between b. SS between + SS within c. SS within - SS between d. none of these 12-23. The sums of squares components relates such that SS between is equal to a. SS within b. SS total - SS within c. SS total + SS within d. none of these 12-24. The sum of squares components relate such that SS within is equal to a. SS total b. SS total + SS between c. SS total - SS between d. SS between - SS total 12-25. When the between sum of squares is relatively large, then the shape of the sampling distribution is necessarily a. leptokurtic b. platykurtic c. mesokurtic d. none of these 12-26. When the within sum of squares is relatively large, then the shape of the sampling distribution is necessarily a. leptokurtic b. platykurtic c. mesokurtic d. none of these 12-27. The greater the value of the F ratio, a. the more the sample distributions overlap b. the less the sample distributions overlap c. the larger the sum of squares within groups d. none of these 12-28. The greater the sum of squares within groups a. the more the sample means deviate from the total mean b. the more the sample means deviate from each other c. the more the individual scores deviate from the total mean d. none of these

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12-29. The less overlap there is among the various sample-group distributions a. the larger the F ratio b. the smaller the F ratio c. the greater the degrees of freedom d. none of these 12-30. An F ratio of 5, tells us specifically that the variance between groups is a. 5 times smaller than the variance within b. 5 times larger than the variance within c. 5 times smaller than the total variance d. 5 times larger then the total variance 12-31. If the F ratio were performed on a two-group design and F were equal to 9, then t would be equal to a. 81 b. 9 c. 3 d. none of these 12-32. If the F ratio were performed on a four-group design and F were equal to 4, then t would be equal to a. 16 b. 4 c. 2 d. none of these, since t is only appropriate on a two-group design 12-33. The mean square (MS) is always equal to (when based on SS/df) a. the square of the total mean b. the sum of the squares of the separate group means c. the variance squared d. the variance 12-34. The larger the F ratio a. the more likely the samples represent a common population b. the less likely the samples represent a common population c. the more likely that the between sum of squares is small d. the more likely that the within sum of squares is large 12-35. The greater the value of the MS between groups, a. the more likely Ho will be rejected b. the less likely Ho will be rejected c. the more likely the various group means will equal each other d. none of these

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12-36. If a significant F ratio is obtained, then a. Ho must be accepted b. Ho must be rejected c. Ha must be rejected d. there must be at least four sample means 12-37. When four leptokurtic sampling distributions show no overlap whatever, then a. the MS within is likely to be large b. the MS between is likely to be small c. the F ratio is likely to be significant d. there must be zero degrees of freedom 12-38. When doing an F ratio on a four-group design, 4 subjects in each group, the between degrees of freedom will equal a. the MS between b. 4 c. 3 d. 12 12-39. When doing an F ratio on a three-group design, 5 subjects in each group, the within degrees of freedom will equal a. the MS within b. 2 c. 5 d. none of these 12-40. When doing an F ratio, subtracting the number of groups from the total number of subjects yields the a. alpha error b. between degrees of freedom c. within degrees of freedom d. total degrees of freedom 12-41. When doing an F ratio, subtracting the constant, 1, from the total number of sample groups yields the a. alpha error b. between degrees of freedom c. within degrees of freedom d. total degrees of freedom

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12-42. Dividing the sum of squares (SS) by the degrees of freedom (df), yields the a. F ratio b. alpha error c. the mean square d. ANOVA 12-43. Dividing the MS between by the MS within, yields the a. F ratio b. alpha level c. variance d. ANOVA 12-44. When the calculated F ratio is equal to or greater than the critical, tabled value of F for a given number of degrees of freedom, then a. alpha is inflated b. There is a failure to reject Ho c. Ho is rejected d. none of these 12-45. The F ratio can never be significant unless a. the MS between is significantly smaller than the MS within b. the MS between is equal to the MS within c. the MS between is significantly larger than the MS within d. none of these 12-46. When analyzing experimental research via the F ratio, the action of the independent variable is most clearly shown in the value of the a. MS total b. MS within c. MS between d. degrees of freedom 12-47. When analyzing experimental research via the F ratio, random error and individual differences among the subjects is most clearly shown in the value of the a. MS total b. MS within c. MS between d. degrees of freedom

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12-48. When analyzing experimental research via the F ratio, the number of sample groups, and the number of subjects in each sample group, are most clearly shown in the value of the a. MS total b. MS within c. MS between d. degrees of freedom 12-49. If the MS between equals 10, and the MS within equals 5, then the F ratio equals a. 15.00 b. .50 c. 2.00 d. cannot tell from the above information 12-50. If the MS between equals 10, and the MS within equals 10, the total degrees of freedom are equal to a. 1 b. 20 c. 0 d. cannot tell from the above information 12-51. The one-way ANOVA may only be used when a. the direction of the difference is specified b. the F ratio is significant c. there are more than 3 sample groups d. there is only one independent variable 12-52. Tukey's HSD is used in order to a. establish the significance of the overall F ratio b. allocate the degrees of freedom c. establish the alpha error d. none of these 12-53. Once an F ratio has been found not to be significant, we must immediately a. apply Tukey's HSD b. increase the alpha level to a point where the F is significant c. re-allocate the degrees of freedom d. none of these 12-54. Once an F ratio is found to be significant, then a. each sample mean must represent a different population b. each sample mean must represent a single population c. each sample mean deviates by the same amount d. none of these 207

12-55. When analyzing a significant ANOVA, in order to establish precisely where the sample differences may have occurred, we may a. apply the HSD b. reduce the alpha error c. calculate the coefficient of determination d. none of these, since a significant F indicates precisely where the sample differences can be found 12-56. If the tabled value for F, for a given number of degrees of freedom, were equal to 5.14, and the calculated value of F also was equal to 5.14, then a. There is a failure to reject Ho b. Ho is rejected c. Ho is equal to Ha d. cannot tell from the above information 12-57. When doing an ANOVA with one degree of freedom between groups, then a. only one sample group has been selected b. exactly two sample groups have been selected c. more than two sample groups have been selected d. cannot tell without knowing the within degrees of freedom 12-58. An F ratio which equals exactly 1.00, indicates that a. the total degrees of freedom are equal to 1 b. the between degrees of freedom are equal to 1 c. the within degrees of freedom are equal to 1 d. the MS between has exactly the same value as the MS within 12-59. In order to calculate an F ratio, there must be a. more than one sample group b. interval data c. homogeneous variances among the sample groups d. all of these 12-60. The factorial ANOVA is only applied when a. there is more than one independent variable b. there is more than one dependent variable c. there is exactly one independent variable d. there is 1 degree of freedom between groups 12-61. When doing a factorial ANOVA, the main effects are found from the a. rows b. columns c. both of these d. neither of these 208

12-62. A factorial ANOVA must have a minimum of a. two cells b. four cells c. eight cells d. none of these, since the factorial ANOVA may be performed on any number of cells 12-63. On a factorial ANOVA, the number of different variations in treatment conditions, determines the number of a. cells b. dependent variables c. subjects within each group d. all of these 12-64. A factorial ANOVA may show the significance of a. treatment A b. treatment B c. the interaction between A and B d. all of these 12-65. On a factorial ANOVA, a significant interaction indicates a. the cumulative effect of two or more independent variables b. the partial effect of several independent variables taken separately c. the interactive effect of several dependent variables d. none of these 12-66. When several independent variables are manipulated simultaneously, an appropriate analysis would be the a. one-way ANOVA b. factorial ANOVA c. partial ANOVA d. none of these 12-67. In a factorial ANOVA, the row variability, column variability and rxc variability, are all components of the a. MS within b. MS total c. MS between d. interaction effect

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12-68. In order to calculate an F ratio, there must be a. interval data b. a normal distribution in the population from which the samples were selected c. similarity among the within variances for each sample d. all of these 12-69. When the data form is nominal, the appropriate F ratio is a. the one-way b. the two-way c. the factorial d. none of these, since F should not be calculated from nominal data

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C: True or False: For the following, indicate T (True) or F (False) 12-70. The F ratio may compare two or more sample means.

12-71. The F ratio may only be calculated when the independent variable has exactly two levels.

12-72. The F ratio, like the t ratio, can be used to predict the direction of the mean difference.

12-73. The F ratio cannot be used on post-facto data.

12-74. Total variability results from the accumulated differences between each individual score and the total mean.

12-75. The variance, or mean square, results from dividing the SS by the degrees of freedom.

12-76. Between variability results from the accumulated differences between each sample mean and the total mean.

12-77. Within variability results from the accumulated differences between each individual score and the total mean.

12-78. The between SS plus the within SS equals the total SS. 12-79. A significant F ratio can only occur when the between variability is less than the within variability.

12-80. An ANOVA cannot be done unless the sample sizes are unequal.

12-81. When an F ratio is significant, the null hypothesis is accepted.

12-82. An F ratio of 10 means that the MS within is 10 times larger than the MS between.

12-83. With a sample mean of 10, the MS must equal 100.

12-84. The factorial ANOVA should not be used unless there is more than one independent variable.

12-85. A within MS of zero, indicates that all the sample scores are identical.

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D: For the following questions, calculate the values.

For questions 86 through 94 use the data on the following page: A researcher is interested in establishing whether memory is a function of both IQ and time of testing. A random sample of 40 university students was selected and each student was given an IQ test. Half of the students scored above 115 and were assigned to the high IQ category. The students were than all given a series of paragraphs to read, followed by a memory test on the content of what had been read. Half the students took the memory test immediately after reading the material, while the other half waited an hour before testing. The results were as follows (high scores indicating better memory retention):

Lo IQs

Hi IQs

TIME OF TESTING DELAY IMMEDIATE a 3.0 b 6.5 4.1 5.3 3.6 7.1 3.8 6.7 5.0 5.8 4.3 6.1 3.3 7.0 2.9 5.5 4.5 6.2 4.2 6.6 c 5.2 5.4 5.1 7.0 6.8 5.7 6.1 4.9 5.2 4.6

d

7.3 7.1 8.2 7.9 8.5 7.9 7.8 8.5 8.8 7.0

12-86. Find the MS within.

12-87. Find the MS for rows.

12-88. Find the MS for columns.

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12-89. Find the MS for interaction.

12-90. Find the F ratio for rows, and if significant, state the alpha level.

12-91. Find the F ratio for columns, and if significant, state the alpha level.

12-92. Find the F ratio for the interaction, and if significant, state the alpha level.

12-93. Was the row independent variable manipulated or assigned?

12-94. Was the column independent variable manipulated or assigned?

12-95. Calculate the F ratio on the following sets of scores.

Group 1 Group 2 Group 3 4 6 7 3 3 5 1 2 1

12-96. Calculate the F ratio on the following sets of scores.

Group 1 Group 2 Group 3 5 7 10 3 6 10 3 6 9 2 4 10

12-97. Calculate the F ratio on the following sets of scores.

Group 1 Group 2 Group 3 5 8 10 4 6 10 1 6 12 1 5 8 213

12-98. Three groups of subjects were randomly selected, four subjects in each group. The SS between was 132.17, and the SS within was 19.50. a. Calculate the F ratio b. Should Ho be rejected ? If rejected, state the probability of a Type 1 error.

12-99. Three groups of subjects were randomly selected, 10 subjects in each group. The SS between was 668.47, and the SS within was 923.40. a. Calculate the F ratio b. Should Ho be rejected? If rejected, state the probability of alpha error.

12-100. Four groups of subjects were randomly selected, and subjected to different experimental treatments. Their scores were: Group 1 Group 2 Group 3 Group 4 5 7 8 15 4 5 7 14 3 6 6 18 6 8 8 12 a. Calculate the F ratio b. Should Ho be rejected ? If rejected, state the probability of alpha error.

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Questions 101 through 108 are based on the following: (This is similar to an item in the text at the back of chapter 12 but WITH A DIFFERENT DATA SET).

A researcher is interested in whether a certain amphetamine drug will affect the retention of learned behavior among rats as a function of how much time is allowed to elapse between the training and the retention test. Four groups, ten subjects in each group, were randomly selected from a colony of Sprague-Dawley rats. All the rats were given the same amount of training on a simple maze.

Groups A and C were given the amphetamine, whereas Groups B and D were given a placebo. Groups A and B were tested for retention immediately following the training, whereas Groups C and D were tested 4 hours later. The retention measure consisted of a scaled score based on how many correct responses the rats would exhibit while running the maze, the higher the score the higher the retention.

FACTOR A Amphetamine Placebo Group A Group B 15 13 9 9 9 12 Immediate 11 7 Test 10 7 6 8 10 10 12 9 14 5 7 11 FACTOR B Group C Group D 1 8 5 6 2 9 Delayed 1 4 Test 4 4 5 5 6 6 5 6 2 6 6 7 12-101. Calculate the F ratio for rows

12-102. Calculate the F ratio for columns 215

12-103. Calculate the F ratio for the interaction

12-104. Is the row F ratio significant?

12-105. Is the column F ratio significant?

12-106. Is the interaction significant?

12-107. Calculate the means for each of the four groups (If you graph them you will discover that the two graph lines are not parallel).

12-108. What conclusions might you draw?

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CHAPTER 13 NOMINAL (CATEGORICAL) DATA AND THE CHI SQUARE Non-Parametrics Writ Large A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Identifying Nominal Data Once they get the hang of nominal data, students usually find the chi square fairly easy to calculate. After completing the section on t tests and factorial ANOVAS, chi square actually comes as almost a pleasant diversion.

The main problem seems to be in learning to clearly identify the nominal case.

Stress the fact that nominal data must come in the form of whole numbers, or that the frequency observed actually comes from the counting of the number of persons, things or events that fall within a given category. These are literally whole numbers, not integers because nominal data CANNOT be negative. .We can't count out the fact that 3.17 persons agreed or disagreed with a certain item (tell them you'd like to see that .17 person). The frequencies observed are nose-counting data, as in a political poll. It may be that American families have an average of 2.3 children, but nobody ever observed either 2.3 kids or minus 2.3 kids.

Once this hurdle is overcome, the nominal case problems begin to jump right off the page for them. One interesting twist occurs when some members of the class, after conquering factorial ANOVAS, try that procedure on nominal data, only to discover that the sum of squares is very hard to calculate. They see the contingency table, and assume that this must be another ANOVA, even when it looks like the following: a. = ONE WHOLE NUMBER

b. = ONE WHOLE NUMBER

c. = ONE WHOLE NUMBER

d. = ONE WHOLE NUMBER

Point out again that had this been an ANOVA, there would have to be several scores in each cell, not just a single whole number. For the ANOVA it would look like this:. a. b.

X1 X1 X2 X2 X3 X3 X4 X4 c.

X1 X2 X3 X4

d.

X1 X2 X3 X4 217

Activity: Nominal and Interval Data: A Comparison This is also a good opportunity to stress again some of the differences between interval and nominal data, especially the fact that with nominal data we can't identify any single individual, other than to say that the person was in a group of people who all shared a given trait. With interval data, we have a separate score for each individual being tested, and that interval score can be easily found and identified.

Activity: Chi Square and the Hypothesis of Difference The message to get across is that chi square is a test of differences, that it is typically used to test the hypothesis that two or more groups differ with respect to some trait or characteristic. We simply count the number of observations in each group that fall in these independent categories, and then compare the proportion of cases in each group with the proportion of cases in the other groups. After the chi square has been calculated and if the difference is found to be significant, then it can also be used to estimate the strength of a possible correlation. Explain that the examples can be phrased either way, either as final tests of difference or final tests of correlation.

For example, a researcher may wish to discover if there is a DIFFERENCE in leadership qualities between tall and short men. The data could be as follows, with the tall and short groups being categorized on whether the men were either under or over 5'9. Judges could then be used to determine if in a leaderless group situation, the men were perceived as "leaders" or "followers".

Tall

Leader 40

Follower Totals 20 60

Short

15

35

50

Totals

55

55

110

Chi Square = 14.666: The difference is significant at P < .01.

Thus our data show that there is a difference regarding the perception of leadership as a function of height. Now, you could set up the same problem as a correlational question. Is there a correlation between height and leadership?

Activity: Chi Square and Correlation Since the Chi Square in the previous problem was significant, we could then run the data through C, the coefficient of contingency, and in this case we get a C of .344. Now this may not seem like a strong relationship, but remind your students that C has an upper limit that depends on the number of cells in the 218

contingency table. When the contingency table has an equal number of rows and columns, such as a 2 by 2, or a 3 by 3, then the upper limit for C is equal to the square root of (k-1)/k.

For a 2 by 2 chi square, then, the upper limit is the square root of (21)/2, which is .707.

For a 3 by 3 chi square, the upper limit is equal to the square root of (31)/3, which is .816.

What this means is that we can't directly compare C values from different sized contingency tables, nor can they ever be directly compared to the Pearson r (which always has an upper limit of +1.00). If your students demand a rough comparison estimate, a C value of .344 established from a 2 by 2 chi square is approximately as strong as a Pearson r of .64 (.34+.30).

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B: Multiple Choice Items 13-1. When measuring individuals with the nominal scale, the key question is, a. how much of the given trait does the individual possess? b. does individual A possess more of the given trait than individual B? c. does the individual possess or not possess the trait? d. all of these are key questions when using the nominal scale 13-2. All nominal categories a. must share the same trait b. must overlap c. must be totally independent of each other d. must be rank-ordered 13-3. When using the nominal scale to assess political-party affiliation, a. the strength of each individual's party loyalty must be established b. the subjects within each category must be rank-ordered with respect to affiliative strength c. each and every subject must receive a different scale number d. the individuals must be independently grouped as to party preference 13-4. The major mathematical rule underlying nominal scaling is a. the greater-than, or less-than rule b. the equality versus non-equality rule c. the equal-appearing-intervals rule d. none of these, since nominal scaling is not mathematical 13-5. If during a presidential campaign, a certain random sample of voters indicated 30 persons for Clinton, 25 for Bush, and 10 for Perot, the resulting scale would be a. nominal b. ordinal c. interval d. ratio 13-6. If a certain nominal scale, applied to a single group of subjects, showed observed frequencies of 30, 25 and 20, the frequencies expected on the basis of chance would be a. 30, 25 and 20 b. 30, 30 and 25 c. 25, 30 and 25 d. 25, 25 and 25

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13-7. When a single group of subjects is allocated among any number of categories, the chi square analysis is a. r by k b. r by 1 c. 2 by 2 d. 1 by k 13-8. When a single group of subjects is allocated among any number of categories, the chance fe's for the chi square are a. equal to the size of the sample b. equal to the fo's, divided by the sample size c. equal to the sample size, divided by the fo's d. equal to the fo's

13-9. The null hypothesis for chi square states that the a. means for each group are equal b. means for each group are equal to the fo's c. fo's are equal to the fe's d. population mean is equal to the sample mean 13-10. For any chi square, the degrees of freedom are equal to a. the size of the sample b. sample size, minus 1 c. sample size, minus 2 d. none of these 13-11. When the chi square fo's deviate significantly from the fe's, then a. the independent rule has been violated b. the data are not nominal c. the chi square cannot be calculated d. null is rejected 13-12. With chi square, the more categories there are a. the fewer the degrees of freedom b. the greater the degrees of freedom c. the higher the fo values d. the higher the fe values 13-13. With chi square, the more categories there are, then a. the fe's must each equal the fo's b. the fe's must all equal each other c. no single fe may ever equal a given fo d. none of these 221

13-14. When the fe's are determined, not on the basis of strict chance, but on the basis of some previous theory, then the tested hypothesis is a. accepted b. rejected c. no longer appropriate to a chi square analysis d. a priori 13-15. When the obtained value of chi square is equal to the critical, table value for a given number of degrees of freedom, then a. Ho is rejected b. There is a failure to reject Ho c. the alternate hypothesis is accepted d. none of these, since no conclusions are possible when the obtained value equals the tabled value 13-16. On a 1 by k chi square with a sample size of 100, the degrees of freedom must always equal a. 100 b. 99 c. k-1 d. none of these 13-17. The degrees of freedom for a 2 by 2 chi square must always equal a. 4 b. 2 c. 1 d. none of these 13-18. On a 1 by k chi square, the tabled value needed for significance, at a given alpha level, cannot be found unless a. the number of subjects is known b. the number of categories is known c. the number of rows is known d. all of these 13-19. When the chi square value is negative, it means a. the relationship is inverse b. the second group is higher than the first group c. the fo's are greater than the fe's d. none of these, since obtaining a negative chi square is impossible

222

13-20. The degrees of freedom on a 3 by 3 chi square are equal to a. 9 b. 4 c. 3 d. cannot tell from the above information 13-21. A researcher obtains a 2 by 2 chi square value of 3.83. The appropriate degrees of freedom indicate required table values of 3.84 at .05, and 6.64 at .01. The researcher should a. Fail to reject Ho b. reject Ho at .05 c. reject Ho at .01 d. only test the a priori hypothesis 13-22. Whenever a single group of subjects is being allocated among four independent categories, the chi square analysis would be a. 1 by 3 b. 1 by 4 c. 2 by 2 d. cannot tell from the above information 13-23. In the chi square table of critical values, as degrees of freedom increase a. so too do the critical values b. the critical values decrease c. the critical values remain constant d. none of these, since for chi square the critical table values are independent of the degrees of freedom 13-24. When several groups of subjects are being compared with regard to some observed frequencies, the appropriate chi square analysis would be a. 1 by k b. r by k c. N by df d. none of these

13-25. When fe's are obtained by multiplying row totals by column totals, and then dividing by N, the chi square is a (n) a. 1 by k b. r by k c. 1 by r d. all of these, depending on the degrees of freedom

223

13-26. When comparing two or more groups with regard to some observed frequencies, the independent variable should be set in the a. rows b. columns c. marginal totals d. lower, right-hand corner 13-27. When comparing two or more groups with regard to some observed frequencies, the dependent variable should be set in the a. rows b. columns c. marginal totals d. lower, right-hand corner 13-28. When doing a 2 by 2 chi square, when any expected frequency is less than 10, the Yates correction suggests that the researcher should a. reduce the fo-fe value by .5 b. increase the fe by .5 c. reduce the fe by .5 d. not calculate a chi square because of the small fe 13-29. The Yates correction may only be applied a. when any fo is less than 5 b. when any fo is less than fe c. when chi square equals 0 d. none of these 13-30. Regardless of the fo or fe values, the Yates correction need only be applied when a. the chi square is significant b. the difference between a given fo and fe is less than 10 c. the df equals 1 d. none of these 13-31. The effect of the Yates correction on the chi square value is that chi square is a. slightly decreased b. slightly increased c. greatly increased d. none of these, since Yates does not influence the chi square value

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13-32. When Yates is applied, the likelihood of obtaining a significant chi square is a. slightly decreased b. slightly increased c. greatly increased d. eliminated Questions 33 through 42 are based on the following: A researcher randomly selects two sample groups and provides one group with daily doses of vitamin C and the other group with a placebo. Later, the two groups are compared as to whether or not the subjects caught the flu.

13-33. The measurement scale employed in the study was a. nominal b. ordinal c. interval d. ratio 13-34. The independent variable in the study was a. whether the subjects caught the flu b. whether the subjects received the vitamin C c. the control group d. the placebo 13-35. The dependent variable in the study was a. whether the subjects caught the flu b. whether the subjects received the vitamin C c. the control group d. the placebo 13-36. This is an example of a. post-facto research b. experimental research c. correlational research d. none of these 13-37. The independent variable in this study was a. assigned b. not allowed to vary c. manipulated d. significant 13-38. The appropriate analysis of these data would involve a. a 1 by k chi square b. a 1 by r chi square c. a 2 by 2 chi square d. an F ratio, followed by a chi square 225

13-39. In setting up the contingency table, whether the groups received the vitamin C should be placed in the a. columns b. rows c. marginal totals d. lower, right-hand corner 13-40. In setting up the contingency table, whether the groups caught the flu should be placed in the a. columns b. rows c. marginal totals d. lower, right-hand corner 13-41. In evaluating these data, the chi square's degrees of freedom would equal a. 1 b. 4 c. 2 d. cannot tell from the information provided 13-42. In analyzing the data from this research, the Yates correction should only be applied if a. one or more of the fo's was less than 10 b. one or more of the fo's was greater than 10 c. one or more of the fe's was less than 10 d. none of these 13-43. In order to use the chi square a. all data must be in nominal form b. the categories must be independent of each other c. the sum total of all fo's must equal N d. all of these 13-44. The chi square statistical test may handle data from a. experimental research b. post-facto research c. both of these d. neither of these 13-45. Chi square, like t and F, always tests the hypothesis of a. association b. correlation c. difference d. a priori events 226

13-46. With chi square, the larger the sample size a. the smaller the alpha error b. the larger the alpha error c. the larger the critical, tabled value needed for significance d. none of these 13-47. The use of (C), the correlational technique cited in the text for nominal data, first requires a. a manipulated independent variable b. an a priori hypothesis c. that the constant, .50, be subtracted from the chi square d. that the chi square be significant 13-48. The coefficient of contingency (C) may be applied when a. using a 2 by 2 contingency table b. using a 3 by 4 contingency table c. both of these d. neither of these 13-49. The McNemar test may be used on nominal data taken from a. a repeated-measures design b. a matched-subjects design c. both of these d. neither of these 13-50. In order to use the McNemar test, the samples should be a. independent of each other b. dependent on each other c. free of sampling error d. normally distributed 13-51. In order to calculate a chi square, it is assumed that a. the underlying distributions are normal b. the population mean is a known value c. the mean of the sample does not deviate significantly from the population mean d none of these 13-52. The chi square is a a. parametric test of mean differences b. a non-parametric test of frequency differences c. a parametric test of frequency differences d. an interval test of rank differences

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13-53. Whenever the chi square is applied to interval data, the resulting value will be a. inflated b. lowered c. the same as when applied to nominal data d. none of these, since chi square may not be applied to interval data 13-54. The McNemar test may only be used when the contingency table is a. 1 by k b. 2 by 2 c. 2 by 3 d. none of these, since the McNemar test may be applied to contingency tables of any size 13-55. The chi square test may not be used when a. the underlying distributions are skewed b. the variability within the sample differs c. the intervals between successive scale points are not equal d. none of these 13-56. When the 2 by 2 chi square contingency table contains expected frequencies of less than 5, most statisticians agree that a. the Yates correction must be applied b. coefficient of contingency must be applied c. the degrees of freedom then must increase by 5 d. the chi square should not be calculated 13-57. When using chi square, the larger the alpha level a. the more difficult it becomes to reject Ho b. the easier it becomes to reject Ho c. the greater the degrees of freedom d. none of these

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C: True or False: For the following, indicate T (True) or F (False) 13-58. Chi square is a non-parametric statistical test.

13-59. Chi square may never be used with nominal data.

13-60. The null hypothesis for chi square states that the observed frequencies are equal to the expected frequencies.

13-61. The degrees of freedom for chi square are based on the sample size.

13-62. Chi square may not be used when comparing two or more samples.

13-63. When any expected frequency is less than 10, some statisticians argue that Yates should be applied when df=1).

13-64. Every 2 by 2 chi square, regardless of sample size, contains only 1 degree of freedom.

13-65. Unlike t of F, as the degrees of freedom increase for chi square, the critical, table values also increase.

13-66. When applying chi square to a 2 by 2 contingency table, each fo must be at least equal to 10.

13-67. When applying chi square to a 3 by 5 contingency table, Yates must be used whenever any fe is less than 10.

13-68. Because of the independence restriction, the McNemar test cannot be used on dependent samples.

13-69. As long as the data are in nominal form, the McNemar test may be applied to results from matched-subjects' designs.

13-70. When chi square equals zero, there is a failure to reject Ho 13-71. In order to use chi square, the underlying distribution must be normal.

13-72. Applying the Yates correction, makes it somewhat easier to accept Ho. 13-73. Other things being equal, the larger the chi square value, the greater the likelihood of rejecting Ho. 13-74. The Yates correction is designed to reduce the value of chi square.

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13-75. In a 1 by k chi square, the expected frequency is always equal to k-1.

13-76. The chi square test demands that cell entries be independent of each other.

13-77. Chi square may only be used to compare frequencies, never sample means.

13-78. When analyzing research data via the chi square, the independent variable should be placed in the rows.

13-79. Chi square may only be used for the analysis of post-facto research data.

13-80. Correlations may never be established when the data form is nominal.

13-81. The coefficient of contingency (C) demands that there be only 4 independent cells.

13-82. Chi square is a far more powerful test than is either t or F. 13-83. If one group of 35 subjects is being sorted into 8 independent categories, the chi square must have 7 degrees of freedom.

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D. For the following questions, calculate the values.

13-84. Calculate the chi square on the following nominal data.

CATEGORIES A B C D E fo 30 20 20 15 15

13-85. Calculate a chi square on the following nominal data.

CATEGORIES A B C fo 35 25 10

For questions 86 through 91, use the following: Hypothesis: Police citations for traffic violations are a function of the age of the driver. A random sample of 50 drivers was selected from the population of cited traffic violators during a one-month period. The drivers were categorized according to age, and the frequencies for each category were established.

AGE OF DRIVER 18-27 28-37 38-47 fo 35 25 15

48-57 13

58 and over 12

13-86. Calculate the chi square.

13-87. Indicate the number of degrees of freedom.

13-88. Indicate the tabled value for chi square at the .05 level.

13-89. Indicate the table value for chi square at the .01 level

13-90. Should Ho be rejected?

13-91. State the probability value of alpha.

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13-92. Using the nominal data in the following 2 by 2 contingency table:

a40

b20

c10

d20

a. Calculate the chi square. b. Should Ho be rejected ? If rejected, state probability value of alpha.

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CHAPTER 14 REGRESSION ANALYSIS A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Regression and Secular Trend If the students get a little bored predicting IQ scores or introversionextraversion scores, or they can't seem to see any practical application in calculating regression lines, perhaps an example from their own immediate collective life space might perk up the interest. Explain to them they might find regression analysis useful when it comes time to interview for a job, especially a job in the corporate world. Corporate interviewers have been known to become thoroughly impressed by the applicant who is motivated enough to take the time to run a secular trend line on the corporation's financial history. Explain to students that these data are readily available in the company's annual report (which they can either get from the company or from a local stock broker). To do this, they can use regression techniques for predicting trends over a period of time. The term "secular trend" simply refers to data that are spread over a time frame and are not overly cyclical in nature. Thus, any time we use past and current data to predict future results, we may use a time series called secular trend analysis. Let's take some data from the XYZ corporation, and ask what the corporation should earn in the year 2015.

Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Earnings per share of common stock .90 1.10 1.00 1.20 1.40 1.50 1.60 1.90 2.30 2.40

When using a hand calculator, it might be wise to have them code the years, using equal time intervals, by simply converting the years into sequential digits beginning with 1, so that 2003 = 1, 2004 = 2, and so on. Then have them set up the coded years as the X distribution, and the earnings-per-share as the Y distribution. (If they prefer to use the computer program, the years can be plugged in directly.) When coded, the data appear as follows"

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Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Coded Year X 1 2 3 4 5 6 7 8 9 10

Earnings per share Y .90 1.10 1.00 1.20 1.40 1.50 1.60 1.90 2.30 2.40

The means of the two distributions are 5.50 for X and 1.53 for Y. The standard deviations are 2.872 for X and .498 for Y. Summation XY is then equal to 385 and the Pearson r is .969, which is significant at p<.01.

The regression line is then equal to .168X+.606, giving us a predicted profit of 2.79 in 2015 (year 13).

It might be a good idea to run through the values in the regression line at this point, showing that the intercept of .606 is what the company is assumed to have earned in the zero year, which in this case is 2002 (since we coded 2003 as year 1). This helps to reinforce the concept of the intercept, which of course is the value of Y when X equals zero.

Also, the slope value of +.168 indicates the amount of earnings increase we anticipate for each succeeding year. You might even set up a table with some predictions for successive years and hopefully the students might notice that the yearly increase is indeed the slope value of .168 (the amount of change in Y we expect for a given unit change in X).

Activity: Secular Trend and the Computer If the students try doing some of these trend lines on the computer and put the years in directly (rather than in coded form), they may come to you in shocked disbelief when they see some amazing intercepts, such as for the above problem a value of - 335.487. Usually the students that don't get it will rant and rave and ask how any company could lose $335 per share of common stock (or perhaps try to blame it on big Wall Street bankers). At that point they will get a better understanding of the intercept when you gently chide them that if they are using the noncoded years, then the zero year is no longer 2002 but goes back at least to the birth of Christ.

Activity: Trend Analysis and the Rearview Mirror Explain to the students that trend analysis is an extremely useful tool, but they must exercise a certain degree of caution whenever future trends are being extrapolated. Trend analysis always assumes that the present trend will continue, that other things will remain relatively equal. Although a trend 234

that has existed up to the present time is certainly a fact, the future trend must remain a probability estimate and the farther into the future the prediction, the more chance there is for other things not to remain the same.

Another problem results from the fact that each year's value is not really independent of the adjoining year, but may instead be correlated to some extent. This is called an autocorrelation. If the autocorrelation is extremely high, the accuracy of the forecast is somewhat reduced. Still another issue occurs when the two series are increasing each year, as they were for most of the years in the problem above. This tends to produce a Pearson r which may overestimate the degree of correlation, since both of the values are smaller in the early years and larger in the later years. Secular trend analysis, you might point out, is rearview-mirror research -one is always looking behind to predict what is ahead. A brief look back can sometimes be dangerous as it may lead one to make blind extrapolations. Also, if one stops looking back and thereby misses the new images appearing in the mirror, major accidents may result. New data must constantly be inserted as soon as it becomes available. In one method, called the "moving average", the addition of the latest data entry is accompanied by the deletion of the oldest entry, which of course "moves" or changes the average. In this way, the secular trend line is always being updated. Despite all these warning, the regression line can be an extremely useful forecasting tool. Extrapolations can of course be dangerous, but in the area of forecasting they are, alas, necessary.

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B: Multiple Choice Items 14-1. The major goal of correlational research is to a. establish cause-and-effect relationships b. analyze the effects of an actively manipulated independent variable c. evaluate the results of experimental research d. make better-than-chance predictions 14-2. The higher the correlation between X and Y, then a. the higher the probability of having isolated the causal factor b. the higher the probability of having sampled from a single population c. the higher the probability of inflating the alpha error d. the more accurate the resulting predictions 14-3. The regression of Y on X allows for a. predictions of Y, given information on X b. predictions of X, given information on X, Y, and Z c. predictions of both X and Y, given information on a third variable d. none of these 14-4. The regression line a. is drawn through the points on a scatter plot b. is a straight line c. comes closet to the majority of the scatter points d. all of these 14-5. The regression equation of Y on X a. produces a straight line b. produces a curvilinear relationship c. always slopes from upper left to lower right d. all of these 14-6. In order to produce the regression line, the researcher must know a. how much the scatter points deviate from the line b. the slope of the line c. where the line crosses the ordinate d. all of these 14-7. In the regression equation for Y on X, the "a" term symbolizes a. the slope of the line b. the amount of scatter around the line c. the point of intercept d. the correlation coefficient

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14-8. The slope of the regression line is designated as the a. "a" term b. "b" term c. "y" term d. "r" term 14-9. The regression line's point of intercept designates a. the slope of the line b. the amount of change in Y which accompanies a given change in X c. the value of Y when X is equal to zero d. the strength of the correlation 14-10. In regression analysis, the amount of change in Y which accompanies a given unit change in X is designated by a. the slope of the regression line b. the Y intercept c. the "a" value d. the magnitude of the difference between X and Y 14-11. The higher the correlation between X and Y a. the more the scatter points deviate from the regression line b. the more the scatter points cluster around the regression line c. the more the scatter points create a circular configuration d. none of these 14-12. In regression analysis, when Y increases by two units for each equal single-unit increase in X, then a. the intercept equals +2.00 b. the intercept equals +.50 c. the slope equals +2.00 d. the slope equals +.50 14-13. The slope of the regression line a. may never be negative b. may never be greater than +1.00 c. may never be greater than the correlation value d. none of these 14-14. When high scores on X associate with low scores on Y, the regression line a. has a negative slope b. crosses the ordinate at the zero point c. runs parallel to the abscissa d. runs parallel to the ordinate

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14-15. Whenever the correlation has a positive value, so too will the a. slope of the regression line be positive b. Y intercept be positive c. mean of the Y distribution d. mean of the X distribution be positive 14-16. Whenever the value of the correlation is negative a. the resulting predictions will be inaccurate b. the amount of information about Y being contained in X is zero c. both of these d. neither of these 14-17. When Y decreases by one unit for each equal unit increase in X, then a. the intercept value will be negative b. the slope value will be negative c. the correlation value will be positive d. both a and b, but not c 14-18. The higher the value of the Y intercept a. the higher the correlation value b. the higher the slope value c. the higher the value of the mean of X d. none of these 14-19. When the correlation between X and Y is zero, then the safest prediction of Y which can be made, give information on X, a. is the mean of X b. is the mean of Y c. is the mean of the combined distributions of X and Y d. none of these 14-20. When the correlation between X and Y is equal to zero, then the only possible value that the Y pred. may take on is a. zero b. a value equal to the slope c. a value equal to the mean of X d. a value equal to the mean of Y 14-21. When the standard deviation of both the X and Y distributions are equal, then a. the value of the slope equals the value of the correlation b. the value of the Y intercept equals the mean of Y c. the value of X equals the value of Y d. none of these 238

14-22. When the correlation between X and Y equals zero, then the slope of the regression line of Y on X is equal to a. the mean of Y b. the point of intercept c. zero d. none of these 14-23. When the standard deviation of the Y distribution equals zero, then a. the slope of the regression line of Y on X equals zero b. the mean of X equals zero c. the mean of Y equals zero d. none of these 14-24. The higher the correlation between X and Y, then a. the more likely it is to be positive b. the less likely that null will be rejected c. the less the predicted value of Y may deviate from the mean of Y d. The more the predicted value of Y may deviate from the mean of Y 14-25. The discovery of the regression phenomenon is credited to a. Karl Pearson b. Sir Francis Galton c. Sir Ronald Fisher d. Dr. Sigmund Gestalt 14-26. According to the regression concept, the children of very tall parents a. should be taller than average, but shorter than their parents b. should be at least as tall as their parents c. should be below average in height d. should just as likely be tall as short 14-27. The lower the value of the correlation between X and Y, the more the predicted Y value a. will regress toward the mean of Y b. will deviate from the mean of Y c. will assume a mean of zero d. both b and c, but not a 14-28. Whenever the calculated value of r equals the critical, tabled value of r for a given number of degrees of freedom, then a. There is a failure to reject Ho b. Ho is rejected c. the mean of Z equals the mean of Y d. the alpha error is rejected 239

14-29. When the correlation in a sample is found to be significant, then it is assumed that a. the mean of Y differs from the mean of X in the population b. the mean of Y equals the mean of X in the population c. the regression of Y on X cannot be determined d. the correlation also exists in the population 14-30. When the correlation between X and Y equals zero, the standard error of estimate a. equals 0 b. equals the mean of Y c. equals the standard deviation of Y d. equals 1.00 14-31. When the correlation between X and Y equals 1.00, the standard error of estimate a. equals 0 b. equals the mean of Y c. equals the standard deviation of Y d. equals 1.00 14-32. The value of the standard error of estimate may never be greater than a. 1.00 b. 0 c. the mean of Y d. the standard deviation of Y 14-33. The value of the standard error of estimate may never be less than a. 1.00 b. 0 c. the mean of Y d. the standard deviation of Y 14-34. The most accurate predictions are made when the standard error of estimate equals a. 1.00 b. 0 c. the mean of Y d. the standard deviation of Y 14-35. When using the standard error of estimate for establishing the confidence interval at .95, then a t ratio for an alpha of a. +1.00 should be used b. .05 should be used c. .95 should be used d. .01 should be used 240

14-36. When using the standard error of estimate for establishing a confidence interval of .99, then a t ratio for an alpha of a. +1.00 should be used b. .05 should be used c. .99 should be used d. .01 should be used 14-37. When the slope of the regression line equals -3.00, then a. the Y intercept equals -3.00 b. there is a 3 equal-unit gain in Y for each unit gain in X c. there is a 3 equal-unit loss in Y for each unit gain in X d. there is a one-third, equal-unit loss in Y for each unit gain in X 14-38. When the Y intercept equals +1.00, then a. the correlation between X and Y equals +1.00 b. the slope of the regression line equals +1.00 c. an X score of +1.00 corresponds with a Y score of 0 d. a Y score of +1.00 corresponds with an X score of 0 14-39. When predicting a student's GPA on the basis of the Verbal SAT score, the regression line was equal to .01X + .10. This means that there was a. a .01 increase in GPA for each point increase in SAT b. a .01 increase in SAT for each point increase in GPA c. an SAT value of .10 when the the GPA was equal to zero d. a and c, but not b 14-40. When predicting a student's GPA on the basis of the Verbal SAT score, the regression line was equal to .01X + .10. This means that there was a. a .01 decrease in GPA for each point increase in SAT b. a .10 increase in GPA for each point increase in SAT c. a predicted GPA of .10 when the SAT score was at its minimum value d. a predicted GPA of .01 when the SAT score was at its minimum value 14-41. When a correlation value among three or more interval-data variables is required, then the appropriate statistical test is a. the multiple R b. the Pearson r c. the Spearman rs d. the multi-factor V 14-42. In order to use the multiple R, a. there must be at least three internal correlations b. at least four variables must be involved c. the correlation between variables 1 and 2 must not be significant d. Ho must be accepted 241

14-43. When information regarding two or more predictor variables is combined in order to predict performance on Y, then the appropriate statistical test is a. the partial correlation b. the Galton "g" c. the Yates Correction d. the multiple R 14-44. When the slope of the regression line of Y on X is positive, then the array of data points on the scatter plot a. goes from upper left to lower right b. goes from lower left to upper right c. goes parallel to the abscissa d. none of these 14-45. The most efficient use of the multiple R is when a. its value is higher than any of the values of the separate, internal correlations with the y variable b. its value is lower than the values of the separate, internal correlations c. the scores to be predicted are in nominal form d. the research is experimental 14-46. In order to utilize the multiple R in making specific Y predictions a. the scores must be in interval form b. at least two slope values must be determined c. Ho must be rejected for each of the internal r's d. all of these 14-47. When predicting from a three-factor, multiple R, the value of b indicates a. the first intercept b. the criterion variables c. the first predictor variable d. how much change in Y will occur for a given change in the first predictor, when the effects of the second predictor have been held constant 14-48. The technique employed for establishing the correlation that is left over between two variables, when both are known to be correlated with a third variable, is called the a. multiple R b. coefficient of determination c. product-moment d. partial correlation

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14-49. The value for ry1.2 provides information regarding a. the multiple correlation among three variables b. the correlation between variables 1 and 2, with variable Y held constant c. the correlation between variables Y and 1, with variable 2 held constant d. the correlation between variables Y and 2, with variable 1 held constant 14-50. Establishing a three-variable partial correlation, results in a. a multiple R b. a third-order partial c. a second-order partial d. a first-order partial 14-51. In a four-variable correlation, when two variables are held constant, while examining the remaining relationship between the other two, the technique is referred to as a. a multiple R b. a first-order partial c. a higher-order partial d. none of these 14-52. When all the points in a scatter plot lie directly on the regression line of Y and X, then a. X has caused Y b. the correlation must equal 0 c. the correlation cannot be calculated d. the correlation must equal +1.00 14-53. The correlation between a criterion variable and two or more predictor variables, is called the a. partial correlation of X on Y b. partial correlation of Y on X c. Pearson r d. multiple R 14-54. In a two-variable scatter plot, the variable being predicted is placed on a. the abscissa b. the regression line c. the ordinate d. none of these

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14-55. A major difference between the Pearson r and the multiple R is that the multiple R requires the use of a. sample data b. population data c. an accuracy measure d. more than two variables 14-56. The standard error of multiple estimate may be used when making predictions from a. two or more predictor variables b. a multiple R c. a Pearson r d. a and b, but not c 14-57. For the Pearson r, the greater the number of degrees of freedom, the greater the likelihood of a. obtaining significance b. rejecting Ho c. the sample correlation also existing in the population d. all of these

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C: True or False: For the following,indicate T (True) or F (False) 14-58. In a two variable linear regression, the intercept indicates the value of when X is at its minimum value.

14-59. If X correlates .95 with Y, then Y correlates .95 with X. 14-60. The regression line of Y on X must always slope from lower left to upper right.

14-61. The regression line is the single straight line which passes through most of the data points on a scatter plot.

14-62. The regression of Y on X is used to predict X, given information on a third variable.

14-63. The slope of the regression line of Y on X must always assume a positive value.

14-64. The more the value of a Pearson r deviates from zero, the greater its predictive power.

14-65. The major goal of correlational research is to allow for better-than-chance predictions.

14-66. Each single data point on a scatter plot represents a pair of scores.

14-67. In order to obtain significance for the Pearson r, Ho must be rejected.

14-68. For the Pearson r, rejecting Ho means that the obtained correlation does exist in the population.

14-69. In the regression of Y on X, when the Pearson r equals 1.00, so too must the slope value equal 1.00 14-70. If the Y intercept equals 50, then a Y value of 50 corresponds with an X value of 0.

14-71. If the slope value equals +10, then Y increases by 10 units for each equal unit increase in X. 14-72. When the correlation in the sample is negative, it can never be used to extrapolate to the population.

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14-73. The standard error of estimate may never assume a negative value.

14-74. When the standard error of estimate equals zero, it means that all the data points lie right on the regression line.

14-75. The standard error of estimate may never equal zero, unless the Pearson r equals 1.00.

14-76. The standard error of estimate may never assume a value larger than the true standard deviation of the Y distribution.

14-77. In order to calculate the standard error of estimate, the value of the Pearson r must be known.

14-78. The partial correlation may never be used, unless more than two variables are involved.

14-79. The multiple R is used to assess the strength of the combined correlations among three or more variables.

14-80. A first-order partial correlation assesses the strength of the relationship between two variables, with the effects of a third variables held constant.

14-81. When all of the internal correlations are significant, then the multiple R must always be significant.

14-82. The resulting correlation between two or more predictor variables and a criterion variable is called the multiple R. 14-83. If X is independent of Y, the safest Y pred., given information about X, is the mean of the Y distribution.

14-84. The lower the value of the Pearson r, the greater the likelihood of the Y pred. falling near the mean of Y. 14-85. The Pearson r, the multiple R, and the coefficient of determination all demand at least interval data.

14-86. For the Pearson r, the degrees of freedom are a direct function of the number of pairs of scores.

14-87. A negative correlation may never be a significant correlation.

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D: For the following questions, calculate the values.

For questions 88 through 90, use the following: Assume a significant XY correlation of .60 and a sample size of 62. The mean of X equals 14.00, and the mean of Y equals 2.00. The estimated sd of the sample X equals 3.00 and the estimated sd of Y equals .80.

14-88. Predict a Y value for a known X value of 10.00.

14-89. Calculate the standard error of estimate.

14-90. Find the .95 confidence interval for the predicted Y value.

For questions 91 through 94, use the following: Assume a significant XY correlation of .90 and a sample size of 40. The mean of X equals 100.00. The mean of Y equals 500.00. The estimated sd of X equals 10.00. The estimated sd of Y equals 100.00 14-91. Predict a Y value for a known X value of 110.00

14-92. Calculate the standard error of estimate.

14-93. Find the .95 confidence interval for the predicted Y value.

14-94. Find the .99 confidence interval for the predicted Y value.

For questions 95 through 98, use the following: Assume a significant XY correlation of .85 and a sample size of 30. The mean of X equals 800.00, and the mean of Y equals 20.00. The estimated sd of X equals 75.00, and the estimated sd of Y equals 3.70.

14-95. Predict a Y value for a known X value 1100.00.

14-96. Calculate the standard error of estimate.

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14-97. Find the .95 confidence interval for the predicted Y value.

14-98. Find the .99 confidence interval for the predicted Y value.

For questions 99 through 104, use the following: A researcher selects a random sample of 8 high school students. Each student is tested on both Need Achievement and Mechanical Aptitude. The scores were: Subject Need Achievement Mechanical Aptitude 1 12 10 2 2 3 3 5 7 4 9 5 5 11 9 6 10 8 7 4 6 8 1 3 14-99. Find the Pearson r correlation between Need Achievement and Mechanical Aptitude.

14-100. Should Ho be rejected or accepted? And if rejected, state the significance level.

14-101. Predict a Y value for a known X value of 8.00.

14-102. Find the standard error of estimate.

14-103. Find the .95 confidence interval for the predicted Y value.

14-104. Find the .99 confidence interval for the predicted Y value.

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For questions 105 through 110, use the following: A group of 8 elementary school students was randomly selected . Each pupil was tested on both Math Ability and Spelling Ability. Their scores were: Subject 1 2 3 4 5 6 7 8

Math Test 14 12 3 9 8 5 10 2

Spelling Test 15 10 2 10 7 6 9 4

14-105. Find the Pearson r correlation between the Math and Spelling scores.

14-106. Should Ho be rejected ? If rejected, state the significance level.

14-107. Predict a Y value for a given X value of 7.00.

14-108. Find the standard error of estimate.

14-109. Find the .95 confidence interval for the predicted Y scores.

14-110. Find the .99 confidence interval for the predicted Y scores.

14-111. The Pearson r correlation between variables y and 1 is a significant .60. The Pearson r correlation between variables y and 2 is a significant .50. The Pearson r between variables 1 and 2 is a significant .70. Find the multiple R for predicting y from 1 and 2.

14-112. The Pearson r between variables y and 1 is a significant .73. The Pearson r between variables y and 2 is a significant .62. Find the multiple R for predicting y from 1 and 2.

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14-113. A researcher runs Pearson r correlations among three variables, GradePoint Average, IQ and Reading Speed. All correlations are significant.

GPA and IQ = .80 GPA and Reading Speed = .85 IQ and Reading Speed = .60 Find the multiple R for predicting Grade-Point average from IQ and Reading Speed.

14-114. A corporate researcher runs Pearson r correlations among three variables, net income, gross income and amount spent on advertising. All correlations are significant.

Net Income and Gross Income = .60 Net Income and Advertising = .53 Gross Income and Advertising = .35 Find the multiple R for predicting net income from both gross income and advertising.

For questions 116 through 119, use the following: A researcher collects the following measures: Y mean = 50, with a standard deviation of 10 X1 mean = 100, with a standard deviation of 15 X2 mean = 200, with a standard deviation of 20 Significant correlations among the variables were found to be as follows: y and 1 = .60 y and 2 = .75 1 and 2 = .20 14-115. Find the multiple R for predicting y from the X1 and X2 variables.

14-116. Predict the Y value for a known X1 value of 115 and a known X2 value of 220.

14-117. Calculate the standard error of multiple estimate.

14-118. Find the .99 confidence interval for the predicted Y score.

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For questions 119 through 122, use the following: A random sample of college students was selected from the population of male, physical-education majors. The mean height was 68". The mean weight was 150 lbs. The mean waist-measure size was 32". The standard deviations were, for height, 3" for weight, 30 lbs for waist-size, 2" Significant Pearson r's were, Height and weight, .70 Height and waist-size, .40 Weight and waist-size, .30 14-119. Find the multiple R for predicting height from both weight and waist-size.

14-120. Predict the height of a student whose weight was 120 lbs., and who had a waist-size of 29".

121. Calculate the standard error of multiple estimate.

14-122. Find the .95 confidence interval for the predicted Y value.

14-123. Among elementary school children, the correlation between strength and weight is .60, and the correlation between strength and age is .70. The correlation between weight and age is .80. (All correlations are significant.) What is the partial correlation between strength and weight, holding age constant?

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CHAPTER 15 REPEATED-MEASURES AND MATCHED-SUBJECTS DESIGNS ( INTERVAL DATA) A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Degrees of Freedom and Marital Bliss As a reminder on degrees of freedom, you might tell your students that as a long-ago professor of mine used to say, "going from an independent t to a paired t is like getting married.....you lose half your degrees of freedom".

Explain to those who are still single, that the long-ago professor was actually wrong, that in fact when you get married you'll lose ALL your degrees of freedom. T'was ever thus.

Activity: The Paired t and the Pearson r Sometimes when doing the paired t, students don't quite see the relationship between the value of the resulting t ratio and size of the correlation factor. To show that the resulting value of the t ratio is increased as a function of the size of the r, it might be wise to use two sets of identical scores, and then have them do both types of t ratios on the data.

You can rearrange these same scores by pairing them off differently in order to illustrate the changes in the value of the paired t as you alter the strength of the correlation.

For example, you could begin with no correlation (in effect an independent t): Group 1 Group 2 17 6 15 14 12 8 10 12 10 6 10 4 9 2 8 10 6 12 6 8 Mean 1 = 10.300 Mean 2 = 8.200 SE mean 1 = 1.126 SE mean 2 = 1.209 Pearson r = .011 SE of Difference = 1.643 t = 1.278

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Using the above data as a straight independent t (where the correlation is assumed to be zero) we would get an SE of difference of 1.652, and a t of 1.271. The slight difference from the data shown above is due to the fact that the paired t treated that correlation of .011 as significant and this caused the slight downward adjustment to the standard error of difference and the slight increase in the resulting t ratio. In any event the above t ratio is clearly an accept of Ho. Now if we take the same data, but rearrange the pairings to jump the correlation to a high positive value, we get: Group 1 17 15 12 10 10 10 9 8 6 6

Group 2 14 12 12 10 8 8 6 6 4 2

Mean 1 = 10.300 Mean 2 = 8.200 SE mean 1 = 1.126 SE mean 2 = 1.209 Pearson r = .942 SE of difference = .406 t = 5.172 Now the paired t ratio is producing a clear reject of Ho at P <.01, which again shows the power of the paired t ratio as a function of the size of the correlation Finally, take the same data, but rearrange the pairings to create a high negative correlation: Group 1 Group 2 17 2 15 4 12 6 10 6 10 8 10 8 9 10 8 12 6 12 6 14 253

Mean 1 = 10.300 Mean 2 = 8.200 SE mean 1 = 1.126 SE mean 2 = 1.209 Pearson r = -.951 SE of difference = 2.306 t = .911 The t of .911 is, of course, now an accept of Ho. With the negative correlation, the standard error of difference is INCREASED and the resulting t ratio severely decreased. Although it might be hard to imagine a matching variable so bad as to create a significant negative correlation, at least the students might better understand the workings of the paired t by being exposed to this final arrangement.

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B Multiple Choice 15-1. Of the following research designs, the one(s) in which the sample scores must be independent of each other, is (are) the a. between-subjects b. repeated-measures c. matched-subjects d. all of these 15-2. Of the following research designs, the one(s) requiring active manipulation of the independent variable is (are) the a. between-subjects b. repeated-measures c. matched-subjects d. all of these 15-3. Of the following experimental designs, the one(s) in which sample scores are correlated with each other is (are) the a. between-subjects b. repeated-measures c. matched-subjects d. both b and c, but not a 15-4. When experimental research yields a significant correlation between pairs of scores, the experimental design was a. either between-subjects, or repeated-measures b. either between-subjects, or matched-subjects c. either repeated-measures, or matched subjects d. all of these 15-5. When subjects are measured in the "pre" condition, then the independent variable is inserted, and the subjects are measured again in the "post" condition, the experimental design is a. test-retest b. repeated-measures c. post-facto d. none of these 15-6. In comparing the between-subjects with the matched-subjects designs, the matched-subjects design should yield a. a significant positive correlation between the paired scores b. no correlation between paired scores c. a significant negative correlation between the paired scores d. a partial correlation 255

15-7. Whenever the repeated-measures design is used a. the same subjects are used as their own controls b. there should be a positive correlation between the pre and post scores c. the researcher is using the post-facto method d. both a and b, but not c 15-8. The paired t is to the independent t, as a. chi square is to ANOVA b. the Pearson r is to the Spearman rs c. the Pearson r is to the chi square d. the repeated-measures is to the between-subjects design 15-9. With interval data and random assignment of subjects, the appropriate analysis of data from an between subjects (two group) design would be a. the paired t b. the Pearson r c. the independent t d. the Spearman correlation 15-10. The between-subjects design, with random assignment of subjects, should yield between the scores in the two groups. a. a significant positive correlation b. a significant negative correlation c. no correlation d. all of these, depending on the degrees of freedom 15-11. For a given number of scores, the independent t has degrees of freedom as the paired t. a. one-half b. twice c. the same d. cannot tell from the above information

the

15-12. In a paired t, when the correlation is significant, a. the degrees of freedom become less b. the alpha error becomes smaller c. the estimated standard error of difference is increased d. the estimated standard error of difference is reduced 15-13. Whenever the value of the estimated standard error of difference is increased, then a. the positive correlation was significant b. the zero correlation was not significant c. the repeated-measures design was used d. none of these 256

15-14. With any t test, independent or paired, the more subjects there are being measured a. the greater the number of degrees of freedom b. the fewer the degrees of freedom c. the greater the alpha level d. none of these 15-15. The final goal of the paired t, is to a. test the hypothesis of association b. test the hypothesis of difference c. match the independent and dependent variables d. match the independent and control variables 15-16. The final goal of the independent t is to a. test the hypothesis of association b. test the hypothesis of difference c. match the independent and dependent variables d. match the independent and control variables 15-17. A t ratio which is calculated on data from a repeated-measures design, using 20 subjects, should yield a. 19 degrees of freedom b. 38 degrees of freedom c. 18 degrees of freedom d. cannot tell from the above information 15-18. When a paired t is applied to between subjects data with independently assigned subjects then a. the positive correlation should be significant b. the correlation should approximate zero c. the negative correlation should be significant d. the multiple R should be used 15-19. With any t test, paired or independent, the final comparison is always drawn between measures of the a. independent variable b. dependent variable c. control variable d. none of these

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15-20. Whenever Ho is accepted, when, in fact, it should have been rejected, the error has been committed. a. Type 1 b. Type 2 c. standard d. sampling 15-21. The more powerful a given statistical test is, the greater its likelihood of a. rejecting Ho b. failing to reject Ho c. detecting the standard error d. reducing the degrees of freedom 15-22. Subtracting beta from the constant 1.00, yields a. the sampling error b. the deviation error c. bias d. power 15-23. Whenever the value of the estimated standard error of difference is reduced, a. the value of alpha is increased b. the value of the sampling error is increased c. the value of the t ratio is increased d. the value of the t ratio is reduced 15-24. The relationship between the size of the values for the estimated standard error of difference and the t ratio, is a. inverse b. positive c. curvilinear d. impossible to determine

15-25. When the paired t is used on data which are really independent, there is an increased likelihood of a. Failing to reject Ho b. rejecting Ho c. committing the Type 1 error d. none of these

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15-26. When the independent t is used on data which are really correlated, there is an increased likelihood of a. rejecting Ho b. failing to reject Ho c. committing the Type 1 error d. none of these 15-27. The more powerful a given statistical test is, the more apt it is to a. reject Ho when Ho should be rejected b. failing to reject Ho when Ho should be rejected c. inflate beta d. trigger a statistical decision 15-28. Accepting the null hypothesis, when null should have been rejected causes the a. Type 1 error b. Type 2 error c. loss of degrees of freedom d. all of these, depending on the sample size 15-29. Rejecting the null hypothesis when it should have been accepted, causes the a. Type 1 error b. Type 2 error c. loss of degrees of freedom d. all of these, depending on the sample size 15-30. The probability of rejecting null always increases when the a. Type 1 error is decreased b. sample size is decreased c. sample size is increased d. power of the test is reduced 15-31. Compared to the independent t, the paired t, on data which are really correlated, is a. a less powerful test b. a more powerful test c. equal in power with the independent t d. none of these, since power is never a function of which statistical test is being used

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15-32. When attempting to establish differences among three sets of matched interval scores, the appropriate test is the a. paired t b. factorial ANOVA c. within-subjects ANOVA d. multiple R 15-33. When the correlation between two sets of interval measures is significant and positive, the value of the paired t will always be a. higher than the independent t b. lower than the independent t c. equal to the independent t d. none of these 15-34. In order to calculate a within-subjects ANOVA, one must first a. establish the value of the multiple R b. establish the values of the separate t tests c. establish the value of the Pearson r d. none of these 15-35. When equating subjects on a matched-subjects design, the researcher should choose some measure which is correlated with the a. independent variable b. dependent variable c. measurement scale d. confounding variable 15-36. Using a repeated-measures design with a separate control group, helps to prevent a. sampling error b. score dispersion c. individual differences d. confounding variables 15-37. Compared to the independent t, the paired t for a given sample size always a. has fewer degrees of freedom b. has more degrees of freedom c. uses degrees of freedom which are independent of the sample size d. none of these, since when sample sizes are fixed, so too are the degrees of freedom

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15-38. When analyzing interval data from a repeated-measures design, with a separate control group, the t ratio must be established between a. the "pre" and "post" scores b. the "post" scores, without reference to the "pre" scores c. the "pre" scores, without reference to the "post" scores d. the "change" scores of the experimental and control groups 15-39. In a repeated-measures design, with a separate control group that is not matched to the experimental group, the "change" scores should be a. correlated with each other b. independent of each other c. positive in sign d. none of these 15-40. The within-subjects ANOVA should never be used a. on interval data b. on a matched-group design c. when more than two groups have been equated d. none of these 15-41. When the correlation is not significant, the major difference between the paired and independent t ratios, is in the a. degrees of freedom b. the value of the estimated standard error of difference c. the value of the estimated standard error of the mean d. the value of the mean difference 15-42. Dividing the mean difference by the estimated standard error of difference always yields the a. standard error of the mean b. standard error of estimate c. standard error of proportion d. t ratio 15-43. When using the independent t on interval data from a one-group, beforeafter design, there is an increased likelihood of a a. sampling error b. Type 2 error c. Type 1 error d. rejection of Ho

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15-44. When a researcher manipulates two independent variables simultaneously, the appropriate analysis must always utilize the a. independent t b. paired t c. the one-way ANOVA d. none of these 15-45. Regardless of the size of the sample, or the scale of measurement, when correlated samples are involved, the researcher may never use a. the paired t b. the within-subjects ANOVA c. the appropriate degrees of freedom d. none of these 15-46. When the probability of beta increases, then a. so, too, does alpha b. so, too, does sampling error c. so, too, does the power of the test d. power decreases 15-47. When the calculated value of a paired t, for a given number of degrees of freedom, equals the critical, table value of t, then a. there is a failure to reject Ho b. Ho is rejected c. no statistical decision can be made d. the alpha error is reduced 15-48. With the within-subjects ANOVA, when null is rejected at an alpha of .05, then a. the alpha becomes .10 b. the alpha becomes .01 c. the alpha becomes .95 d. none of these 15-49. With any t test, independent or paired, the loss of degrees of freedom causes a. an increased likelihood of rejecting Ho b. a decreased likelihood of rejecting Ho c. an increase in the standard error of difference d. the mean difference to assume a negative value

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15-50. For the t ratio, the only time a negative t value may occur is when a. sample mean 1 is smaller than sample mean 2 b. sample mean 2 is smaller than sample mean 1 c. the sample means are equal d. none of these, since t may never be negative 15-51. For the t test, other things being equal, the smaller the estimated standard error of difference, the greater the likelihood of a. committing the beta error b. failing to reject Ho c. rejecting Ho d. eliminating the test's power 15-52. Accepting Ho when it should have been accepted causes an increase in the a. Type 1 error b. Type 2 error c. sampling error d. none of these 15-53. With ratio data, calculating the t ratio is identical with the procedures used on a. nominal data b. ordinal data c. interval data d. none of these, since with ratio data the t test cannot be used 15-54. With a within-subjects ANOVA, a significant difference implies that a. the difference occurs only in the sample b. the difference occurs in the population, but not in the sample c. the difference occurs both in the sample and in the population d. there is no difference 15-55. For a given number of scores, the within-subjects ANOVA a. always has more degrees of freedom than does the independent F b. always has the same number of degrees of freedom as the independent F c. always has fewer degrees of freedom than does the independent F d. has more degrees of freedom than the independent F when Ho is rejected 15-56. Sampling error is assumed a. only when the F ratio is calculated b. only when the t ratio is calculated c. only when the Pearson r is significant d. whenever samples are measured

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15-57. The only time the Type 2 error may be committed is when the a. degrees of freedom are infinite b. degrees of freedom are less than 30 c. degrees of freedom are calculated on the basis of sample size d. none of these

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C. True or False: For the following, indicate T (True) or F (False) 15-58. The paired t should not be used for the analysis of data from independent sample groups.

15-59. For a given number of scores, the paired t has one-half the degrees of freedom of an independent t.

15-60. With an equal mean difference, when the correlation is significant, the paired t results in a higher t ratio than does the independent t.

15-61. With a paired t, a significant correlation has the effect of reducing the value of the estimated standard error of difference.

15-62. Degrees of freedom are a direct function of the sample size for all statistical tests except the paired t.

15-63. When there is a correlation across the rows, the within-subjects ANOVA results in a higher value than when there is no correlation across the rows.

15-64. With interval data and a two-group, matched-subjects design, the appropriate analysis is via the independent t.

15-65. When analyzing data from a repeated-measures design, if the correlation is significant, the paired t increases the test's power.

15-66. An F ratio may never be used on data from a matched-subjects design.

15-67. For a given mean difference, a significant correlation increases the value of a paired t ratio.

15-68. The power of a test is equal to the summation of both the alpha and beta probabilities.

15-69. The lower the value of beta, the higher the power of any statistical test.

15-70. The beta level defines the probability of accepting null when, in fact, null should be rejected.

15-71. With an extremely powerful test, the probability of a Type 1 error is eliminated.

15-72. With a paired t, the greater the number of paired scores, the greater the number of degrees of freedom.

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15-73. The use of the repeated-measures design with a separate control group, prevents the possibility of manipulating the independent variable.

15-74. With interval data from a three-group, matched design, the appropriate test would be the within-subjects ANOVA. 15-75. Failing to reject Ho can never occur when alpha has been set at .01

15-76. Other things being equal, the smaller the value of the estimated standard error of difference, the less likely the t ratio will be significant.

15-77. The paired t may never result in a negative value.

15-78. The within-subjects ANOVA may never result in a negative value.

15-79. When analyzing interval data from a repeated-measures design with a separate control group, an independent t may be applied to the two sets of "change" scores.

15-80. With 20 pairs of interval scores, the paired t has 38 degrees of freedom.

15-81. The paired t may only be applied to experimental data when there are at least two independent variables.

15-82. For the paired t, increasing the sample size always increases the Type 1 error.

15-83. The higher the probability of accepting Ho, the greater the likelihood of committing a Type 2 error.

15-84. For the paired t, the fewer the degrees of freedom, the easier it becomes to reject the null hypothesis.

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D. For the following questions, calculate the values. We suggest that all paired t's be done via the long method, where the correlation is first checked for significance. Using the direct-difference method, because of rounding differences, may yield slightly different answers.

15-85. Calculate a paired t on the following scores: Group 1 Group 2 8 6 7 6 4 4 8 2

15-86. For question 85, should Ho be rejected as a two-tail t? If rejected, state the alpha level.

15-87. Using the data from question 85, calculate an independent t.

15-88. For question 87, should Ho be rejected as a two-tail t? If rejected, state the alpha level.

15-89. Calculate a paired t on the following scores: Group 1 Group 2 10 6 4 4 12 9 5 3 4 3 6 5 8 7 7 4

15-90. For question 89, should Ho be rejected as a two-tail t? If rejected, state the alpha level.

15-91. Using the data from question 89, calculate an independent t.

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15-92. For question 91, should Ho be rejected as a two-tail t? If rejected, state the alpha level.

15-93. Calculate a paired t on the following scores: Group 1 Group 2 13 21 10 19 6 15 1 13 15-94. For question 93, should Ho be rejected as a one-tail t? If rejected, state the alpha level.

15-95. Using the data from question 93, calculate an independent t.

15-96. For question 95, should Ho be rejected as a one-tail t? If rejected, state the alpha level.

15-97. Calculate a paired t on the following scores: Group 1 Group 2 15 22 12 21 8 17 3 15

15-98. For question 97, should Ho be rejected as a one-tail t? If rejected, state the alpha level.

15-99. Using the data from question 97, calculate an independent t.

15-100. For question 99, should Ho be rejected? If rejected, state the alpha level.

268 For questions 101 through 105, use the following:

A researcher predicts that training on the balance beam reduces reading errors among learning disabled youngsters. A random sample of 9 children was selected from the population of learning-disabled children from a large metropolitan school district. They were all given a reading test and their errors recorded. They were then trained on the balance beam for one year, and again their error scores on the reading test recorded. The data follows: Subject # Before After 1 12 10 2 12 10 3 2 1 4 6 4 5 6 5 6 7 7 7 10 9 8 9 8 9 11 8 15-101. Calculate the appropriate statistical test.

15-102. As a one-tail test, should Ho be rejected? If rejected, state the alpha level.

15-103. Had the researcher made an error and calculated an independent t, what value would have been obtained?

15-104. For question 103, should Ho be rejected or accepted as a one-tail, And also should Ho be rejected as a two-tail? If rejected, state the alpha level.

15-105. What is the major research error in the design of this experiment?

15-106. Calculate a within-subjects ANOVA on the following matched-subjects data: Group 1 Group 2 Group 3 1 2 3 3 4 5 5 6 7 7 8 9 10 10 11 269

15-107. For the above data (question 106), should Ho be rejected? If rejected, state the alpha level.

15-108. Calculate a within-subjects ANOVA on the following matched-subjects data: Group 1 Group 2 Group 3 6 3 5 2 5 4 1 7 3 5 3 6 4 4 6

15-109. For question 108 should Ho be rejected ? If rejected, state the alpha level.

15-110. Calculate a within-subjects ANOVA on the following matched-subjects data: Group 1 Group 2 Group 3 1 5 6 3 4 7 5 3 8 4 6 10 2 2 4

15-111. For question 110, should Ho be rejected? If rejected, state the alpha level.

15-112. Calculate a within-subjects ANOVA on the following matched-subjects data: Group 1 Group 2 Group 3 1 2 3 3 3 6 5 6 11 4 5 9 2 4 6 15-113. For question 112, should Ho be rejected? If rejected, state the alpha level.

270 For questions 114 through 116, use the following:

A researcher suspects that increasing illumination increases motoric speed. A group of 7 subjects was randomly selected and tested on the Digit Symbol subtest of the WAIS-R IQ test. The same subjects were tested under three illumination conditions, low light, medium light and high light. Their scores under each condition were as follows: Subject # Low Light Medium Light High Light 1 7 10 11 2 8 7 10 3 6 8 10 4 5 6 9 5 4 4 7 6 3 5 6 7 8 10 13 15-114. Calculate the appropriate F ratio.

15-115. For question 114, should Ho be rejected? If rejected, state the alpha level.

15-116. What is the major research error in this study?

15-117. A researcher sets up a repeated-measures design with an independent control group. Calculate the appropriate t ratio.

EXPERIMENTAL GROUP CONTROL GROUP Pre Post Pre Post 100 109 100 106 98 110 110 115 106 113 95 102 108 116 105 111 110 116 102 107

15-118. Should Ho be rejected as a two-tail test? If rejected, state the alpha level.

271 15-119. A researcher sets up a repeated-measures design with an independent

control group. Calculate the appropriate t ratio.

EXPERIMENTAL GROUP Pre Post 10 20 6 14 11 20 8 18 5 12

CONTROL GROUP Pre Post 8 16 12 16 6 12 9 12 8 10

15-120. Should Ho be rejected as a one-tail test?

15-121. A researcher sets up a repeated-measures design with a matched control group. Calculate the appropriate t ratio.

EXPERIMENTAL GROUP CONTROL GROUP Pre Post Pre Post 10 17 10 20 11 17 10 15 12 16 8 12 10 15 9 14 11 15 13 15 10 14 11 13

15-122. Should Ho be rejected? If rejected state the alpha level.

272 For questions 123 through 126, use the following:

A researcher devised a special training program for high school students in order to increase their verbal SAT scores. A random sample of 10 high school juniors was selected. They all took the SAT and their verbal scores are recorded. The following June, they were all given a three-month summer training program, entitled "Beating the SATs". During November of their senior year they again took the SAT. Their verbal SAT scores follow: Junior Year Scores Senior Year Scores Pre Post 510 520 500 530 550 580 550 580 550 570 480 500 450 470 300 320 600 620 400 420 15-123. Using the paired t, find out if there was a significant improvement in the SAT scores of those who had the summer coaching.

Their junior year SAT scores were checked and then compared with those received during November of their senior year. The data follow: Junior Year Scores Senior Year Scores Pre Post 520 540 540 580 560 580 540 570 490 510 450 470 300 320 600 620 650 670 400 420 15-124. Using the paired t, find out if there was a significant improvement in the SAT scores of those who were not coached.

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15-125. Calculate a paired t ratio on the change scores between the coached and non-coached groups.

15-126. Should Ho be rejected as a two-tail test? If rejected, state the alpha level.

15-127. Which group did the results favor?

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CHAPTER 16 NON-PARAMETRICS REVISITED: THE ORDINAL CASE A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: When Normality Fails Point out that in this chapter, which is devoted to ordinal data, the student is presented with statistical tests that can of course be used when the data are in ranked form, but also can be used when interval or ratio (scale) data don't meet the requirements for parametric testing. Perhaps the interval-score sampling distribution is significantly skewed, or perhaps the underlying distribution in the population clearly violates the assumption of normality in other ways. These are the kinds of situations when the non-parametric tests shown in this chapter can come to the rescue. It should be stressed, however, that there are parametric alternatives to these ordinal tests that can be used even if the homogeneity of variance assumption has been violated. The advantage of these alternatives is that they retain their parametric power. Advise your students to read Special Unit 2 on the issue of unequal variance.

Activity: Parametric Restrictions Whereas t, F and r are parametric tests, that is tests that demand certain, sometimes stringent, assumptions about population values, they are often inappropriate to a given set of research data. With the parametric tests we assume that the samples represent fairly normal populations, and in the case of both Student’s t and Fisher’s F, share the same variance. With the "distribution free" or nonparametric tests these assumptions are no longer a burden. The student must realize, however, that switching from parametric tests to non-parametric tests does involve a loss of statistical power, when the parametric restrictions are observed. Thus, when the data do conform to parametric assumptions, the parametric tests SHOULD BE used, since they offer an increased probability of rejecting the null hypothesis. This chapter has been saved until late in the book because the student confronting non parametric tests for the first time must have some familiarity with descriptive statistics, such as means, medians, modes and standard deviations, as well as at least a preliminary knowledge of distribution shapes, normal, skewed and bimodal. The student should also have had acquaintance with parametric correlations, at least the Pearson r, and also with the basics of hypothesis testing, especially as applied to the t tests and ANOVA. Finally, a reminder that some adjusted parametric tests are still available, even when the homogeneity of variance restriction has not been fully met (see Special Unit 2).

Activity: Nonparametrics and their Parametric Counterparts Perhaps the best way to introduce the nonparametric tests is to constantly point out their parametric counterparts, and that ordinal tests should be used only when the parametric assumptions cannot be met. For example you might begin by asking students to review the use of the independent, two275

sample t test, including its basic assumptions. Then create a situation where the t test would be inappropriate because of skewed distributions, such as comparing two sets of income distributions or age distributions. For example, to take blatant case, and sometimes it takes blatant cases to make the point, suppose we were comparing the ages of two groups of rats assumed to represent a common population of age values. One group is injected daily with a certain carcinogenic substance, while the other group is given a placebo. Both groups were otherwise treated the same, same diet, same amount of activity and so forth. Both groups are then compared with regard to longevity. Their ages at death (in days) follow: Experimental Group Control Group A. B. 180 205 175 210 175 205 160 215 180 200 190 210 185 200 195 205 200 200 310 100 Both distributions are badly skewed, the experimental group skewed to the right and the control group to the left. The means for the two groups are 195 for Group A and 195 for Group B. Applying a t test to the data gives us a t ratio of 0. (Some students will immediately call out this answer as soon as they discover that the two sample means are equal.) For those students who work out all the details of the t, the results are as follows: Group A Group B Mean = 195 Mean = 195 SE of mean = 13.271 SE of mean = 10.672 SE of difference = 17.030 t = 0 Accept Ho. Difference is not significant Now show the students that despite the lack of normality the data can still be saved and analyzed through the use of the Mann-Whitney U test. This presents the opportunity to show your students that each of the major parametric tests has its nonparametric counterpart: for the one-way ANOVA they could use the Kruskal Wallis H, for the paired t they could use the Wilcoxon t, and for the within-subjects F, they can use the Friedman ANOVA by ranks.

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In the above case, we use the Mann-Whitney U, and using the same data, but converting the interval scores (ages) to ordinal ranks, (and accounting for ties), we get Ranks for A. Ranks for B. 15.5 6.0 17.5 3.5 17.5 6.0 19.0 2.0 15.5 9.5 13.0 3.5 14.0 9.5 12.0 6.0 9.5 9.5 1.0 20.0 Sums 134.50

75.50

Notice that the rat that ranked highest for longevity appears in the A group, while the rat ranked lowest is in the B group, hence the skewed distributions.

Applying the Mann-Whitney U test to these ranked data, we get U = 20.500 Zu = -2.230. Reject Ho, the difference is significant at P < .01.

With the appropriate statistical analysis the null hypothesis is now clearly rejected. There is a significant longevity difference between the two groups, a difference that, because of the skewed distributions, was obscured when doing the t test. This provides a good opportunity to repeat to your students that power is not just a function of the statistical test, but also results in part from the use of the specific statistical test that is appropriate to the both the research data and the research situation.

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B. Multiple Choice Items 16-1. With the ordinal data, the important information being provided concerns a. equality versus non-equality b. greater-than, or less-than c. the distance between the successive scale points d. an absolute zero 16-2. The most appropriate measure of central tendency on ordinal data is the a. mean b. median c. mode d. interquartile range 16-3. Ordinal tests of significance are all a. parametrics b. non-parametrics c. distribution free d. both b and c, but not a 16-4. Of the following correlation coefficients, which requires ordinal data? a. the Pearson r b. the coefficient of contingency c. the Spearman rs d. all of these 16-5. When the conversion is appropriate, it is always possible to convert a. nominal into interval data b. ordinal into interval data c. nominal into ordinal data d. interval into ordinal data 16-6. In terms of the amount of information being contained in the measures, the following measurement scales rank a. first nominal, then ordinal, and then interval b. first ordinal, then nominal, and then interval c. first interval, then nominal, and then ordinal d. first interval, then ordinal, and then nominal 16-7. In order to calculate an ordinal test of significance, the following must be true a. the scores must be normally distributed in the population b. the variability within each sample must be homogeneous c. the population means must be assumed to be equal d. none of these 278

16-8. The ANOVA by ranks was created by a. Wilcoxon b. Mann and Whitney c. Kruskal and Wallis d. Friedman 16-9. The T for correlated samples was created by a. Wilcoxon b. Mann and Whitney c. Kruskal and Wallis d. Friedman 16-10. The U test was created by a. Wilcoxon b. Mann and Whitney c. Kruskal and Wallis d. Friedman 16-11. The H test was created by a. Wilcoxon b. Mann and Whitney c. Kruskal and Wallis d. Friedman 16-12. The ordinal test for assessing the difference between two correlated samples is the a. Wilcoxon T b. Mann-Whitney U c. Kruskal-Wallis H d. Spearman r 16-13. The ordinal test for assessing the difference between three or more independent samples, is the a. Wilcoxon T b. Mann-Whitney U c. Kruskal-Wallis H d. Spearman r 16-14. The ordinal test for assessing the difference between two independent samples, is the a. Wilcoxon T b. Mann-Whitney U c. Kruskal-Wallis H d. Spearman r 279

16-15. The ordinal test for assessing the difference among three or more correlated samples, is the a. Wilcoxon T b. Mann-Whitney U c. Kruskal-Wallis H d. Friedman ANOVA by ranks 16-16. The Wilcoxon T test is the ordinal equivalent of which interval test? a. independent t b. paired t c. independent F d. within-subjects ANOVA 16-17. The Mann-Whitney U is the ordinal equivalent of which interval test? a. independent t b. paired t c. independent F d. within-subjects ANOVA 16-18. The Kruskal-Wallis H is the ordinal equivalent of which interval test? a. independent t b. paired t c. independent F d. within-subjects ANOVA 16-19. The Friedman ANOVA by ranks is the ordinal equivalent of which interval test? a. independent t b. paired t c. independent F d. within-subjects ANOVA 16-20. The Pearson r is to the Spearman rs , as the a. F is to t b. chi square is to V c. H is to U d. independent F is to H 16-21. When doing the ordinal T test, it is important to establish a. the signed ranks b. frequency of occurrence c. the difference between the means d. all of these 280

16-22. The ordinal T test shown in the text requires at least a. 12 pairs of scores b. 16 independent scores c. 6 pairs of scores d. 25 independent scores 16-23. For the ordinal T on small samples, with a calculated value of 4.13 and a critical, tabled value of 6.00, then a. Ho is rejected b. There is a failure to reject Ho c. Ho cannot be tested d. the sample mean equals parameter mean 16-24. For the ordinal T on small samples, with an obtained value of 17, and a critical, tabled value of 6.00, then a. Ho is rejected b. There is a failure to reject Ho c. Ho cannot be tested d. the sample mean equals the parameter mean 16-25. Homogeneous variability within the various sample groups is required in order to calculate the a. U test b. H test c. Spearman r d. none of these 16-26. Homoscedasticity is required in order to calculate the a. U test b. H test c. T test d. none of these 16-27. If ordinal data are obtained from a two-group, between-subjects design (with random assignment of all subjects), the appropriate test would be a. U b. T c. Spearman r d. Friedman ANOVA by ranks

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16-28. If ordinal data were obtained from a three-group, matched-subjects design, then the appropriate test would be the a. T b. H c. U d. Friedman ANOVA by ranks 16-29. If ordinal data were obtained from a one-group, repeated-measures design, then the appropriate test would be the a. T b. H c. U d. chi square 16-30. If ordinal data were obtained from a four-group, between-subjects design, then the appropriate test would be the a. T b. H c. U d. Friedman ANOVA by ranks 16-31. The Spearman r should only be used to test the hypothesis of a. difference b. association c. equivalence d. none of these 16-32. Ordinal tests may be used to test data from the a. experimental method b. post-facto method c. both of these d. neither of these 16-33. If the calculated value of zu were 3.15 and the critical, tabled value were 1.96, then Ho would be a. rejected b. a failure to reject Ho c. untestable d. none of these

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16-34. Of the ordinal tests presented in the text, the only one which demands a reject of Ho when the obtained value is smaller than the critical, tabled value, is the a. T b. Spearman r c. U d. none of these 16-35. Establishing whether the difference between two independently selected sample means is significant, requires the a. U test b. H test c. T test d. none of these 16-36. Compared to the parametric tests, the non-parametrics have a. more assumptions b. fewer assumptions c. the same number of assumptions d. all of these, depending on the scale of measurement 16-37. All non-parametric tests demand a. nominal data b. ordinal data c. interval data d. none of these 16-38. When all the assumptions of the interval tests are not met, the researcher should use a. parametric test b. non-parametric tests c. tests of association d. none of these 16-39. Compared to the Pearson r, the Spearman r is a a. more powerful test b. less powerful test c. test of equal power d. none of these, since power cannot be established for the r

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16-40. The ordinal test, cited in the text, for testing the hypothesis of association is the a. Pearson r b. H c. T d. Spearman r 16-41. Regardless of the scale of measurement, the researcher analyzing data from a three-group, between-subjects design, must use the a. T b. U c. H d. none of these 16-42. Using a non-parametric test, when a parametric test should have been used, tends to a. reduce the likelihood of rejecting Ho b. increase the likelihood of rejecting Ho c. over-estimate the difference between the sample mean and the parameter mean d. none of these 16-43. If a Spearman r were used on data requiring a Pearson r, then in order to obtain significance a. a larger sample would be required b. a smaller sample would suffice c. the means of the two samples would have to deviate significantly from each other d. the population mean must be a known value 16-44. For a given sample size, the more powerful a test is, the more it is likely to produce a. significance b. a lack of significance c. a larger beta d. none of these 16-45. The assumption of a normal distribution within the underlying population of scores, is required with the a. Spearman r b. U test c. chi square d. none of these

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16-46. The hypothesis of difference may be tested by using a. the U test b. the H test c. Friedman ANOVA by ranks d. all of these 16-47. In order to test experimental data via the U test, the design must be a. between-subjects b. repeated-measures c. matched-subjects d. none of these, since U may not test experimental data 16-48. In order to test experimental data via the H test, the design must be a. between-subjects b. repeated-measures c. matched-subjects d. both b and c, but not a 16-49. Of the following, which may be used for the analysis of post-facto data? a. Spearman r b. H c. U d. all of these 16-50. In order to test experimental data via the ordinal T test, the design must be a. between-subjects with independent assignment of subjects b. repeated-measures c. matched-subjects d. both b and c, but not a 16-51. In order to test experimental data via the Friedman (ANOVA by ranks), the design must be a. between-subjects with independent assignment of subjects b. repeated-measures c. matched-subjects d. both b and c, but not a 16-52. Whenever there is an increased probability of accepting null when null should have been rejected, then a. the Type 2 error is increased b. the power is decreased c. there is a decreased likelihood of obtaining significance d. all of these 285

16-53. The more powerful a given statistical test is, the more likely it is to a. increase the Type 2 error b. detect significance c. produce non-chance results d. both b and c, but not a 16-54. Of the following, which does not belong with the others? a. U test b. H test c. Pearson r d. T test 16-55. Of the following, which does not belong with the others? a. Pearson r b. Spearman r c. Coefficient of contingency d. U test 16-56. Whenever non-parametrics are used instead of parametrics, and the parametric assumptions are met, then a. power is increased b. power is reduced c. power is eliminated d. power becomes equal to .95 16-57. Under which of the following conditions does an increase in beta lead to an increase in power? a. between-subjects design with ordinal data b. repeated-measures design with interval data c. matched-subjects design with nominal data d. none of these, since the premise is false 16-58. Of the following, which test has the least power? a. independent t b. Within F c. paired t d. U test

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B. True or False: For the following, indicate T (True) or F (False) 16-59. All parametric tests are less powerful than the non-parametric.

16-60. Conditions which tend to increase beta, also increase power.

16-61. Since it has so few assumptions, chi square may always be used on ordinal data.

16-62. Ordinal test require more assumptions than nominal tests.

16-63. Interval data may always be converted into ordinal data.

16-64. Ordinal data may always be converted into interval data.

16-65. All non-parametric tests are also distribution-free.

16-66. The concept of "greater than" or "less than" is at the heart of the ordinal case.

16-67. The more power a test has, the more likely it is to produce significant results.

16-68. If a distribution of interval scores is badly skewed, the scores should be ranked and ordinal tests performed.

16-69. Ordinal data may never be skewed.

16-70.For a given sample size, the Spearman r is more powerful than the Pearson r 16-71. The Friedman ANOVA by ranks does for ordinal data what the U test does for interval data 16-72. The Spearman r does for ordinal data what the Pearson r does for interval data.

16-73. The U test makes no assumptions regarding the population parameters.

16-74. The Type 2 error may only occur when the null hypothesis is accepted.

16-75. Non-parametric tests may never be used on data from post-facto research.

16-76. The Type 2 error is the probability of being wrong whenever the null hypothesis is rejected.

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16-77. The ordinal T for small samples only rejects null when the obtained value is equal to or less then the tabled value.

16-78. One should not use the Spearman r on skewed interval data, unless the data could be converted into ordinal form.

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D. For the following questions, calculate the values.

16-79. For the following distribution of interval scores, indicate, in order, the correct rankings. 110, 55, 55, 50, 42, 42, 42, 30, 27

16-80. For the following distribution of interval scores, indicate, in order, the correct rankings. 799, 522, 515, 515, 515, 515, 515, 500.

For the questions 81 through 86, use the following: A researcher used a one-group, repeated-measure design, and established the following data: Scores Ranked by Rows Subject # Measure 1 Measure 2 Measure 3 1 1 2 3 2 1 2 3 3 2 1 3 4 2 1 3 5 3 2 1 6 1 2 3 7 1 2 3 8 3 1 2 9 1 2 3 10 2 3 1 16-81. Which statistical test should be used?

16-82. Calculate the appropriate statistical vale.

16-83. Indicate the number of degrees of freedom.

16-84. Indicate the tabled value at the .05 level.

16-85. Indicate the tabled value at the .01 level.

16-86. Should Ho be rejected?.

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For questions 87 through 90, use the following A researcher used a three, matched-subjects experimental design, and establishes the following data: Scores Ranked by Rows Group 1 Group2 Group 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 16-87. Which statistical test should be used?

16-88. Calculate the appropriate statistical value.

16-89. Indicate the number of degrees of freedom.

16-90. Should Ho be rejected ?

For questions 91 through 95, use the following: A researcher sets up a one-group, repeated-measures experimental design.

Because the original, interval measures were badly skewed the data were converted into ordinal form.

Subject # Pre-Rank Post-Rank 1 1 1 2 3 2 3 2 7 4 5 3 5 7 4 6 6 8 7 8 9 8 10 5 9 9 6 10 4 10 16-91. Which statistical test should be used?

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16-92. Calculate the appropriate statistical value.

16-93. Indicate the critical, tabled value of T at .05

16-94. Indicate the critical, tabled value of T at .01

16-95. Should Ho be rejected?

For questions 96 through 100, use the following: A researcher sets up a two, matched-subjects experimental design, and collects ordinal measures on the subjects. The data follow: Group 1 Group 2 1 4 2 5 3 6 7 8 9 11 10 12 13 13 8 7 11 9 12 10 4 3 5 2 6 1 16-96. Which statistical test should be used?

16-97. Calculate the appropriate statistical value.

16-98. Indicate the critical, tabled value at .05.

16-99. Indicate the critical, tabled value at .01.

16-100. Should Ho be rejected?

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For questions 101 through 105, use the following: A researcher sets up a three-group, between-subjects design with random assignment of all subjects. Because the interval measures on the dependent variable are badly skewed, the measures are rank-ordered. The data follow: Group 1 Group 2 Group 3 1 3 5 2 4 9 6 8 13 7 12 16 10 14 17 11 15 18 16-101. Which statistical test should be used?

16-102. Calculate the appropriate statistical value.

16-103. Indicate the critical, tabled value at .05.

16-104. Indicate the critical, tabled value at .01.

16-105. Should Ho be rejected?

For questions 106 through 108, use the following: A researcher sets up a three-group, repeated-measures design with random assignments of all subjects. All measures were originally taken in ordinal form. The data follow: Group 1 Group 2 Group 3 1 3 5 2 4 6 7 9 11 8 10 12 13 14 15 16 18 17 16-106. Which statistical test should be used?

16-107. Calculate the appropriate statistical value.

16-108. Should Ho be rejected?

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For questions 109 through 112, use the following: A researcher wishes to find out if a speed-reading course increases reading speed. Sixteen subjects were randomly selected from the adult population and then randomly divided into two groups. Group A was given a speed-reading course, and then, because the interval scores were badly skewed, each subject was ranked according to reading speed. Their ranks were 1, 3, 4, 6, 8, 9, 10, 15. Group B received no training, and their ranks on the same test were 2, 5, 7, 11, 12, 13, 14, 16.

16-109. Identify the type of research.

16-110. Which statistical test should be used?

16-111. Calculate the appropriate statistical value.

16-112. Should Ho be rejected?

For questions 113 and 114, use the following: A researcher believes that athletic skill in basketball is a function of "handedness". A random sample of 24 professional basketball players was selected from the N.B.A., and divided into two groups according to preferred hand for shooting foul shots. The players were than ranked according to ability by a panel of sports writers. The data follow: Prefer Right Hand Prefer Left Hand 4 1 5 2 7 3 10 6 11 8 13 9 14 12 17 15 18 16 19 21 20 22 23 24 n1 = 13

n2=11

16-113. Identify the type of research.

16-114. Which statistical test should be used, and what are the resulting values?

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CHAPTER 17 TESTS AND MEASUREMENTS A: SUGGESTIONS FOR CLASS ACTIVITIES Activity: Library Sources and the Internet One of the best ways to acquaint your students with the basics of measurement theory, including test reliability and validity is to let them see both the theoretical and the practical side of the testing enterprise by reading some of the volumes edited by O.K. Buros. Send your students to the library to look up some of the major test data contained in your library's latest edition of the Mental Measurements Yearbook, prepared by O.K. Buros and published by Gryphon Press of Highland Park, New Jersey. Over the years Buros has also prepared a volume on Vocational Tests and Reviews as well as one on Personality Tests and Reviews, all published by Gryphon Press. The Buros volumes are a rich resource for discovering the research results and clinical uses of so many of our standard tests. It can become very interesting to the students to see in print some of data supporting these various tests, tests that many of them have already heard of or have even taken at some point in their lives. To be able to see first hand that a given test was standardized using a test-retest reliability coefficient, or a split-half correlation (dutifully corrected with the Spearman-Brown) puts the text book pages in a far more realistic light. In fact it may come as a shock to some of them when they discover that certain famous tests, such as the Rorschach ink blot test, are not always blessed with unequivocally respectable reliability or validity values.

Another treasure trove of test information comes from a book by Spreen and Strauss, titled "A Compendium of Neuropsychological Tests", published by Oxford University Press in 1991. The coverage in this book provides information on the following: test name, purpose of the test, source of the test, description of the test, administration, approximate time for administration, scoring, comment and evaluation, and normative data (including the available studies of the test's validity and reliability). Students not only learn about some of the standardizing techniques used in the Psych and Ed Biz, but also become acquainted with and often interested in the large number of tests available for evaluating neuropsychological problems. A student-led class discussion of one or two of these tests often provides fertile soil for planting the seeds of psychological interest among motivated students.

Finally, an incredibly powerful tool for digging out test information is the internet.

Psychological tests of all types can be found with just a few key strokes, perhaps only a “Google” away. Care must be taken, however, to always check the references in order to ensure that you’re not just being blogged.

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Activity: Creating a Test An exceptionally powerful technique for generating interest and understanding of the basic principles of testing is to have your students, perhaps in groups, create a new test. The test can be designed to evaluate virtually anything their creative minds can dream up. In one class, two of my students set about to evaluate what they perceived to be possible personality differences among various position players on the college football team. Admittedly not all the personal profile items included in the test met the stringent standards of the Psychological Corporation. For instance, they decided to test for a personal dimension they called "sloppiness", and incidentally attempted to validate it by having a judge rate the neatness of each player's dorm room and locker.

This led to some interesting arguments between player and judge (who happened to be one of the assistant coaches). In any case, the overall activity of making up the questions, administering the test, scoring the test and attempting to assess both its reliability and validity was an eye-opening educational experience. They couldn't wait to run the stats and they even confessed to having visions of Psych. Corp. knocking on their door and offering them a big cash advance for the rights to their test.

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B. Multiple Choice Items 17-1. When a test has been shown to be reliable, it means that its results a. are stable and consistent b. are measuring what the test was designed to measure c. are automatically assumed to be valid d. both a and b, but not c 17-2. The test-retest method can be used to assess a test's a. predictive validity b. reliability c. construct validity d. all of these, depending on what statistical tests are used 17-3. The split-half method can be used to assess a test's a. predictive validity b. construct validity c. reliability d. all of these, depending on what statistical tests are used 17-4. The Spearman-Brown prophecy formula should not be used unless the correlation has been obtained on the basis of the a. test-retest technique b. alternate forms technique c. predictive validity technique d. split-half technique 17-5. When a test has been shown to be measuring what it purports to measure, then we may assume that the test is a. reliable b. valid c. containing only homogeneous items d. containing only heterogeneous items 17-6. All tests that have been shown to be statistically reliable, must also be a. valid b. norm-referenced c. criterion-referenced d. none of these

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17-7. When test scores are correlated with an already established and accepted measure of the construct under study, then the procedure was used to assess a. predictive validity b. construct validity c. concurrent validity d. none of these, since the procedure outlined above is used to assess reliability, not validity 17-8. When a correlation is established between the odd and even-numbered items, then the procedure was used to assess a. predictive validity b. construct validity c. concurrent validity d. none of these, since the procedure outlined above is used to assess reliability, not validity 17-9. The Spearman-Brown prophecy formula is used to correct for a. an over-estimate of the original correlation b. an under-estimate of the original correlation c. the lack of any degrees of freedom d. having previously eliminated the alpha error 17-10. When test scores have been shown to accurately predict later performance, then the test is considered to have a. predictive reliability b. predictive validity c. internal consistency d. both a and b, but not c 17-11. Unless the Spearman-Brown is used, split-half reliability values tend to be a. lower than test-retest values b. higher than test-retest values c. identical with test-retest values (for the same test) d. sometimes higher and sometimes lower, depending on which trait is being measured 17-12. An example of an internal consistency reliability method would be the a. test retest b. split half c. alpha omega d. alpha beta

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17-13. IRT, or Item Response Theory, attempts to evaluate a test on the basis of a. item difficulty b. item discrimination c. item length d. both a and b, but not c.

17-14. When each item in a test is evaluated on the basis of the proportion of persons in the standardizing group who answered the question correctly, then the researcher is attempting to establish a. item difficulty b. item discrimination c. item generalization d. both a and b, but not c.

17-15. When engaging in item analysis techniques, the researcher treats each item as a. a criterion based score b. a criterion against which all other items must be compared c. a subtest in a larger composite test. d. a robust correlation 17-16. If the proportion of persons answering a given item correctly were to reach 95%, then the item would be assigned a. a .95 confidence level b. a .95 alpha error c. a .95 discrimination index d. a .95 difficulty index 17-17. For a given item, when the performances of those whose overall scores put them among the highest third of the group are compared with those who scored among the lowest third of the group, the resulting difference is called the a. discrimination index (DI). b. item difficulty level (IDL) c. item response test (IRT) d. discrimination of performance evaluation (DOPE) 17-18. An item would be considered extremely difficult if it had a difficulty index of a. .99 b. .50 c. .01 d. none of these, since item difficulty is not a quantified value.

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17-19. For the majority of the items in any given test, the difficulty index should be kept at approximately a. .99 b. .50 c. .01 d. none of these, since item difficulty is not a quantifiable value.

17-20. One popular statistical technique for evaluating the level of item discrimination is the a. standard error of measurement b. standard error of the mean c. standard error of estimate d. point biserial correlation 17-21. An attempt to determine whether a given item separates the high from the low performers on the overall test is called the a. discrimination index (DI). b. item difficulty level (IDL) c. item response test (IRT) d. discrimination of performance evaluation (DOPE) 17-22. The Kuder-Richardson formula (KR21) presented in the chapter is used for establishing a. test validity b. internal consistency reliability c. test retest reliability d. predictive validity 17-23. A method for assessing the overall reliability of a test by splitting it into all possible halves is called the a. point biserial b. discrimination index (DI) c. vix formula 44 d. Kuder-Richardson formula (KR21) 17-24. When predicting all possible scores a given individual could have obtained around the actual score that was obtained, we can use the a. standard error of measurement b. standard error of the mean c. standard error of estimate d. point biserial correlation

299

17-25. In order to predict how much an individual's score might change on a retest, we can use the a. standard error of measurement b. standard error of the mean c. standard error of estimate d. point biserial correlation 17-26. When the internal consistency of a test is being determined on the basis of the Kuder-Richardson formula (KR21), then it should NOT be followed by a. any attempts to determine predictive validity b. any attempts to determine concurrent validity c. the Spearman-Brown prophecy formula d. none of these, since the KR21 is NOT an internal consistency measure 17-27. If a certain test had a reliability value of .95, then its validity must be at least a. .95 b. .05 c. .50 d. none of these, since the reliability of a test tells us nothing about its validity.

17-28. If a certain test had a reliability value of .95, then we know that a. its validity must also be high b. its validity must be low c. the test is measuring something very consistently, even though we don't necessarily know what it is that's being measured d. none of these, since a test could never have a reliability of .95

17-29. If a certain test had a reliability value of 2.95, then we know that a. its validity must also be high b. its validity must be low c. the test is measuring something very consistently, even though we don't necessarily know what it is that's being measured d. none of these, since a test could never have a reliability of 2.95 17-30. In order to use the Spearman-Brown prophecy formula, the original Pearson r must be a. significant b. obtained on a test retest correlation c. based on at least 18 degrees of freedom d. both a and b, but not c.

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True or False 17-31. The correlation between the odd and even numbered items for a given test provides an estimate of the test's reliability.

17-32. The correlation between the odd and even numbered items for a given test underestimates the test's reliability unless corrected for by a technique such as the Spearman-Brown prophecy formula.

17-33. The test re-test method is used to assess a test's validity.

17-34. The split-half method is used to assess a test's construct validity.

17-35. Correlating test scores with an independent measure of the trait being measured provides an estimate of the test's validity.

17-36. Test reliability provides an estimate of the stability and consistency of the measures.

17-37. Test validity provides an estimate of whether or not the test is measuring what it has been designed to measure.

17-38. Correlations that are used to assess reliability typically result in lower values than those used to assess validity.

17-39. A given test may be reliable, and yet still not be valid.

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For items 40 through 43, find the values.

17-40. A psychologist wishes to establish the reliability of a new IQ test that is based on a variety of brain measures (including neural conduction speed and the rate at which glucose in the brain is metabolized). These brain measures are factored together to provide a single scaled score for each subject. The subjects were randomly selected from a population of young adults and given the same brain test twice. The data are as follows:

TEST RETEST Subject # Brain Scaled Score Brain Scaled Score 1.

50 55 2.

60 57 3.

65 64 4.

40 45 5.

35 40 6.

40 38 7.

60 57 8.

55 54 9.

50 48 10.

45 45 11.

52 50 12.

48 50 13.

51 52 14.

49 50 15.

50 50 16.

50 49 17.

37 40 18.

63 60 19.

50 50 20.

50 50 What is the reliability of this new test.

17-41. What type of reliability procedure was used in this example?

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17-42. The researchers then wanted to assess the test for validity, and so each subject was given the WAIS-R IQ test and their Test (not the retest) scores were compared with their WAIS-R IQ scores.

Subject # Brain Scaled Score WAIS-R IQ 1.

50 100 2.

60 115 3.

65 120 4.

40 96 5.

35 100 6.

40 85 7.

60 115 8.

55 105 9.

50 103 10.

45 90 11.

52 120 12.

48 105 13.

51 110 14.

49 95 15.

50 100 16.

50 100 17.

37 95 18.

63 112 19.

50 100 20.

50 100 What is the Pearson correlation between the two sets of test scores?

17-43. Was the correlation significant?

17-44. Was the test statistically valid?

17-45. What type of validity procedure was used in this example?

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17-46. A researcher in the area of cognitive functioning has designed a new test of creativity, called the CIA test (Cognitive Information Assessment). A random sample of ten, five-year-old children is selected and given the test. A week later the same test was administered for a second time.

Estimate the reliability of this test.

Subject # Test Retest X Y 1 .

30 33 2 .

10 15 3 .

20 18 4 .

18 20 5 .

22 16 6 .

23 21 7 .

17 15 8 .

20 22 9 .

25 28 10 .

15 15 17-47. A researcher wishes to estimate the reliability of the new test of a person’s sense of humor, the LOL test (Lambda Orientation Level). The test is composed of a total of 40 items, and the researcher splits the scoring in order to compare each subject's score on the odd numbered items with those on the even numbered items. A total of 10 adult males was randomly selected and their scores were as follows: Odd Even Subject # X Y 1 15 17 2.

10 13 3.

5 8 4.

8 8 5.

12 12 6.

13 13 7.

7 10 8.

10 8 9.

11 12 10.

9 9 Estimate the reliability of the whole test.

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17-48. A researcher wishes to estimate the reliability of the FYI (Functional Youth Indicator) - a new test of adolescent shyness. The test is composed of a total of 30 items, and the researcher splits the scoring in order to compare each subject's score on the odd numbered items with those on the even numbered items. A total of 10 teen-age males was randomly selected and their scores were as follows:

Subject # 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Odd X 10 8 12 10 11 9 10 15 5 10

Even Y 9 11 12 9 10 10 9 14 6 10

Estimate the reliability of the whole test.

17-49. Find the whole test reliabilities for the following split-half correlations, all of which were calculated on the basis of a sample of 30 persons. a. .63 b. .93 c. .33 d. .78 17-50. A test constructor is interested in discovering how long a set of sub-tests must be to reach a reliability value of .85. For each sub-test, the following significant correlations were found: a. .739 b. .653 c. .585 d. .790

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17-51. A researcher wishes to establish the DI (Discrimination Index) for a series of five items contained in a new test being developed to assess Whole Language Readiness (WLR). The data are as follows: Item Proportion Correct Proportion Correct For Top Third For Bottom Third a. .70 .60 b. .70 .50 c. .50 .20 d. .95 .70 e. .45 .50

17-52. A test constructor is interested in determining the reliability of a certain item on a new test of aggressiveness, the ASAP (Aggressive Syndrome Assessment Profile). The items were scored 1 for showing aggressiveness and 0 for passivity. For each subject, the total score on the test is reported, high scores indicating higher levels of aggressiveness.

Subj Item Score Total Score X Y 1.

1 10 2.

1 12 3.

1 10 4.

1 11 5.

1 15 6.

0 4 7.

0 5 8.

0 6 9.

0 9 10.

0 7 Find the point-biserial reliability value for the selected item.

306

17-53. A researcher is interested in determining the reliability of a certain item on a test of one’s potential for learning a foreign language, the SMELL test (Sensory Modality for Elevated Language Learning). The item was scored 1 for showing a high potential and 0 for a non-potential response. For each subject, the total score on the test is reported, high scores indicating higher levels of language potential.

Subj. # Item Score X 1.

1 2.

1 3.

1 4.

1 5.

1 6.

1 7.

0 8.

0 9.

0 10.

0

Total Score Y 40 35 29 25 25 28 15 24 22 10

17-54. A new test, the NCAA (Number-Concept-Awareness-Ability) is being standardized for college athletes majoring in math, and it has been found to have a mean of 85 and an SD of 15. The test is composed of 125 items. Find its KR21 reliability.

17-55. Another researcher attempted to shorten the NCAA test shown above and reduced the number of items to only 20. The mean on this revision was 10 with an SD of 3. Find its KR21 reliability.

17-56. A researcher wants to assess the reliability of a new test for diagnosing a teenager’s predisposition for drinking wine, the BYOB (Behavioral Youth Oenology Battery). A random sample of 10 sixteen-year-old females was selected and given the BYOB test, higher scores indicating higher levels of receptivity to wine. Six months later they were given the test again. Was the test reliable? Their test and retest scores follow: Subj# 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

BYOB Test 5 2 3 3 1 2 4 3 3 4

BYOB Retest 9 3 7 4 3 3 5 7 4 5 307

a. Find the rtt reliability for this test. b. Calculate a paired t between the tow sets of scores.

17-57. A researcher wants to assess the validity of a test of "Childhood Separation Anxiety" (CSA). A random sample of 10 four-year-old children was selected and given the CSA test, an oral test given by a trained examiner (higher scores indicating more anxiety). Later that same day, a group of three judges observed the children in staged interactions with their respective mothers and rank ordered each child with regard to his or her perceived separation anxiety. The data follow: Subj.# CSA Score Anxiety Rank 1.

100 3 2.

38 1 3.

35 8 4.

35 6 5.

10 7 6.

28 10 7.

37 5 8.

33 9 9.

30 4 10 .

35 10 a. Is the test statistically valid? b. What type of validity would this be?

17-58. For a given administration of the welcome-back-to-school test, the KABC (Kotter Assessment Battery for Children), the mean was 100 with an SD of 15. The test-retest reliability was .90. Find the standard error of measurement.

17-59. Using the standard error of measurement found above, find the confidence interval at the .95 level for the following KABC scores: a. 80 b. 90 c. 100 d. 110 e. 120 17-60. For a given administration of the KISS (Kelly Impulsivity Standards Scale), the mean was 50 with an SD of 10. The test-retest reliability was .85. Find the standard error of measurement.

308

17-61. Using the standard error of measurement found above, find the confidence interval at the .95 level for the following KISS scores : a. 40 b. 45 c. 50 d. 55 e. 57 17-62. The standard error of measurement on a certain test has been found to be 10 with an SD of 15.00. Using the Rulon formula, find the rtt (reliability value) for the test.

17-63. A certain test has been found to have an SD of 50 and a standard error of measurement of 32. Using the Rulon formula, find the rtt (reliability value) for the test.

17-64. Suppose that you as a statistical consultant were given the following test data from a researcher: M = 400, SD = 32 and the standard error of measurement = 100.

You are asked to use the Rulon formula for finding the test's reliability.

17-65. The reliability of a certain test is .80, whereas the reliability of the criterion is only .30. What is the highest possible validity value for this test?

17-66. The reliability of a certain test is .90, whereas the reliability of the criterion is .85. What is the highest possible validity value for this test?

17-67. A testing specialist has created a four item test of aggressiveness and given it to a random sample of 5 subjects. The test is scored only on the basis of whether the item indicates aggressiveness or whether it does not. It was scored as 1 for aggressiveness and 0 for no aggressiveness. Using the Cyril Hoyt ANOVA, find the internal consistency reliability of the test.

Items Subjects a b c d 1 1 1 1 1 2 0 1 1 1 3 0 0 0 1 4 0 1 1 0 5 0 1 1 0

309

17-68. A new test, designed to test parental permissiveness among parents of K through 4th grade children, was developed and you as the statistical consultant have been asked to assess its reliability. The test is composed of 4 items and was given to a random sample of 5 parents. The items on the test were scored from 1 to 10, with 10 showing the most permissiveness. Using the Cyril Hoyt ANOVA, find the internal consistency reliability of the test.

Items Subjects a b c d 1 8 10 9 8 2 10 8 9 10 3 6 6 7 4 4 3 3 7 5 5 1 2 5 8

17-69. A forensic psychologist is attempting to assess the reliability of a new test of psychopathy. A random sample of inmates from a large state prison, all of whom had been convicted of violent offenses was selected and given the test. Assume that the test is composed of five items. The test items were scored from 1 to 10, with a 10 assumed to define the highest level of psychopathy.

Items Subjects a. b. c. d e.

1 1 3 2 6 2 2.

7 9 8 10 9 3.

5 6 7 4 5 4.

8 8 9 10 8 5.

3 4 5 4 4 6.

2 6 5 3 2 7.

7 6 8 6 7 8.

9 10 10 10 9 9.

5 5 6 5 4 10.

7 8 5 7 6 Calculate coefficient alpha and determine the test’s level of reliability.

17-70. Is the test proving to be reliable? .

310

17-71. A test specialist is attempting to find the reliability of a test designed to assess drug dependence. Ten subjects were randomly selected from a group of individuals. all of whom on a waiting list for a treatment program for drug abuse. The subjects were all given the test and then a week later were retested.

The scores were as follows: Subject# 1 2 3 4 5 6 7 8 9 10

Test 45 35 50 67 45 55 65 50 60 45

Retest 50 32 54 70 40 60 62 48 70 40

Find the test-retest reliability value.

17-72. Is it significant?

17-73. Is the test proving to be reliable? .

311

COMPUTER PROBLEM A forensic psychologist is attempting to assess the validity of a new test of police aptitude. The test is given to a random sample of first-day trainees during their orientation session at the police academy. At the end of the 2 month training program each subject is evaluated by the academy’s director and rank-ordered on the basis of overall proficiency. The data follow.

Subj# 1 2 3 4 5 6 7 8 9 10

Test Score 55 60 45 75 43 34 48 62 50 46

Ranking 8 3 6 2 5 10 7 1 4 9

17-74. Find the Spearman correlation.

17-75, Is it significant? .

17-76. Is this a valid test? .

Point out to your students that this is actually a positive correlation as far as the validity is concerned. The only reason the negative sign appears is because in the ranking process, the highest score gets a 1 (a metrically low number) and the lowest score gets a 10 (a metrically higher number). Ask your students to rank order the test scores, and then again compute the Spearman, using only the two sets of ranks. At that point, that correlation of -.733 now has a positive sign.

312

COMPUTER PROBLEM A psychologist wants to assess the reliability of a new test of Self Control, the higher the scores the higher the level of control. A random sample of college students is selected and given the test. In order to tap the reliability, the test is scored on the basis of comparing the results of the odd items versus the even items. The data follow: Subject# Odd Even 1 45 57 2 50 48 3 35 57 4 62 55 5 55 60 6 38 50 7 55 50 8 45 38 9 65 70 10 55 50 17-77. What is the correlation between the odd and even items?

17-78. Is it significant? .

17-79. What is the correlation when corrected by the Spearman-Brown?

17-80. Is it significant? .

17-81. Is this a reliable test? .

313

Special Unit 2 Questions SU2-1When the Levene test shows a significance level of greater than .05, a. the sample groups have equal variablility. b. the sample groups have unequal variablility. c. The sample groups have significantly different means. d. The sample means do NOT differ significantly.

SU2-2 When the Levene test shows a significance level of .05 or less, a. the sample groups have equal variablility. b. the sample groups have unequal variablility. c. the sample groups have significantly different means. d. the sample means do NOT differ significantly.

SU2-3 When using the Fw (Welch) ANOVA, rather than the traditional F ratio, the degrees of freedom a. are always increased. b are always reduced. c are not affected by the choice of F ratios. d none of these, since the F ratio does not utilize degrees of freedom.

SU2-4. Statistical tests that don’t have to adhere to all the stringent parametric criteria, are called a. random. b. significant. c. meaningful d. robust SU2- 5. The scores used to compute the Levene test are always based on measures of the a. independent variable b. dependent variable c. controlled variable d. none of these, since the variables used by the Levene are unknown.

SU2-6. In general, parametric tests are preferred over non-parametrics because they are a. less affected by the lack of strict requirements b. more affected by random error.. c. less affected by random error d. more powerful

314

SU2-7. As degrees of freedom are reduced, parametric tests become .a. less likely to establish significance b. less likely to reject the null hypothesis c. less powerful d. all of these SU2-8 With parametric tests, large sample sizes a. produce more degrees of freedom b. produce fewer degrees of freedom c. do not influence the degrees of freedom. d. all of these, depending on whether the results are significant.

SU2-9. When degrees of freedom are reduced, significant differences become a. less likely b. more likely c. equally likely d. either b or c, but not a.

SU2-10 When a researcher changes the type 1 error (alpha) from .05 to .01. significant differences become a. less likely b. more likely c. equally likely d. either b or c, but not a.

SU2-11. Using t’, rather than Student’s t, accounts for the possibility of a. unequal group variances b. equal group variances c. equal group variances as long as the sample sizes differ d. None of these, since t’ does not include a variance estimate.

SU2-12. Using Fw (the Welch F ratio) rather than the traditional F ratio accounts for the possibility of a. unequal group variances b. equal group variances c. equal group variances as long as the sample sizes differ d. None of these, since F ratios never include a variance estimate.

SU2-13. When trying to determine equal variability between two groups, it is best to compare a. the two means b. the two medians c. the two modes d. the two squared standard deviations 315

SU2-14 The lack of equal variability becomes more of a threat when a. the sample sizes differ and the larger sample has more variability b. the sample sizes differ and the smaller sample has more variability c. the sample sizes are equal d. none of these, since the lack of equal variability can only occur when sample sizes reach an N of at least 30.

SU2-15 When a Levene probability reaches .01 or LESS a. the samples have equal variability b. the samples have unequal variability c. the sample sizes are significantly too small. d. the sample sizes are significantly too large.

316

CHAPTER 1 ANSWERS 1-1 c 1-2 a 1-3 d 1-4 d 1-5 b 1-6 d 1-7 a 1-8 d 1-9 a 1-10 b 1-11 d 1-12 d 1-13 b 1-14 b 1-15 a 1-16 d 1-17 F 1-18 T 1-19 T 1-20 T

1-21 1-22 1-23 1-24 1-25

F T T T F

317

CHAPTER 2 ANSWERS 2-1 d 2-21 2-2 a 2-22 2-3 a 2-23 2-4 c 2-24 2-5 d 2-25 2-6 d 2-26 2-7 b 2-27 2-8 a 2-28 2-9 a 2-29 2-10 c 2-30 2-11 d 2-31 2-12 a 2-32 2-13 a 2-33 2-14 c 2-34 2-15 a 2-35 2-16 d 2-36 2-17 b 2-37 2-18 c 2-38 2-19 d 2-39 2-20 c 2-40 2-61 2-62 2-63 2-64 2-65 2-66 2-67 2-68 2-69 2-70 2-71 2-72 2-73 2-74 2-75

a b d a d c c c d c a a a b a b a a a a

2-41 2-42 2-43 2-44 2-45 2-46 2-47 2-48 2-49 2-50 2-51 2-52 2-53 2-54 2-55 2-56 2-57 2-58 2-59 2-60

T M=5.33, Mdn = 4.50, Mo=3.00 M=13.13, Mdn= 12.00, Mo = 10.00 M = 5.83, Mdn = 6.00, Mo = 7.00 M = 6.33, Mdn = 5.00, Mo = 5.00 M = 7.17, Mdn = 7.50, Mo = 8.00 M = 5.83, Mdn = 5.50, Mo = None M = 10.20, Mdn = 10.00, Mo = 10.00 M = 6.97 Mdn = 5.50 Mo = 5.00 The Mean The Mode Skewed to the right Sk+ M = 5.85, Mdn = 5.50

318

a d c a b d c d a F F T T F T T T F T T

CHAPTER 3 ANSWERS 3-1. d 3-21. d 3-2. c 3-22. b 3-3. d 3-23. d 3-4. b 3-24. a 3-5. a 3-25. d 3-6. a 3-26. d 3-7. d 3-27. d 3-8. b 3-28. a 3-9. d 3-29. d 3-10. c 3-30. d 3-11. a 3-31. c 3-12. b 3-32. c 3-13. d 3-33. a 3-14. d 3-34. b 3-15. d 3-35. b 3-16. c 3-36. b 3-17. b 3-37. b 3-18. d 3-38. a 3-19. d 3-39. d 3-20. a 3-40. a

3-41. d 3-42. d 3-43. b 3-44. a 3-45. b 3-46. a 3-47. b 3-48. c 3-49. c 3-50. c 3-51. c 3-52. a 3-53. c 3-54. b 3-55. b 3-56. F 3-57. T 3-58. F 3-59. T 3-60. T

3-61, F 3-62. F 3-63. T 3-64. T 3-65 T 3-66. F 3-67. T 3-68 F 3-69. T 3-70. T 3-71. T 3-72. F 3-73. T 3-74. 9.05 3-75 9 3-76. 9 3-77 .16 3-78. 3.442 3-79. Sk = 0 3-80. 2.67

3-81. Slightly platykurtic 3-82. Two scores 3-83. M = 12.00 3-84. Mdn = 11.50 3-85. Mo = 12.00 3-86. R = 12 3-87. SD = 2.79 3-88. V = 7.78 3-89. M = 7.88, R = 10, SD = 3.43 3-90. M = 7.25, SD = 3.27 3-91. M = 11.86, SD = 2.46 3-92. M = 6.10, SD = 2.66 3-93. M = 12.00, SD = 3.643 3-94. M = 12.222, SD = 4.685 3-95. M = 50.000 3-96. Mdn = 49.000 3-97. Sk = 1.867 3-98. Ku = 4.744 3-99. M = 5.000 3-100. Mdn = 5.000 3-101. Sk = .117 3-102. Ku = -.394 319

CHAPTER 4 ANSWERS 4-1 d 4-2 d 4-3 c 4-4 a 4-5 d 4-6 a 4-7 c 4-8 b 4-9 d 4-10 c 4-11 a 4-12 d 4-13 c 4-14 d 4-15 d 4-16 a 4-17 d 4-18 b 4-19 d 4-20 c

4-21 4-22 4-43 4-24 4-25 4-26 4-27 4-28 4-29 4-30 4-31 4-32 4-33 4-34 4-35 4-36 4-37 4-38 4-39 4-40

4-61 4-62 4-63 4-64 4-65 4-66 4-67 4-68 4-69 4-70 4-71 4-72 4-73 4-74 4-75 4-76 4-77 4-78 4-79 4-80

4-81 T 4-82 43.83% 4-83 45.82% 4-84 21.23% 4-85 99.48% 4-86 92.51% 4-87 40.52% 4-88 7.49% 4-89 74.86% 4-90 57.62% 4-91 20.29% 4-92 50.14% 4-93 11.04% 4-94 18.92% 4-95 7.82% 4-96 32.02% 4-97 16.73% 4-98 21.85% 4-99 27.84% 4-100 16.25%

T T T T F T T F F F F T F T T F F T F F

d c a a c a b d a a d d d c a c a d a c

320

4-41 4-42 4-43 4-44 4-45 4-46 4-47 4-48 4-49 4-50 4-51 4-52 4-53 4-54 4-55 4-56 4-57 4-58 4-59 4-60

d d c d a a d d a d d a c d b d a d a a

Chapter 4 Continued 4-101 74.86% 4-102 46.55% 4-103 61.79 or 62nd percentile 4-104 88.69 or 89th percentile 4-105 38.21 or 38th percentile 4-106 3.44 or 3rd percentile 4-107 21.48 or 21st percentile 4-108 99.68 or 100th percentile 4-109 57.62% 4-110 75.80 or 76th percentile 4-111 84.14 or 84th percentile 4-112 42.07 or 42nd percentile 4-113 69.15 or 69th percentile 4-114 96.41 or 96th percentile 4-115 99.39 or 99th percentile 4-116 2.8137, .00000, .00000, -2.81357, -2.11018, -.28136, .28136, -.14068, .14068, .00000, .42204, -.42204, .14068, -.14068, .28136, -.28136, 1.40679, -1.40679, -.28136, .28136, .00000, .42204, -.42204, .14068, -.14068, .00000, .84407, -.84407, 1.12543, 1.12543, .00000, .14068, -.14068, .00000, 4-117 .00000, -.20755, .20755, -.10377, .10377, .41510, -1.55662, 1.55662, 2.07550, -2.07550, .20755, .00000, .20755, .00000, Point out to your students that the z scores they found that were identical, save for sign, (such as -.20755 and .20755) were caused by the fact that they are based on raw score values that are exactly the same distance from the mean (as in the case just shown, which has a mean of 500 and is based on raw scores of 480 and 520). This helps reinforce the z score definition as a value found by subtracting the raw score from the mean and then dividing by the standard deviation.

321

CHAPTER 5 ANSWERS 5-1 d 5-2 c 5-3 b 5-4 a 5-5 a 5-6 c 5-7 c 5-8 a 5-9 c 5-10 d 5-11 d 5-12 d 5-13 a 5-14 a 5-15 b 5-16 b 5-17 b 5-18 a 5-19 b 5-20 a 5-61 5-62 5-63 5-64 5-65 5-66 5-67 5-68 5-69 5-70 5-71 5-72 5-73 5-74 5-75 5-76 5-77 5-78 5-79 5-80

F T T T F F T F T T X=254 X=113.40 X=81.55 5.48 or 5th X=87.80 X=77.15 X=83.40 X=114.32 X=30.94 X=79.00

5-21 5-22 5-23 5-24 5-25 5-26 5-27 5-28 5-29 5-30 5-31 5-32 5-33 5-34 5-35 5-36 5-37 5-38 5-39 5-40

d d c a d d d b d a d a d d c a d a c d

5-41 5-42 5-43 5-44 5-45 5-46 5-47 5-48 5-49 5-50 5-51 5-52 5-53 5-54 5-55 5-56 5-57 5-58 5-59 5-60

5-81 5-82 5-83 5-84 5-85 5-86 5-87 5-88 5-89 5-90 5-91 5-92

X=99.75 53.96 inches M=100.88 M= 69.12 M = 159.20 M=69.26 M=13.52 SD= 8.85 SD = 46.43 SD= 17.86 SD=35.00 SD=6.25 5-93 T= 62.80 T= 49.00 5-95 T= 75.80 5-96 T= 33.50 5-97 T= 50.00 T=67.60 5-99 T=40.20 5-100 T=38.40

5-101 T=73.60 5-102 X=77.15 5-103 X=102.36 5-104 X=635.00 5-105 X=55.09 5-106 M=60.18 5-107 M=142.20 5-108 M=100.00 5-109 M=26.75 5-110 M=58.00 5-111 M=500.50

5-94

5-98

322

c a a d c c d d a d T F F T F T T T T F

CHAPTER 6 ANSWERS 6-1 c 6-2 a 6-3 b 6-4 d 6-5 d 6-6 a 6-7 a 6-8 d 6-9 c 6-10 c 6-11 a 6-12 b 6-13 d 6-14 d 6-15 a 6-16 b 6-17 d 6-18 a 6-19 b 6-20 d 6-61 .74 6-62 .02 6-63 .05 6-64 .41 6-65 .38 6-66 .08 6-67 .35 6-68 .29 6-69 .15 6-70 .03 6-71 .32 6-72a .0120 6-72b .0159 6-72c .0119 6-72d .0018 6-73a .012 6-73b .022 6-73c .024 6-73d .022 6-74 a

6-21 6-22 6-23 6-24 6-25 6-26 6-27 6-28 6-29 6-30 6-31 6-32 6-33 6-34 6-35 6-36 6-37 6-38 6-39 6-40

c a b b b c a b c d d d c a c d d d a b

6-75 6-76 6-77 6-78 6-79 6-80

d b a b a d

323

6-41 F 6-42 T 6-43 F 6-44 T 6-45 T 6-46 T 6-47 F 6-48 F 6-49 F 6-50 T 6-51 T 6-52 T 6-53 F 6-54 F 6-55 F 6-56 T 6-57 T 6-58 T 6-59 F 6-60 .12

CHAPTER 7 ANSWERS 7-1 c 7-2 d 7-3 a 7-4 d 7-5 a 7-6 b 7-7 b 7-8 d 7-9 d 7-10 a 7-11 b 7-12 b 7-13 a 7-14 d 7-15 d 7-16 a 7-17 d 7-18 a 7-19 c 7-20 a 7-61 7-62 7-63 7-64 7-65 7-66 7-67 7-68 7-69 7-70 7-71 7-72 7-73 7-74 7-75 7-76 7-77 7-78 7-79 7-80

F T F T T T F T F F T p=.09 p=.12 p=.15 p=.22 p=.31 p=.16 SEm=2.25 SEm=4.75 R = 600

7-21 7-22 7-23 7-24 7-25 7-26 7-27 7-28 7-29 7-30 7-31 7-32 7-33 7-34 7-35 7-36 7-37 7-38 7-39 7-40

b b a d c a c d b d a d d d a b a d d c

7-41 7-42 7-43 7-44 7-45 7-46 7-47 7-48 7-49 7-50 7-51 7-52 7-53 7-54 7-55 7-56 7-57 7-58 7-59 7-60

7-81 SEm=.42 7-82a SEm=2.91 7-82b z=3.45 Reject Ho p<.05 7-83a SEm=2.37 7-83b z=2.11 Reject Ho p<.05 7-84a SEm = .29 7-84b z=1.72 Accept Ho ns 7-85a SEm= .67 7-85b z=-4.10 Reject Ho p<.01 7-86a SEm= .20 7-86b z=10.00 Reject Ho p,.01 7-87a SEm=.15 7-87b z=13.33 Reject Ho p,.01 7-88a SEm=1.74 7-88b z=1.17 Accept Ho ns 7-89 34.57 or 35 children 7-90 96.04 or 96 children 7-91 59.908 or 60 children 7-92 166.41 or 166 children 7-93a SEm=.145 7-93b z=13.793 Reject Ho p<.01

324

a c a b c c a c d c a F T T F F T T F T

CHAPTER 8 ANSWERS 8-1 b 8-2 c 8-3 a 8-4 d 8-5 d 8-6 a 8-7 d 8-8 c 8-9 d 8-10 d 8-11 d 8-12 a 8-13 a 8-14 b 8-15 a 8-16 a 8-17 d 8-18 a 8-19 d 8-20 d

8-21 8-22 8-23 8-24 8-25 8-26 8-27 8-28 8-29 8-30 8-31 8-32 8-33 8-34 8-35 8-36 8-37 8-38 8-39 8-40

8-61 8-62 8-63 8-64 8-65 8-66 8-67 8-68 8-69 8-70 8-71 8-72 8-73 8-74 8-75 8-76 8-77 8-78 8-79 8-80

8-81 T 8-82 T 8-83 F 8-84 F 8-85 F 8-86 F 8-87 T 8-88 F 8-89 T 8-90 F 8-91 T 8-92 F 8-93 F 8-94 F 8-95 T 8-96 1.940 8-97 .610 8-98 1.790 8-99 1.800 8-100 .9450

T F T T T F F F F T F T T F F F F T T T

325

b a a a a d b a a a a b a c b d d a b d

8-41 8-42 8-43 8-44 8-45 8-46 8-47 8-48 8-49 8-50 8-51 8-52 8-53 8-54 8-55 8-56 8-57 8-58 8-59 8-60

a a a c a c b b b c F T F T T F T F F T

8-101 .618 8-102 1.605 8-103 .479 8-104 1.017 8-105 .962 8-106 From 3.728 to 6.272 8-107 From 12.755 to 19.645 8-108 From 8.758 to 12.442 8-109 From 33.152 to 46.848 8-110 Mean = 35.300 8-111 SEm= 0.543 8-112 From 34.164 to 36.436 8-113 From 33.746 to 36.854 8-114 t = -2.100, Reject Ho p<.05 8-115 t=-2.100, Fail to reject 8-116 t=.960, Fail to reject 8-117 t=-1.235 Fail to reject 8-118 t=-2.100 Reject Ho p<.05 8-119 t= -1.490 Fail to reject 8-120 t=-2.140 Fail to reject

Chapter 8 Continued 8-121 Mean of the sample is 53.333 8-122 t= 3.262 Significant at p=.006 8-123 No it is not representative, since the scores are significantly higher than the norm.

8-124 Mean of the sample is 25.333 8-125 t = .712 and is not significant.

8-126 This sample is representative of the population at large.

8-127 The mean of the sample is 227.857 months 8-128 t =.097 8-129 This sample is representative of the population.

326

CHAPTER 9 ANSWERS 9-1 d 9-2 d 9-3 a 9-4 b 9-5 a 9-6 b 9-7 a 9-8 b 9-9 d 9-10 d 9-11 b 9-12 c 9-13 a 9-14 a 9-15 b 9-16 c 9-17 a 9-18 a 9-19 c 9-20 d

9-21 9-22 9-23 9-24 9-25 9-26 9-27 9-28 9-29 9-30 9-31 9-32 9-33 9-34 9-35 9-36 9-37 9-38 9-39 9-40

9-61 9-62 9-63 9-64 9-65 9-66 9-67 9-68 9-69 9-70 9-71 9-72 9-73 9-74 9-75 9-76 9-77 9-78 9-79 9-80

9-81 d 9-82 d 9-83 a 9-84 b 9-85 d 9-86 b 9-87 d 9-88 d 9-89 c 9-90 c 9-91 b 9-92 d 9-93 a 9-94 b 9-95 c 9-96 a 9-97 d 9-98 b 9-99 d 9-100 d

a d d b c a a a b b c a a c b b a b a b

d d a b b c d b d a d a a b a a b a b a

327

9-41 9-42 9-43 9-44 9-45 9-46 9-47 9-48 9-49 9-50 9-51 9-52 9-53 9-54 9-55 9-56 9-57 9-58 9-59 9-60

a c c a c b c c c d d d a a d d b d c d

9-101 c 9-102 c 9-103 d 9-104 b 9-105 a 9-106 d 9-107 d 9-108 d 9-109 d 9-110 c 9-111 d 9-112 d 9-113 a 9-114 c 9-115 b 9-116 b 9-117 a 9-118 a 9-119 b 9-120 b

Chapter9 Continued 9-121 a 9-122 a 9-123 d 9-124 d 9-125 d 9-126 a 9-127 b 9-128 a 9-129 a 9-130 b 9-131 T 9-132 F 9-133 T 9-134 F 9-135 T 9-136 F 9-137 F 9-138 F 9-139 T 9-140 T

9-141 F 9-142 F 9-143 F 9-144 F 9-145 F 9-146 F 9-147 T 9-148 F 9-149 T 9-150 T 9-151 F 9-152 T 9-153 F 9-154 F 9-155 T 9-156 T 9-157 T 9-158 F 9-159 F 9-160 F

328

9-161 T 9-162 F 9-163 F 9-164 T 9-165 F 9-166 T 9-167 F 9-168 T 9-169 F 9-170 T 9-171 T 9-172 T 9-173 T 9-174 T

CHAPTER 10 ANSWERS 10—1 d 10 21 10—2 c 10--22 10—3 d 10--23 10—4 b 10--24 10--5 a 10--25 10- 6 b 10--26 10--7 a 10--27 10—8 b 10--28 10—9 b 10--29 10 10 b 10--30 10--11 b 10—31 10- 12 d 10--32 10--13 c 10--33 10--14 a 10--34 10--15 c 10--35 10--16 a 10--36 10--17 d 10--37 10--18 d 10--38 10--19 d 10--39 10--20 d 10--40 10 – 61 10--62 10--63 10--64 10--65 10--66 10--67 10--68 10--69 10--70 10--71 10--72 10--73 10--74 10--75 10--76 10--77 10--78 10--79 10--80

T T T F T F T T T T F T T F T T T T T F

a c a b c d b b c d c a b c a b b d b b

10-81 F 10-82 F 10-83 F 10-84 T 10-85 T 10-86 F 10-87 T 10-88 F 10-89 T 10-90 F 10-91 T 10-92 F 10-93 T 10-94 T 10-95 F 10-96 T 10-97 F 10-98 T 10-99 F 10-100 F

329

10-41 a 10-42 b 10-43 a 10-44 d 10-45 d 10-46 d 10-47 d 10-48 a 10-49 c 10-50 a 10-51 b 10-52 a 10-53 d 10-54 c 10-55 a 10-56 d 10-57 a 10-58 c 10-59 b 10-60 b

Chapter 10 Continued 10-101 T 10-102 Ma= 9.700 Mb = 7.300 10-103 SDa= 2.052 SDb= 2.722 s = 2.163 s = 2.869 10-104 SEm A = .684 SEm B = .907 10-105 SEd= 1.136 10-106 t= 2.113 10-107 Yes. Sig at p<.05 10-108 Ma= 5.300 Mb= 7.700 10-109 Simply that the scores in Group B were on the average higher than those in Group A 10-110 Reject Ho. Sig at p<.05 10-111 SEd= 1.201 10-112 SEd= 1.129 10-113 t= 2.372 10-114a t=4.608 10-114b Reject Ho. Sig at p<.01 10-114c In all probabilty they do not 10-115a t = .490 10-115b Fail to reject 10-115c Yes 10-116a t=1.510 10-116b Fail to reject 10-116c Yes 10-117a t=1.364 10-117b Fail to reject 10-117c Yes 10-118a t=1.543 10-118b Fail to reject 10-118c Yes 10-119a t=-0.441 10-119b Fail to reject 10-119c Yes 10-120a t=1.863 10-120b Reject Ho. Significant as a one-tail test.

10-120c No 10-121 d 10-122 M for cocaine was 65.500 and the M for methamphetamine 57.100 10-123 Yes, the Levene test was not significant, so we assume homogeneity of variance.

The t ratio = 4.102, and with 18df was significant at p<.001.

10-124 The IV was the drug of choice 10-125 The DV was based on their CAQ, Factor Q1 scores.

10-126 Post facto 10-127 M for psychopaths = 40.400, and M for non-psychopaths = 50.800 330

10-128

Yes. Levene was not significant, so we may assume homogeneity of variance.

The t ratio was -4.028 and with 18df was significant at p<.001 10-129 The IV was the diagnosis 10-130 The DV was based on their IRI scores 10-131 Post-facto 10-132 M for non-recidivists = 41.083 months and for the recidivists M =57.083 months 10-133 Yes. Levene test was not significant, so we may assume homogeneity of Variance. The t ratio was -2.612 and with 22 df was significant at p<.016 10-134 The IV was whether a recidivist 10-135 The DV was based on the length of the sentence/ 10-136 Post facto 10-137 The results really don’t answer the basic question. On the surface it may show that the longer the sentence the greater the risk of recidivating, which of course would negate the deterrence hypothesis. However, it may be that those who had longer sentences had more serious convictions or had more previous incarcerations (which could have been the reason for the longer sentence).

Perhaps they were a less law-abiding group to begin with.

10-138 IV was whether a recidivist 10-139 DV was the sentence length 10-140 Post facto

331

CHAPTER 11 ANSWERS 11-1 d 11-21 d 11-2 d 11-22 c 11-3 c 11-23 b 11-4 b 11-24 d 11-5 d 11-25 d 11-6 d 11-26 c 11-7 d 11-27 b 11-8 c 11-28 b 11-9 b 11-29 b 11-10 b 11-30 c 11-11 b 11-31 d 11-12 a 11-32 a 11-13 a 11-33 b 11-14 c 11-34 d 11-15 c 11-35 d 11-16 b 11-36 a 11-17 a 11-37 c 11-18 d 11-38 a 11-19 a 11-39 d 11-20 d 11-40 b 11-61 d 11-62 a 11-63 d 11-64 d 11-65 c 11-66 d 11-67 d 11-68 d 11-69 b 11-70 F 11-71 T 11-72 T 11-73 T 11-74 T 11-75 F 11-76 F

11-77 T 11-78 F 11-79 T

11-41 d 11-42 b 11-43 b 11-44 b 11-45 b 11-46 c 11-47 a 11-48 d 11-49 c 11-50 d 11-51 d 11-52 b 11-53 c 11-54 c 11-55 c 11-56 d 11-57 b 11-58 b 11-59 d 11-60 a

11-80 F 11-81 F 11-82 T 11-83 F 11-84 T 11-85 T 11-86 F 11-87 T 11-88 T 11-89 F 11-90 F 11-91 r=.920 11-92 Reject Ho Sig p<.01 11-93 The two traits tend to occur together, but it does NOT prove a cause-and-effect relationship 11-94 r=-.632 11-95 Reject Ho. Sig p<.05. Students may miss this since this value ties the tabled value.

11-96 The two traits correlate inversely. Persons who are neat and orderly tend not to be rebellious .

But no cause-and-effect is proven.

11-97 The interval scale 11-98 The ordinal scale 332

11-99 Spearman r, since one variable is in ordinal form.

11-100 Spearman r = .780 11-101 Reject Ho Sig p<.05 11-102 Rank in class and IQ tend to cluster together, but it does not show IQ to be a causal factor in determining one’s rank in class (although it does not rule out that possibility).

11-103 r=1.00 11-104 r=-1.00 11-105 a r=.867 11-105 b Reject Ho. Sig p<.01 11-106 a Spearman r = .619 11-106 b Fail to reject 11-107 Spearman r =.648 11-108 Reject Ho 11-109 Sig p<.05 11-110 Spearman r = .988 11-111 Reject Ho Sig p<.01 11-112 Spearman r = -.624 11-113 Fail to reject 11-114 Pearson r = .080 11-115 8df 11-116 Fail to reject 11-117 Fail to reject Ho not rejected.

11-118 Type 2 (Beta error) 11-119 No 11-120 a t=1.39. Correlation is NOT significant.

11-120 b t=2.001 Reject Ho. Sig 11-120 c t=3.750 Reject Ho Sig 11-120 d t=3.535 Reject Ho Sig 11-121 a t=2.004 Reject Ho Sig 11-121 b t=4.064 Reject Ho Sig 11-121 c t=1.594 Fail to reject 11-121 d t=2.095 Reject Ho Sig 11-122 z=7.161 Reject Ho p<.01. The two correlations differ significantly 11-123 z=1.960 Reject Ho p<.05 The two correlations differ significantly 11-124 z=2.581 Reject Ho p<.01. The two correlations differ significantly 11-125 M for drugs = 50.050, M for alcohol = 50.100 11-126 r=.196 11-127 It is NOT significant 11-128 Post facto 11-129 M age for first arrest = 22.400, M yrs of Education = 10.700 11-130 r=.922 11-131 Yes, significant at p<.001 11-132 Post facto 333

11-133 11-134 11-135

All three variables correlated at a perfect 1.00 All the scores are the same (being presented in different metrics) Spearman correlation =.052 and is NOT significant.

334

CHAPTER 12 ANSWERS 12-1 b 12-21 d 12-2 d 12-22 b 12-3 b 12-23 b 12-4 d 12-24 c 12-5 b 12-25 d 12-6 b 12-26 b 12-7 c 12-27 b 12-8 c 12-28 d 12-9 d 12-29 a 12-10 d 12-30 b 12-11 c 12-31 c 12-12 a 12-32 d 12-13 b 12-33 d 12-14 d 12-34 b 12-15 c 12-35 a 12-16 c 12-36 b 12-17 b 12-37 c 12-18 a 12-38 c 12-19 c 12-39 d 12-20 b 12-40 c

12-41 b 12-42 c 12-43 a 12-44 c 12-45 c 12-46 c 12-47 b 12-48 d 12-49 c 12-50 d 12-51 d 12-52 d 12-53 d 12-54 d 12-55 a 12-56 b 12-57 b 12-58 d 12-59 d 12-60 a

Chapter 12 Answers Continued 12-61 c 12-81 F 12-62 b 12-82 F 12-63 a 12-83 F 12-64 d 12-84 T 12-65 a 12-85 T 12-66 b 12-86 MSw= .463 12-67 c 12-87 MSr= 28.057 12-68 d 12-88 MSc= 55.461 12-69 d 12-89 MSrxc= .029 12-70 T 12-90 Fr= 60.598 Reject Ho. Sig at p<.01 12-71 F 12-91 Fc= 119.786 Reject Ho Sig at p<.01 12-72 F 12-92 Frxc= .063 Accept Ho ns 12-73 F 12-93 Assigned to rows on basis of IQ 12-74 T 12-94 Manipulated IV in the columns (time of testing) 12-75 T 12-95 F = .396 12-76 T 12-96 F = 37.752 12-77 F 12-97 F = 18.561 12-78 T 12-98a F= 30.496 12-79 F 12-98b Reject Ho. Sig at p<.01 12-80 F 12-99a F= 9.773 12-99b Reject Ho Sig at p<.01 335

Chapter 12 continued 12-100 a F= 30.667 12-100 b Reject Ho Sig at p<.01 12-101 Fr= 44.594 12-102 Fc= 0.697 12-103 Frxc= 6.271 12-104 Fr is Sig at p<.01 12-105 Fc is NOT significant ns 12-106 Frxc is Sig at p<.05 12-107 a GroupA M= 10.300 12-107 b GroupB M = 9.100 12-107 c GroupC M = 3.700 12-107 d GroupD M= 6.100 12-108 Because of the significant F ratio for rows, it seems clear that the time of testing had an impact on the retention scores, with the immediate groups scoring higher than the delayed groups. The non-significant F ratio for columns appears to show that the drug is having no effect on the retention, but since the interaction is significant and since both the highest and the lowest scoring groups were on the drug, it appears that the drug in ombination with time of testing is having an effect. It appears thaat the drug aids in retention when the testing is done immediately, but the drug seems to hinder retention when the drug wears off. Perhaps the learning was state dependent.

336

CHAPTER 13 ANSWERS 13-1 c 13-21 a 13-2 c 13-22 b 13-3 d 13-23 a 13-4 b 13-24 b 13-5 a 13-25 b 13-6 d 13-26 a 13-7 d 13-27 b 13-8 b 13-28 a 13-9 c 13-29 d 13-10 d 13-30 c 13-11 d 13-31 a 13-12 b 13-32 a 13-13 d 13-33 a 13-14 d 13-34 b 13-15 a 13-35 a 13-16 c 13-36 b 13-17 c 13-37 c 13-18 b 13-38 c 13-19 d 13-39 b 13-20 b 13-40 a 13-61 F 13-62 F 13-63 T 13-64 T 13-65 T 13-66 F 13-67 F 13-68 F 13-69 T 13-70 T 13-71 F 13-72 T 13-73 T 13-74 T 13-75 F 13-76 T 13-77 T 13-78 T 13-79 F 13-80 F

13-81 F 13-82 F 13-83 T 13-84 chi sq = 7.500 13-85 chi sq = 13.580 13-86 chi sq = 19.400 13-87 df=4 13-88 9.49 13-89 13.28 13-90 Reject 13-91 p<.01 13-92a chi sq=9.000 13-92b Reject Ho. Sig p<.01

337

13-41 a 13-42 c 13-43 d 13-44 c 13-45 c 13-46 d 13-47 d 13-48 c 13-49 c 13-50 b 13-51 d 13-52 b 13-53 d 13-54 b 13-55 d 13-56 d 13-57 b 13-58 T 13-59 F 13-60 T

CHAPTER 14 ANSWERS 14-1 d 14-21 a 14-2 d 14-22 c 14-3 a 14-23 a 14-4 d 14-24 d 14-5 a 14-25 b 14-6 d 14-26 a 14-7 c 14-27 a 14-8 b 14-28 b 14-9 c 14-29 d 14-10 a 14-30 c 14-11 b 14-31 a 14-12 c 14-32 d 14-13 d 14-33 b 14-14 a 14-34 b 14-15 a 14-35 b 14-16 d 14-36 d 14-17 b 14-37 c 14-18 d 14-38 d 14-19 b 14-39 a 14-20 d 14-40 c 14-61 T 14-62 F 14-63 F 14-64 T 14-65 T 14-66 T 14-67 T 14-68 T 14-69 F 14-70 T 14-71 T 14-72 F 14-73 T 14-74 T 14-75 T 14-76 T 14-77 T 14-78 T 14-79 T 14-80 T 14-81 T 14-82 T 14-83 T

14-41 a 14-42 a 14-43 d 14-44 b 14-45 a 14-46 d 14-47 d 14-48 d 14-49 c 14-50 d 14-51 c 14-52 d 14-53 d 14-54 c 14-55 d 14-56 d 14-57 d 14-58 T 14-59 T 14-60 F

14-84 T 14-107 y= 7.090 14-85 T 14-108 SEest = 1.456 14-86 T 14-109 3.048 to 11.13 2 14-87 F 14-110 1.693 to 12.487 14-88 y = 1.360 14-111 R = .611 14-89 SEest= .645 14-112 R = .847 14-90 .070 to 2.650 14-113 R = .925 14-91 y = 590.000 14-114 R = .690 14-92 SEest = 44.159 14-115 R = .880 14-93 500.622 to 679.378 14-116 y = 61.255 14-94 470.247 to 709.759 14-117 SEmultest= 4.750 14-95 y = 32.600 14-118 49.001 to 73.509 14-96 SEest = 1.984 14-119 R = .728 14-97 27.937 to 36.063 14-120 65.138 inches 14-98 26.518 to 37.482 14-121 SEmultest = 2.057 14-99 r = .867 14-122 61.107 to 69.169 14-100 Reject HoSig at p<.01 14-123 Partial r = .093 14-101 y = 7.039 14-102 SEest = 1.407 14-103 3.133 to 10.945 14-104 1.823 to 12.255 14-105 r = .943 14-106 Reject Ho Sig at p<.01 338

CHAPTER 15 ANSWERS 15-1 a 15-21 a 15-2 d 15-22 d 15-3 d 15-23 c 15-4 c 15-24 a 15-5 b 15-25 a 15-6 a 15-26 b 15-7 d 15-27 a 15-8 d 15-28 b 15-9 c 15-29 a 15-10 c 15-30 c 15-11 b 15-31 b 15-12 d 15-32 c 15-13 d 15-33 a 15-14 a 15-34 d 15-15 b 15-35 b 15-16 b 15-36 d 15-17 a 15-37 a 15-18 b 15-38 d 15-19 b 15-39 b 15-20 b 15-40 d 15-61 T 15-62 F 15-63 T 15-64 F 15-65 T 15-66 F 15-67 T 15-68 F 15-69 T 15-70 T 15-71 F 15-72 T 15-73 F 15-74 T 15-75 F 15-76 F 15-77 F 15-78 T 15-79 T 15-80 F 15-81 F 15-82 F 15-83 T

15-41 a 15-42 d 15-43 b 15-44 d 15-45 d 15-46 d 15-47 b 15-48 d 15-49 b 15-50 a 15-51 c 15-52 d 15-53 c 15-54 c 15-55 c 15-56 d 15-57 d 15-58 T 15-59 T 15-60 T

15-85 t=2.457 15-86 Fail to reject 15-87 t= .654 15-88 Fail to reject 15-89 t= 3.906 15-90 Sig p<.01 15-91 t= 1.488 15-92 Fail to reject 15-93 t= 10.983 15-94 Sig p<.01 15-95 t= 2.991 15-96 Sig p< .05 15-97 t= 8.989 15-98 Sig p<.01 15-99 t= 3.004 15-100 Sig p<.05 15-101 Paired t = 4.895 15-102 Sig p<.01 15-103 t= .960 15-104 Fail to reject (neither is significant) 15-105 Needs a separate control group. The IV is confounded 15-106 F = 60.701 339

15-107 Sig p<.01 15-108 F = .473 15-109 Fail to reject 15-110 F = 12.998 15-111 Sig p<.01 15-112 F = 23.628 15-113 Sig p<.01 15-114 Within F = 36.531 15-115 Sig p<.01 15-116 Practice was not controlled 15-117 Independent t on change scores 15-118 Sig p<.05

340

CHAPTER 16 ANSWERS 16-1 b 16-21 a 16-2 b 16-22 c 16-3 d 16-23 a 16-4 c 16-24 b 16-5 d 16-25 d 16-6 d 16-26 d 16-7 d 16-27 a 16-8 d 16-28 d 16-9 a 16-29 a 16-10 b 16-30 b 16-11 c 16-31 b 16-12 a 16-32 c 16-13 c 16-33 a 16-14 b 16-34 a 16-15 d 16-35 d 16-16 b 16-36 b 16-17 a 16-37 d 16-18 c 16-38 b 16-19 d 16-39 b 16-20 d 16-40 d 16-61 F 16-62 T 16-63 T 16-64 F 16-65 T 16-66 T 16-67 T 16-68 T 16-69 T 16-70 F 16-71 F 16-72 T 16-73 T 16-74 T 16-75 F 16-76 F 16-77 T 16-78 T 16-79 1,2,5,2,5,4,6,6,6,8,9 16-80 1,2,5,5,5,5,5,8 16-81 Friedman ANOVA by ranks 16-82 = 3.800 16-83 2df 341

16-41 d 16-42 a 16-43 a 16-44 a 16-45 d 16-46 d 16-47 a 16-48 a 16-49 d 16-50 d 16-51 d 16-52 d 16-53 d 16-54 c 16-55 d 16-56 b 16-57 d 16-58 d 16-59 F 16-60 F

16-84 = 5.990 16-85 = 9.210 16-86 Accept Ho ns 16-87 Friedman ANOVA by ranks 16-88 = 15.800 16-89 2df 16-90 Sig p<.01 16-91 Wilcoxon T 16-92 T = 21.500 16-93 =6.000 16-94 = 2.000 16-95 Fail to reject 16-96 Wilcoxon T 16-97 T = 36.500 16-98 = 14.000 16-99 = 7.000 16-100 Fail to reject 16-101 Kruskal-Wallis 16-102 H=4.769 16-103 = 5.990 16-104 = 9.210 16-105 Fail to reject 16-106 Kruskal-Wallis H 16-107 H = .919 16-108 Accept Ho ns 16-109 Experimental Between Subjects 16-110 Mann-Whitney U test 16-111 =1.260 16-112 Fail to reject 16-113 Post-Facto 16-114 U = 1.304 Accept Ho ns

342

CHAPTER 17 ANSWERS 17-1 a 17-21 a 17-2 b 17-22 b 17-3 c 17-23 d 17-4 d 17-24 a 17-5 b 17-25 a 17-6 d 17-26 c 17-7 c 17-27 d 17-8 d 17-28 c 17-9 b 17-29 d 17-10 b 17-30 a 17-11 a 17-31 T 17-12 b 17-32 T 17-13 d 17-33 F 17-14 a 17-34 F 17-15 c 17-35 T 17-16 d 17-36 T 17-17 a 17-37 T 17-18 c 17-38 F 17-19 b 17-39 T 17-20 d 17-40 rtt =.956 Sig p<.01 17-41 test-retest reliability 17-42 r=.771 17-43 Sig p<.01 17-44 Yes the test is valid 17-45 Concurrent validity 17-46 rtt =.842 Sig p<.01 17-47 r=.833 Sig p<.01, Spearman-Brown = .909 17-48 Spearman Brown = .923 17-49a rtt=.773 17-49b rtt=.964 17-49c r not sig (ns) Cannot use Spearman Brown 17-49d rtt = .876 17-50a 2.001 17-50b 3.011 17-50c 4.020 17-50d 1.506 17-51a DI=.100 17-51b DI= .200 17-51c DI=.300 17-51d DI=.250 17-52 rpbis = .833 Sig p<.01 17-53 rpbis = .745 Sig p<.05 17-54 KR21 =.886 343

Chapter 17 Continued 17-55 KR21 = .468 17-56a rtt = .749 Sig p<.05 17-56b Paired t = -4.472 As they got older (6 months) they increased their wine Receptivity scores 17-57a Spearman rs= .685, Sig p<.05 17-57b Predictive validity 17-58 SEM = 4.74 17-59a 70.71 to 89.29 17-59b 80.71 to 99.29 17-59c 90.71 to 109.29 17-59d 100.71 to 119.29 17-59e 110.71 to 129.29 17- 60 SEM = 3.87 17-61a 32.415 to 47.585 17-61b 37.415 to 52.585 17-61c 42.415 to 57.585 17-61d 47.415 to 62.585 17-62 Rulon reliability = .555 17-63 Rulon reliability = .590 17-64 Rulon reliability = -8.76 (an impossible value, since SEM can never be greater than the SD 17-65 .490 17-66 .875 17-67 F = 1.700 ns 17-68 F = 7.116 Sig p<.01, and the rtt = .859 17-69 Alpha = .956 17-70 Test is reliable 17-71 rtt = .932 17-72 Sig p<.01 17-73 Test is reliable 17-74 Spearman correlation = -.733 17-75 The correlation is significant at p<.016 17-76 Yes the test has a high degree of validity. Point out to you students that this is actually a positive correlation as far as the validity is concerned. The only reason the negative sign appears is because in the ranking process, the highest score gets a 1 (a metrically low number) and the lowest score gets a 10 (a metrically higher number). Ask your students to rank order the test scores, and then again compute the Spearman, using only the two sets of ranks. At that point, the correlation -.733 now has a positive sign.

17-77 The correlation between forms is .338 17-78 IT IS NOT SIGNIFICANT 17-79 The Spearman-Brown for equal lengths is .505 17-80 IT IS NOT SIGNFICANT 17-81 The test has been proven not to be reliable.

344

SPECIAL UNIT 2 ANSWERS SU2-1 a SU2-2 b SU2-3 b SU2-4 d SU2-5 b SU2-6 d SU2-7 a SU2-8 a SU2-9 a SU2-10 a SU2-11 a SU2-12 a SU2-13 d SU2-14 b SU2-15 b

345

COMPUTER PROBLEMS CHAPTER 2 C1 The following is a list of the average public elementary and secondary school teachers’ pay for the year 2009 (Source: National Education Association, May 12, 2010).

When entering these data do not use the commas.

Alabama 46,604 South Carolina 45,758 Alaska 56,758 South Dakota 36,674 Arizona 45,772 Tennessee 45,030 Arkansas 45,773 Texas 46,179 California 64,424 Utah 41,615 Colorado 47,248 Vermont 46,593 Connecticut 61,976 Virginia 46,796 Delaware 55,994 Washington 49,884 Dist of Columbia 60,648 ` West Virginia 42,529 Florida 46,930 Wisconsin 49,051 Georgia 51,560 Wyoming 53,074 Hawaii 53,400 Idaho 44,099 Illinois 60,474 Indiana 48,508 Iowa 46,664 Kansas 45,136 Kentucky 47,207 Louisiana 46,964 Maine 43,397 Maryland 60,069 Massachusetts 60,471 Michigan 56,096 Minnesota 50,582 Mississippi 42,403 Missouri 43,206 Montana 42,874 Nebraska 42,885 Nevada 47,710 New Hampshire 47,609 New Jersey 61,277 New Mexico 45,112 New York 62,332 North Carolina 47,354 North Dakota 40,279 Ohio 53,410 Oklahoma 43,551 Oregon 51,811 Pennsylvania 55,833 Rhode Island 57,168 346

a. Find the range, the mean, and the median b. Discuss the shape of this salary distribution with regard to possible skewness.

2-C2 The following lists the 10 largest wind-producing states in the United States. Find the mean and median production values, and indicate if the distribution is skewed, and if so, in which direction. When entering these data, do not use commas.

California New York Washington Oregon Vermont Ohio Georgia Texas New Jersey Kentucky

Thousands of Gallons 422,560 23,180 5,200 1,860 1,550 1,130 1,120 980 840 820

347

COMPUTER PROBLEMS CHAPTER 3 3-C1. For the following set of IQ scores, find the mean, median, and range. Evaluate the distribution for skewness: 100,100,86,100,101,110,100,100,145,114,101,55,99,103, 97,90,100,96,104,100, 101,92,108,119,79 3-C2. For the following list of PSAT scores, find the mean, median and range. Evaluate the distribution for kurtosis: 30,40,45,35,42,65,55,48,56,46,42,54,56,51,47,50, 51, 50, 50, 47, 52, 53, 47, 44, 69,35,40,45,35,42,65,55,48,56,46,42,54,56,51,47, 50, 51, 50, 50, 47, 52, 53, 47, 78,80

348

COMPUTER PROBLEMS CHAPTER 8 8-C1. A science teacher constructs a 100-item, multiplechoice test, each item having four alternatives. Concerned that the sample of students she has this year may not be performing up to their abilities, she decides to check whether her students' scores on the test are any different from what could be expected on the basis of chance. If chance were operating, she assumes that the population mean on the test should equal 25. The scores earned by her students are as follows: 25, 32, 21, 26, 24, 31, 18, 23, 27, 20, 30, 24, 26, 42, 26, 12, 38, 24, 34, 25, 40, 34, 35, 32, 25 a. With alpha set at .05, test the hypothesis that this sample could be representative of a population whose parameter mean is 25. b. Using the t value obtained in part (a) and the same alpha level of .05, determine whether a sample of infinite size could be representative of a population whose mean is 25.

8-C2. A random sample of 25 male college students is selected and given the “ Attitude Toward Women" scale (higher scores indicating a more enlightened attitude). The scores are 15, 12, 11, 3, 11, 10, 12, 9, 8, 9, 13, 10, 11, 12, 8, 9, 9, 12, 13, 11, 13, 18, 14, 15, 12 a. Calculate the estimated standard error of the mean. b. Test the hypothesis that this sample could population whose mean is 10.

represent a

8-C3. The following lists the serum cholesterol levels for a group of 25 adult males. Test the hypothesis that this group could be representative of a population whose mean is 200. These measures are in units called milligrams per deciliter (mg/dl).

195, 165, 150, 245, 200, 194, 285, 200, 180, 210, 193, 188, 205, 185, 197, 193, 196, 194, 189, 196, 201, 194, 195, 195, 175 349

C4. The following is a list of diastolic blood pressure readings for a group of 25 adult females. Test the hypothesis that this group could be representative of a population whose mean is known to be 80. The measures are in units called millimeters of mercury (mm Hg).

85, 84, 83, 87, 88, 90, 60, 112, 110, 100, 75, 84, 85, 85, 86, 90, 80, 87, 83, 86, 84, 85, 90, 85, 82 8-C5. A random sample of eight-year-old students, all of whom tested as learning disabled (LD) and in fact were all receiving special education services, was selected and given the Developmental Indicator for the Assessment of Learning (DIAL) test. The following are the raw scores on the Concepts section of the test. The population mean for this subtest is known to be 20 for the non-LD population.

Test the hypothesis that this sample could be epresentative of the non-LD population.

38, 25, 23, 22, 22, 20, 20, 18, 18, 19, 19, 19, 19, 20, 2, 15, 16, 17, 17, 18 8-C6. The Maryland Addictions Questionnaire (MAQ) was designed to tap the severity of an individual's alcohol and/ or drug abuse problems. The MAQ provides three summary scores, Emotional Distress (EMO), Resistance to Treatment (RES), and Admission of Problems (ADM). A standardizing sample of almost 1000 men, all involved in some form of substance abuse treatment, yielded an EMO mean of 50. A random sample of 24 inmates from a group of confined men in a drug and alcohol rehabilitation facility was selected and their scores on the EMO were as follows: 55, 55, 42, 50, 40, 50, 54, 60, 45, 65, 50, 60, 55, 50, 55, 55, 60, 45, 53, 40, 50, 40, 50, 45 Does this sample represent the standardizing population?

350

COMPUTER PROBLEMS CHAPTER 10 10-C1. A researcher wondered whether there was any difference in pulse rates for smokers versus nonsmokers among female college sophomores. Random samples of 15 students each were selected, and the recorded pulse rates for each group were as follows: Smokers: 64, 68, 92, 94, 68, 72, 80, 99, 90, 64, 76, 78, 80, 78, 82 Nonsmokers: 66, 60, 66, 74, 84, 68, 62, 76, 80, 70, 62, 72, 76, 74, 78 Is there a significant difference between the two groups?

10-C2. Some psychologists theorize that there is a link between personality type and body type. Body types, called somatotypes, are sorted into three general categories: ecto morphs (thin and wiry), mesomorphs (stocky and muscular), and endomorphs(on the heavy side). Further, it has been assumed that mesomorphs are typically more extraverted than either ectomorphs or endomorphs. A group of 21 children, all of whom were rated above average in mesomorphy, was selected, along with another group of 21 children who had been rated below average in mesomorphy. All the selected children were 13-year-old males. Both groups were given the EPIE test, the ” extraversion" scale taken from the Eysenck Personality Inventory. On this test, higher scores indicate a higher degree of extraversion. The scores obtained by the mesomorphs were 12, 13, 11, 12, 14, 10, 9, 15, 20, 4, 12, 12, 10, 14, 11, 13, 12, 13, 12, 13, 10 The scores achieved by the non-mesomorphs were 14, 12, 10, 10, 10, 11, 7, 12, 14, 2, 11, 10, 12, 11, 10, 14, 18, 12, 11, 15, 5 Do the two groups represent the same population with regard to extraversion?

351

10-C3. The Hooper Visual Organization Test (VOT) is composed of a series of pictures of common objects that have been cut into two or more parts and illogically arranged. The subject is asked to name each object. A study was conducted to discover if a difference existed on the VOT between normals and neuropsychological patients who had right-hemisphere cortical lesions. The data are as follows: Normals: 26, 26, 12, 26, 27, 36, 26, 26, 40, 33, 27, 19, 25, 29, 23, 26, 26, 24, 30, 26, 27, 20, 34, 26, 10 Brain damaged:17, 21, 38, 25, 24, 24, 29, 10, 24, 24, 24, 23, 24, 24, 24, 18, 32, 32, 28, 24, 25, 22, 25, 31,27

10-C4. The Shaywitzs, husband and wife psychologists at Yale University, are now saying that the brains of men and women differ for certain language activities. Women appear to be physiologically better at using words and in the speed of their verbal reactions, whereas men are better at visualizing three-dimensional objects in space. Thus, women seem to be constitutionally better able to use and define words, whereas men are better at imagining what objects would look like if they were rotated. When reading a map in a car, for instance, men are less likely to have to turn the map in order to point it in the direction they are going in. Let's assume that a researcher wishes to test a part of this hypothesis on young six-year-old boys and girls using the Three-Dimensional Block Construction (3-D) test. The 3-D test has been used to assess visuoconstructional ability indicated by how well constructions in three-dimensional space are copied.

Subjects reproduce models of increasing complexity using 6, 9, and 15 blocks from an assortment of blocks on a tray. If the Shaywitzs are right, then boys should outperform girls on this test. Test the hypothesis.

Boys: 149, 149, 135, 149, 150, 159, 149, 149, 163, 156, 150, 142, 148, 152, 146, 149, 149, 147, 153, 149, 150, 143, 157, 149, 133 Girls: 153, 149, 132, 145, 146, 146, 141, 160, 146, 146, 146, 147, 146, 146, 146, 162, 138, 147, 142, 146, 145, 148, 145, 139, 143 352

10-C5. Researchers believe that total or partial sleep deprivation has antidepressant benefits in clinically depressed psychiatric patients (Gillin, 1994). Two groups of clinically depressed patients, 31 subjects per group, were randomly selected. Group 1 was sleep deprived during the second half of the night beginning at 3 a.m. Group 2 was allowed to sleep through the night and was not awakened until 7 a.m. At 12 noon, both groups were tested for depression. Test the hypothesis that sleep deprivation affected depression test scores. The scaled scores were as follows (higher scores indicating more depression): Group1:: 15, 13, 17, 14, 12, 16, 15, 14, 17, 13, 15, 15, 16, 14, 15, 15, 12, 18, 14, 14, 15, 14, 13, 16, 16, 15, 15, 16, 14, 14, 14 Group2: 29, 14, 18, 15, 16, 16, 17, 15, 16, 16, 28, 29, 17, 18, 17, 16, 17, 17, 17, 17, 18, 16, 28, 19, 17, 18, 19, 15, 17, 17, 17 10-C6. Researchers at M.I.T. suspected that the hormone melatonin, a natural hormone secreted by the pineal gland at the base of the brain, when given in small doses may be used to combat insomnia. Two groups of subjects were randomly selected; Group 1 received a placebo and Group 2 a small dose of melatonin. The subjects were then put to bed in a darkened room, and the length of time (in minutes) it took each subject to fall asleep was recorded. (The data presented here are simulated, but the overall averages are close to the original data.)

Group1: 15.00, 13.20, 17.50, 14.70, 12.20, 16.50, 15.30, 14.80, 17.00, 13.00, 14.90. 15.10 16.00, 14.00, 15.07, 14.96, 12.00, 18.00, 13.90, 14.60, 15.30, 14.70, 13.90, 16.20, 16.00, 15.80, 15.00, 16.00, 14.80, 14.60, 14.96 Group2: 6.00, 3.80, 7.90, 4.60, 5.30, 5.40, 6.20, 3.80, 4.90, 5.20, 7.00, 6.00, 6.20, 7.60, 6.20, 5.50, 6.50, 6.05, 5.97, 5.50, 6.60, 4.90, 8.00, 8.19, 6.50, 7.30, 8.20, 3.90, 5.50, 5.50, 5.80 353

a. Calculate the independent t ratio. b. Indicate the number of degrees of freedom in this study. c. Setting the alpha error at .01, test the null hypothesis that the two groups represent a common population.

As a sidebar to this sleep research, physiologists have long known that changes in the natural production of melatonin act as an important factor among the hormonal effects that accompany the development of adolescent sexuality. During early child-hood, melatonin inhibits the development of sexual maturity, and yet later, when melatonin production begins to naturally decrease, the stage is set for the sex hormones of adolescence to flourish. Also, melatonin production is inhibited by daylight. In fact, there is little if any difference between children and young adults in the amount of melatonin produced during daylight hours, but at night younger children produced large amounts of melatonin, whereas older children produced less and young adults even less. In fact, some researchers have suggested that humans living in areas where ight exposure is greater, for example, the Mediterranean countries, tend to mature at an earlier age than those in the northern climates because of the sun's influence on melatonin production. So perhaps melatonin will be found to induce sleep, but it may also act to restrain some of the physiological aspects of sexuality. Thus, with increased levels of melatonin you may sleep well, but you may wind up sleeping alone.

10-C7. A researcher is interested in discovering whether scores on the Social Symptomatology Scale of the Holden Psychological Screening Inventory (HPSI) can differentiate between psychopaths and non-psychopaths. A total of 20 psychopaths (diagnosed using DSM-IV criteria during a clinical interview) and 20 non-psychopaths (similarly diagnosed) were randomly selected from a Federal correctional facility (all inmates were serving a minimum of two years). The scores follow (although these scores are fictitious, they are modeled on the research results of Book, Knap, and Holden, 2001). Psychopaths: 17, 12, 22, 13, 21, 17, 19, 10, 17, 25, 17, 10, 27, 15, 19, 16, 15, 16, 19, 17 Non-Psychopaths: 12, 14, 11, 10, 11, 14, 10, 21, 5, 12, 19, 12, 5, 14, 12, 16, 8, 17, 7, 12 354

a. Test the research hypothesis that there will be a difference between the two sets of scores. b. Calculate a confidence interval at the .95 level for establishing the true population difference between the groups.

10-C8. Some theorists link crime to economic issues: the lower the income, the higher the probability of crime. Each year, the FBI discloses the crime rate (based on murder, rape, robbery, aggravated assault, burglary, and car theft) per 100,000 persons for each state. The following is the 2004 list of the 10 safest states, in order, and the 10 least safe, along with each state's median family income.

Is there a significant difference in income levels between the two groups of states? What other variables might be used to explain your result?

Safest

Median Family Income

North 36,717 Dakota New 53,325 Hampshire Maine

Median Family Income

Nevada

46,289

Florida

38,533

Louisia na Marylan Califor nia

37,654

Vermont 41,929 South 38,755 Dakota West 30,072 Virginia Wyoming

Least Safe

Arizona New Mexico Illinoi Tenness Alaska

40,499

Montana 33,900 Iowa 41,827 Nebraska 43,566

33,312 55,912 48,113 41,554 35,251 45,906 36,329 55,412

Remember when using SPSS not to include the commas when entering the incomes.

355

COMPUTER PROBLEMS CHAPTER 11 11-C1. According to Milton Rokeach, there is a positive correlation between dogmatism and anxiety. Dogmatism is defined as a rigidity of attitude that produces a closed belief system (or a closed mind) and a general attitude of intolerance. In the following study, dogmatism was measured on the basis of the Rokeach D Scale (Rokeach, 1960), and anxiety is here measured on the 30-item Welch Anxiety Scale, an adaptation taken from the MMPI (Welch, 1952). A random sample of 30 undergraduate students from a large western university was selected and given both the D Scale and the Welch Anxiety test. The results were as follows: Subj

D Scale

Anxiety Test

1

180

60

2

174

64

3

150

45

4

222

102

5

120

40

6

195

75

7

165

45

8.

150

30

9

245

145

10

200

125

11

194

80

12

285

80

13

200

74

14

180

60

Scores continue on next page 356

15

210

74

16

193

73

17

188

68

18

205

68

19

185

80

20

197

`75

21

193

73

22

196

76

23

194

60

24

180

81

25

186

50

26

201

66

27

194

75

28

165

73

29

195

74

30

170

50

Find the linear correlation between dogmatism and anxiety.

What conclusions might you draw?

357

11-C2. Hypothesis: The use of illegal substances (cocaine, marijuana, or hashish) is a function of, among other things, shyness and a low level of sociability. A group of over 600 adolescent boys (from ninth to twelfth grades) was selected from a representative sample of high schools in a western state. The average age of the respondents was 15.48 years. Subjects were asked how frequently they had used the previously mentioned substances during the past month. An index of substance use was formed by summing the number of times a subject had used an illicit drug during that time period. All subjects were also given the Cheek-Buss Shyness scale, and their scores were recorded. The researchers wanted to establish whether shyness was a risk factor in illicit substance use among adolescent males. The researchers suggested that shy males turn to proactive substances in an effort to alleviate their social discomfort and inhibition. If the correlation turns out to be positive and significant, what conclusions might you draw from these data? Could it be that shy people are drawn to the use of drugs, or could it be that the use of drugs causes persons to become more withdrawn and shy?

Or could some other variable(s) be responsible for the relationship? Your sample will consist of 20 adolescent boys.

Subject Shy 1 2 3 4 5 6 7 8 9 10

X 36 29 27 27 26 26 26 25 25 25

Drug Use

Subject

Shy

Y 30 18 24 18 17 10 12 15 15 16

11 12 13 14 15 16 17 18 19 20

X 25 25 24 24 24 24 23 23 22 14

358

Drug Use Y 16 16 5 7 6 0 10 0 5 0

11-C3. Each year Forbes magazine rates the rich and famous on the basis of their overall celebrity (which consists of Internet presence, press clippings, magazine cover stories, mentions on TV and radio, and, of course, earnings). To help determine how big a role earnings play in celebrity status, here are the top 15 individuals on the list, with their accompanying earnings (in millions of dollars for the most recent year): Celebrity Rank

Earnings ($ millions)

1 . Britney Spears

39.2

2. Tiger Woods

69.0

3. Steven Spielberg

100.0

4. Madonna

43.0

5. Mariah Carey

58.0

6. Michael Jordan

36.0

7. Tom Hanks

45.0

8. George Lucas

200.0

9. Jennifer Lopez

37.0

10. Julia Roberts

20.0

11. Stephen King

52.4

12. Ben Affleck 13. Mel Gibson

40.0 39.9

14. Bruce Willis

46.0

15. Bill Clinton

25.0

Data from Forbes, July 8, 2004.

Is there a significant correlation between celebrity status and earnings?

359

11-C4. A study was conducted to test the hypothesis that birth order impacts intelligence. One theorist suggests that the more children there are in a family, the lower the overall IQs of the children, especially the later-borns (Zajonc, 1986). A random sample of 25 convicted felons was selected from the County House of Correction and their birth orders were compared with their respective IQs. a. Test the hypothesis that birth order correlates significantly with IQ. b. What other variables might be playing a role? c. What conclusions might you draw? (Be very careful.)

Subjec Birth # Order 1.

3

IQ 100

2

2

120

3

7

95

4

2

110

5

2

100

6

3

125

7

1`

115

8

4

105

9

2

105

10

3

100

11

6

80

12

2

135

13

2

110

Scores continue on next page 360

14

3

90

15

2

100

16

3

80

17

5

90

18

1

95

19

3

90

20

2

105

21

1

115

22

6

60

23

3

95

24

4

100

25

3

90

11-C5 A researcher in health care is interested in discovering if there is a correlation between government health spending and the longevity of its citizenry. The following is a list of countries and both the percentages of GDP (gross domestic production) and the life expectancy for that country. (Source: The Economist Pocket World in Figures, 2001 Edition. London: Profile Books, 2001.)

Health Spending (% of GDP)

Life Expect (Years)

United States

14.00

77

Germany

10.50

77

Switzerland

9.80 Data Continue

79

361

France

9.70

78

Canada

9.20

79

Czech Republic

9.10

74

Netherlands

8.60

78

Costa Rica

8.50

76

Portugal

8.20

75

Austria

8.00

77

Iceland

8.00

79

Belgium

7.90

77

Norway

7.90

78

Spain

7.70

78

Italy

7.60

78

a. Compute the correlation and state your conclusions.

The correlation found may be somewhat misleading, since both these distributions are for the countries who were both the highest spenders and had the longest longevity, thus, severely restricting the range. Had the list continued it would have contained the lowest-spending countries, such as

Data on next page 362

Cote d’Ivore

3.50

46

Togo

3.40

49

Zambia

3.30

40

Ethiopia

2.60

43

Kenya

2.60

51

Bangladesh

2.40

58

Ghana

1.50

60

Cameroon

1.40

55

Nigeria

1.30

49

Niger

1.00

49

Sierra Leone

.04

39

Sudan

.03

55

Note: The above includes only the countries where both health spending AND longevity could be found. b. Has adding to the range changed the correlation? If so, what conclusions might you now draw?

What other factors might influence both health spending and longevity? Remember that when evaluating world data, it’s important to consider cultural and ethnic differences. This is especially true when examining health data, which can be distorted on the basis of different methods of tracking and reporting.

363

COMPUTER PROBLEMS CHAPTER 12 12-C1 The Novaco Provocation Inventory (NPI) is either an 80 or 90 item instrument for assessing anger responsiveness. (in order to maximize the variability, the 90-item version was used in this study) The test consists of brief descriptions of situations of provocation, for which the respondent notes the degree of anger that he/she might experience if that event should occur in his/her life. The ratings are on a 5-point Likert-type scale. The main purpose of the NPl is to gauge the range and intensity of a person's anger. Three groups of inmates from a state prison were selected and grouped on the basis of whether this was a first incarceration, a second incarceration, or a third incarceration. Each man took the NPl and the results were as follows: First

Second

Third

223 202 241 234 266 240 292 237 176 201 229 225 147 149 210 319 278 266 176

347 319 310 320 330 300 290 337 310 315 290 295 327 290 320 335 290 310 287

297 310 290 335 320 290 315 310 337 290 300 330 310 300 317 320 300 247 310

Test the hypothesis that anger is a function of the number of incarcerations. If the F is significant, follow it with a Tukey.

364

12-C2. A researcher is interested in whether the phonics method of teaching reading is more or less effective than the sight method, depending on what grade the child is in.

Twenty children were randomly selected from each of three grades: kindergarten (K), first grade (I), and second grade (2). Achievement was measured in terms of reading comprehension, higher scores indicating more comprehension.

Within each grade, 10 children were assigned to each of two methods of teaching reading, phonics or sight.

The data are as follows: Grades Grades

Sight

K

1

2

K

1

2

14

25

49

17

35

34

20

29

49

22

36

33

16

27

46

19

40

34

21

31

46

20

34

39

20

27

44

26

37

38

14

34

43

18

41

33

21

32

50

26

42

35

23

34

43

18

33

42

14

35

48

25

34

42

15

28

52

23

43

38

Phonics

a. Test the hypothesis for both main effects and interactions. b. Specify all variables. c. Write a statement as to your conclusions.

365

12-C3. In a study of hyperactivity among elementary school boys, nine groups of subjects were randomly selected from a school population of attention deficits with hyperactivity, seven-year-old boys. (If left untreated, ADHD can prevent a child from attending to incoming learning stimuli and may also create major disruptions in the classroom.) The researcher wanted to study the classroom effects of both the drug Ritalin and a behavior modification program on the activity levels of the subjects. The drug was varied from no dosage (in the form of a placebo), 10 mg of Ritalin, or 20 mg of Ritalin per day. The behavior modification program consisted of giving the child 10 tokens to start the day and then taking away a token for each hyperactive infraction. The tokens that were saved could then be exchanged for some valued prize. The behavior modification program was varied from no program to the program being used every other day, to the program being in force every day. After four weeks, all the children were evaluated for hyperactivity and were assigned scaled scores ranging from a possible low of zero (no indication of hyperactivity) to a high of 40 (extreme hyperactivity). The nine groups and their hyperactivity scores are as follows: Ritalin 20mg

No Behavior Mod

Ritalin 10mg

Placebo

12

18

21

10

16

22

19

27

33

19

23

35

15

26

27

18

22

34

22

22

32

-----------------------------------

Data continued 366

Every Other Day

28

30

38

15

15

20

20

20

23

12

14

20

14

15

14

14

20

21

17

19

21

--------------------------------

Every Day

3

7

13

10

14

18

11

11

12

8

12

13

11

17

19

5

10

9

13

15

17

a. Conduct the overall analysis. Calculate the means for all nine cells, and write a verbal description of the results. b. Compute the main effects for the drug. c. Compute the main effects for behavior mod. d. Compute the interaction and write a paragraph explaining these results.

367

12-C4. A researcher Is interested In whether a certain amphetamine drug will affect retention of leamed behavior among rats as a function of how much time Is allowed to eIapse between the training and the retention test. Four groups, 10 subjects in each group, were randomly selected from a colony of Sprague-Dawley rats. All the rats were given the same amount of training on a simple maze. Groups A and C were given the amphetamine, whereas Groups B and D were given a placebo. Groups A and B were then tested for retention immediately following the training, whereas Groups C and D were tested four hours later. The retention measures consisted of scaled scores based on how many correct responses the rats would exhibit while running the maze; the higher the score, the greater is the retention.

Test the hypotheses. The data were as follows: Amphetamine

Placebo

Group A

Group B

14

12

8

8

8

11

10

6

9

6

5

7

10 11

9 8

13

4

6

10

Immediate testing

--------------------------

Data continue on next page 368

Delayed testing

Group C

Group D

3

10

2

5

5 6 7

5 6 7

6

7

2

7

7

8

369

7

COMPUTER PROBLEMS CHAPTER 13 Chapter 13 Nominal Data and the Chi Square 13-Cl. Sports physiologists have hypothesized that among skiers a lack of knee strength is a major factor in predicting knee injuries. A random sample of 623 skiers was selected and tested for knee strength on an isokinetic testing device. Of these, 491 showed enough knee strength as to be judged adequate for skiing, while 132 demonstrated deficient strength. At the end of the ski season, 14 of the skiers who had passed the test did have knee injuries, while of those who failed the test, 45 had knee injuries.

Test the hypothesis and state all conclusions.

Injured

Not Injured

Passed the test

14

477

Failed the test

45

87

13-C2. Researchers at the Veterans Studies Center in New Haven, Connecticut, wondered whether there was any difference in the rate of later heart attacks among heart patients based on whether they received bypass surgery or were treated medically with drug therapy. A total of 803 patients were recruited, all of whom suffered from stable angina (chest pains that result when narrowed arteries fail to provide enough blood to the heart). One group of patients had bypass surgery (where the surgeon uses veins from elsewhere in the body to replace the narrowed arteries), and the other group was treated with drugs. All the patients were then monitored and the number who had subsequent heart attacks was tallied. The results were as follows (adapted from the American Heart Association's journal, Circulation, March 1991): Heart Attack Surgery

No Heart Attack

`

141

202

Drug therapy

117

343

Test the hypothesis that there is a difference in the number of later heart attacks based on the mode of therapy.

370

13-C3. Continuing computer problem 2, the researchers then wanted to find whether, among those patients who did have heart attacks, there was a difference in the number of fatal attacks, depending on which mode of therapy had been provided. Of the surgery group, 25 of the patients suffered fatal heart attacks, whereas in the drug-therapy group, 33 had fatal attacks.

Fatal Attacks

Nonfatal Attacks

Surgery

25

116

Drug therapy

33

84

Test the hypothesis that there is a difference in the risk of death from heart attacks, depending on the treatment provided.

13-C4. To assess whether time of day might be a factor in assaults on law enforcement officers, data were collected from the state of Utah for the time period 1991-2000. The number of police officers assaulted during that time was 2370 (Utah Law Enforcement Statistics, 2002). By time of day, the data were as follows: Number of assaults and time of day 10pm-2am 2am.-6am.6am.-10-am,10am-2pm 2pm.-6pm 6pm.-10pm, 965

320

123

171

273

518

Test the hypothesis that assaults on officers differ as a function of time of day.

13-C5. A study was done to determine if the number of murders is in any way connected to the day of the week on which the crime was committed. Data were collected, again from the state of Utah, and the following results were recorded. For the period 1988-2000, there was a total of 665 murders, categorized into the seven separate days (Utah Law Enforcement Statistics, 2002).

371

Number of Murders and Day of the Week Mon Tues

Wed

Thurs

Fri

Sat

Sun

74

94

83

106

101

110

87

Test the hypothesis that the number of murders differs as a function of day of the week.

13-C6. Researchers wished to find out if early referral to the juvenile justice system in any way relates to the chance of that juvenile later being sent to adult prison. A random sample of 476 cases of juveniles adjudicated delinquent in a certain Pennsylvania county was selected from the court records. Of the 243 juveniles adjudicated delinquent on their first referral to juvenile justice, a total of 51 went on to adult prison after the age of 18. Of the 233 juveniles not taken to juvenile Court on their first referral, 99 were imprisoned in adult life after the age of 18 (Brown et al., 1991).

No Prison

Prison

Case adjudicated

192

51

Case not adjudicated

134

99

a. Determine if the difference is significant. b. State all conclusions.

13-C7. The Greenwich Laboratories wished to test their then new antiarthritic drug Acheafectin, and their researchers selected a sample of 200 arthritis sufferers and randomly divided them into two groups. Group A was treated with Therafectin and Group B was given a placebo. Twelve weeks later, all subjects were tested for possible symptom improvement. The data were as follows (from the September 1989 issue of Internal Medicine):

372

Test the hypothesis that Acheafectin leads to arthritic symptom relief.

Improvement

No Improvement

Acheafectin

41

59

Placebo

21

79

13-C8. According to statistics from the National Center for Juvenile Justice, the percentages of adult men and adult women who are reported to have molested children are as follows (Stone, 2004): Molest Boys

Molest Girls

Men

80%

95%

Women

20%

5%

Assume that a sample of 200 child molesters was selected and the numbers in each cell were representative of the percentages shown above. Is there a significant difference between men and women regarding molestation?

Molest Boys

Molest Girls

Men

80

95

Women

20

5

For more on this issue, and on the possibility that therapy with these molesters doesn't work at all, see B. J. Winick and J. Q. LaFond, 2003.

373

13-C9. Researchers wanted to know if obese parents were more likely to have obese children than were parents of normal weight (Fiore et al., 2006). Obesity was defined as being above the 85th percentile on the body-mass index. A total of 1890 families were contacted with children ranging in age from 12 to 16. For this example, assume the following data (modeled after the Fiore study): Neither

One

Parent Obese

Parent Obese

Both Parents Obese

Child Obese

183

240

378

Not Obese

447

390

252

Find the correlation between parental obesity and childhood obesity.

374

COMPUTER PROBLEMS CHAPTER 14 14-Cl. The admissions officer at a small college collected the following SAT data (V, verbal, and M, math) for 47 students who had applied to become computer science majors. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

V 510 470 290 380 640 240 390 330 290 430 510 460 460 650 500 780

M 640 670 420 660 730 540 410 560 310 440 500 500 480 610 430 770

# 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

V 400 420 420 450 420 450 490 780 660 490 610 540 510 530 450 400

M 440 550 550 600 670 470 550 700 490 500 600 590 510 530 470 460

# 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

V 710 710 610 290 780 370 670 460 660 440 510 490 390 610 490

M 780 680 710 420 730 390 670 450 650 440 380 510 400 640 470

a. Find the linear correlation between Math Scores and SAT Scores b Is the correlation significant? c. If significant,predict the math scores for the following SAT scores: 370,610,510 and 700.

375

14-C2. A random sample of 20 inmates from the county correctional center was selected and given three tests: (1) the Novaco Provocation Inventory, a test of the range and intensity of a person's anger; (2) alcohol use-as measured on the Alcohol Scale that are taken from the Substance Abuse Subtle Screening Inventory (SASSI) (Miller, 1990); And (3) drug use-as measured on the SASSI Drug Scale.

Subje Alcohol ct # Score 1 14 2 6 3 5 4 12 5 8 6 15 7 10 8 8 9 11 10 9 11 14 12 6 13 5 14 12 15 8 16 15 17 10 18 8 19 11 20 9

Drug Score 12 10 8 10 7 12 8 10 8 10 12 10 8 10 7 12 8 10 8 10

Anger Score 290 240 180 210 185 230 225 260 290 230 290 240 180 210 185 230 225 260 290 230

a. Find the Pearson r between alcohol use and drug use. Is it significant? b. Find the Pearson r between alcohol use and anger. Is it significant? c. Find the Pearson r between drug use and anger. Is it significant? d. Find the multiple R for predicting anger on the basis of both alcohol use and drug use.

376

14-C3. The following represents distributions of verbal SAT scores and college grade-point averages (GPAs) selected from a random sample of 20 college students, all of whom have completed their first year of study.

Subje SAT ct 1 600 2 550 3 540 4 320 5 480 6 600 7 510 8 480 9 470 1 500

GPA 2.76 2.86 2.77 2.64 2.64 2.70 2.70 2.63 2.62 2.76

11 12 13 14 15 16 17 18 19 20

SAT

GPA

490 500 550 250 400 500 460 500 750 550

2.70 2.61 2.70 1.40 2.54 2.78 2.53 4.00 2.87 2.79

a. For the SAT distribution, find the mean and median. Is the distribution skewed? If so, which way? b. For the SAT distribution, find the range and standard deviation. c. For the GPA distribution, find the mean and median. Is the distribution skewed? If so, which way? d. For the GPA distribution, find the range and standard deviation. e. Find the linear correlation between the SAT verbal scores and GPA. f. Find the regression equation for predicting GPA on the basis of SAT. g. Predict GPAs for the following SAT scores: 470, 560, and 590.

14-C4. In the previous problem, the researcher measured 20 subjects on one independent variable and one dependent variable. Later the researcher wondered if the correlation between SAT and CPA might have been inflated due to a failure to control for IQ. It was decided to give each of the subjects a WAISIV IQ test and their scores were as follows (each subject's number will be followed by the IQ score for that person): 1, 130; 2, 135; 3, 120; 4; 95; 5; 95; 6, 125; 7, 110; 8, 100; 9, 105; 10, 100; 11, 90; 12, 110; 13, 130; 14,.90; 15,105; 16,110; 17,100; 18,140; 19,130; 20,.120.

377

a. For the IQ distribution shown above, find the mean, median, and range. b. Find the correlation between IQ and SAT. c. Find the correlation between IQ and GPA. d. Find the partial correlation between SAT and GPA with the IQ variable being partialled out. (Remember, you already have the original correlation between SAT and GPA from computer problem C3 above).

378

14-C5. A researcher at a correctional facility is interested in predicting an inmate's Level of Security (LSI) score on the basis of his score on the psychopathy score (SS score) from the Holden Personality test and his Drug Abuse (DA) score from the Maryland Addiction Questionnaire. A random sample of 15 inmates was selected from the population of an adult male correctional center.

Subj#

SS

DA

LSI

1

50

65

7

2

60

55

5

3

50

58

3

4

65

68

5

5

45

50

4

6

75

60

8

7

70

60

6

8

50

45

4

9

50

60

3

10

60

67

5

11

60

60

5

12

70

75

7

13

50

50

4

14

65

70

6

15

55

45

4

a. Find the three internal Pearson r values, that is, the correlations between Psychopathy (SS) and Drug Abuse (DA), Psychopathy (SS) and LSI, and Drug Abuse (DA) and LSI. b. Are they all significant?

379

Using the above data, find the multiple R for predicting LSI as the dependent variable on the basis of the SS and DA scores. Is it significant?

380

COMPUTER PROBLEMS CHAPTER 15 15-c1 A researcher wishes to test the hypothesis that the discovery teaching technique leads to higher levels of pupil learning than does the more traditional presentation.

Two groups of sixth-grade students are selected and matched person for person on IQ. Both groups are presented with a geography lesson unit. Group A, the control group, is taught in a fairly traditional manner, whereby the teacher, in covering the subject area, asks a series of theoretical questions but quickly answers them herself. Group B, the experimental group, is exposed to the same material by the same teacher, but in this case she does not answer the rhetorical questions in the hope that this will stimulate pupil thinking and allow the pupils to discover for themselves certain basic relationships. Both groups are then tested on the geography content, and their retention scores (higher scores indicating greater retention) are as follows. Does the discovery teaching technique affect learning?

Subj#

Control

Experimental

1

100

98

17

81

83

2

82

80

18

67

77

3

98

92

19

88

72

4

55

67

20

79

90

5

75

80

21

82

86

6

66

77

22

84

87

7

82

98

23

80

90

8

65

75

24

75

88

9

42

67

25

78

82

10

76

80

26

81

88

Data Continue 381

11

82

95

27

66

72

12

64

74

28

86

88

13

82

80

29

78

77

14

83

87

30

84

86

15

78

75

31

84

88

16

84

88

15-C2. Subjects were asked to take multiple-choice tests under three different conditions of crowding. In one condition (alone), the subject worked alone at a table. In another (three other subjects) each subject was at the same table with three other individuals, all working at the same task. In the third condition (seven other subjects) there were seven other people at the table, all working on the same task. The presentation order of the tasks was counterbalanced, and each subject participated in all three conditions. The dependent variable was the number of items answered correctly out of a total of of 60 questions. Does crowding have an effect on the number of items answered correctly?

The data follow:

382

Subj#

Alone

Three

Seven

1

35

30

20

2

49

34

22

3

33

21

20

4

34

35

30

5

37

29

20

6

29

25

21

7

29

26

24

8

30

43

20

9

17

21

20

10

29

23

10

11

41

30

14

12

50

20

27

13

32

28

14

14

44

28

25

15

39

28

20

383

15-C3. Test the hypothesis that a child’s level of impulsivity can be reduced by training. A large group of 12-year-old children was randomly selected and then split into two groups by equating them, person for person, on an extraversion test.Both groups were then given an Impulsivity Test. Group A, the experimental group, was then given a week of social skill and perception training, while Group B, the control group, was not given this training. At the end of the week both groups were again tested.

Group A

Group B

Pair#

pre

post change

pre

post change

1

55

53

-2

52

50

-2

2

25

20

-5

23

22

-1

3

50

49

-1

45

45

0

4

60

51

-9

52

50

-2

5

56

52

-4

55

54

-1

6

55

48

-7

48

46

-2

7

51

40

-11

42

38

-4

8

76

65

-11

76

66

-10

9

55

52

-3

58

55

-3

10

56

48

-8

60

50

-10

11

60

55

-5

64

60

-4

12

48

45

-3

50

44

-6

13

80

80

0

75

75

0

14

56

47

-9

54

45

-9

15

55

52

-3

51

50

-1

Scores Continue on next page 384

16

58

48

-10

55

49

-6

17

42

40

-2

45

43

-2

18

56

55

-1

60

57

-3

19

58

55

-3

52

50

-2

20

46

45

-1

48

46

-2

3 3

15-C4. A rat study was conducted to find out whether environmental inputs could affect brain growth. Sixteen laboratory baby rat twins were selected and randomly assigned to two groups. The rats were paired off with one rat from each twin pair in the experimental group and the other twin in the control group. The experimental group was then raised in a stimulating environment, in a cage equipped with ladders, running wheels, and other "rat toys." These animals were also let out of their cages for 30 minutes each day and allowed to explore new territory.

They were also trained on a number of learning tasks and in general received a rich and varied array of stimulus inputs. The other group of rats was raised in a condition of extreme stimulus homogeneity. These rats lived alone in dimly lit cages, were rarely handled, and were never allowed to explore areas outside their cages. All animals received exactly the same diet. After 90 days, all the animals were sacrificed and their brains analyzed morphologically and chemically. For one comparison, the weight of each rat's cortex was recorded in milligrams.

Pair# Experimental Control 1 2 3 4 5 6 7 8 9

685 666 678 670 679 663 664 657 675

667 653 662 664 668 656 650 650 680 385

a. Test the hypothesis that the two groups differ significantly regarding cortical weight. b. Was the matching process effective?

15-C5. A researcher is interested in the effects of marijuana on the pulse rate. A group of 20 college students was randomly selected and their pulse rates recorded. They were given (and smoked) one marijuana cigarette and 30 minutes later their pulse rates were again taken. The data follow: Subj#

Pre

Post

1

76

82

2

72

78

3

81

89

4

70

75

5

65

76

6

70

74

7

72

78

8

83

80

9

75

81

10

72

74

11

73

78

12

60

70

13

70

76

14

76

74

Data Continue 386

15

71

79

16

72

80

17

72

78

18

74

80

19

76

86

20

70

74

Test the hypothesis.

As a sidebar to this research, it has been found that marijuana increases the risk of heart attacks for people over 40. Marijuana increases both pulse rate and blood pressure, and from one to two hours after ingestion, heart attack rates are increased by 50%, especially if ingestion is followed by any form of exercise-including sexual activity (Geraci, 2000).

15-C6. A researcher conducts a pre-post study comparing the SAT scores of a random sample of 20 subjects before they took an SAT coaching course and then again after they took the course.

Subje ct Pre # 1 2 3 4 5 6 7 8 9 10

500 510 510 550 370 600 270 700 510 490

Post

#

Pre

Pos t

510 500 510 570 370 620 250 710 520 500

11 12 13 14 15 16 17 18 19 20

500 490 740 510 500 550 450 430 520 500

490 440 760 510 520 560 440 420 530 500

Test the hypothesis that the coaching course had a significant effect on their SAT scores.

387

15-C7. The researcher in the above problem suspects, on the basis of prior research, that perhaps only those who scored above 500 were aided by the coaching course. Thus, only those who scored 510 or better were selected from the original sample and their pre scores were compared with their post scores as follows: Subje Pre ct # 2 510 3 510 4 550 6 600 8 700 9 510 13 740 14 510 16 550 19 520

Pos t 500 510 570 620 710 520 760 510 560 530

Test the hypothesis that coaching is a more important factor for those who already have above-average scores.* "This is a finding that is consistent with the literature, which shows that there is a larger coaching effect for individuals with higher pre-coaching scores than for those with average and below average scores, a fact that argues against the hope that coaching will necessarily narrow the gap between high- and low-scoring students (Sackett et al., 2001).

15-C8. Since dark chocolate contains flavonoids, which are antioxidants, it was suspected that consuming it might help persons with high blood pressure (Grassi, 2005).

Hypothesis: eating dark chocolate may lower blood pressure in people with hypertension. A group of 10 men was randomly selected from a group of patients, all of whom had been diagnosed with hypertension but had not yet been treated, and all of whom were the same age. Diastolic (the lower number in a blood pressure count) measures were taken and the men were then sent home and told to eat a 3.5-ounce bar of dark chocolate each day. Two weeks later, diastolic measures were again collected. The data follow.

388

Pre

Post

110

90

115

87

103

80

105

74

104

88

106

75

105

70

109

82

104

83

109

91

Was there a significant difference between the pre- and post-measures?

389

COMPUTER PROBLEMS CHAPTER 16 16-Cl. A group of inmates in a certain pod at a state correctional facility were rank-ordered on the basis of how much trouble they had caused while in confinement, with the rank of 1 indicating the most trouble and a rank of 12 indicating the least. A researcher wondered if there might be a difference in their trouble-making ranks as a function of their scores on the Holden SS (psychopathy scale).

Twelve inmates were randomly selected and then divided into two groups on the basis of their SS scores. Those over the mean were scored as a 1, and those below as a 2.

Subj#

Trouble

SS score

1

5

1

2

11

1

3

12

2

4

7

1

5

1

1

6

3

2

7

2

1

8

9

2

9

10

2

10

6

2

11

8

2

12

4

1

Test the hypothesis that those above the SS mean are more likely to be rank-ordered as the worst trouble-makers.

390

16-C2. Research was conducted to discover if age at first arrest differed among those subjects who were successful in a reentry program to get them safely back into society as opposed to those who were not successful or were less successful. The subjects were placed into three categories, those first arrested at ages 12 to 15, those first arrested at 16 to 19, and those whose first arrest was at 20 or older. Subjects who were released during the previous three years were rank-ordered on the basis of their parole or probation problems and/or violations. A rank of 1 indicated a subject with the most problems and a rank of 15 indicated a subject with the least.

12-15

16-19

20+

2

4

13

7

12

15

1

3

14

5

8

9

6

19

11

Test the hypothesis that differences in levels of reentry problems were a function of early problems with the law.

16-C3. In a study modeled after the research of Mocan and Tekin (2006), judges evaluated a group of high school yearbook pictures of male students, half of whom had been convicted at least once of a felony during the five years following graduation and the other half who all had clean records. The judges rank-ordered them in terms of their looks, a rank of 1 indicating the best looking and a rank of 20 indicating the worst looking. The felony category was scored as a 1 if convicted and a 2 if the record was clean.

The data follow: a. Test the hypothesis that good-looking males are less likely to have felony convictions.

391

b. Is this experimental or post-facto research? c. Indicate both the IV and DY. Subj#

Looks

Felony

1

5

1

2

10

2

3

12

1

4

16

1

5

3

2

6

7

2

7

11

2

8

2

2

9

14

1

10

4

2

11

15

1

12

13

1

13

1

2

14

6

2

15

8

1

16

9

1

Mocan, H.N., & Tekin, E. (2006). Ugly criminals. American Economic Association presentation, Jan. 6.

392

COMPUTER PROBLEMS CHAPTER 17 17-C1. A random group of 22 elementary school children was chosen and given the Eysenk Venturesomeness Test on a Friday morning at 8:30 a.m. That same afternoon at 3:00 p.m., the children were given the same test again. Their morning scores appear below under "Test" and their afternoon scores under "Retest."

Subj#

Test Retest

Rank

Subj#

Test Retest

Rank

1

60

50

4

12

56

53

9

2

40

35

17

13

51

47

19

3

70

65

1

14

55

50

15

4

53

48

11

15

53

48

22

5

57

52

7

16

57

52

6

6

59

55

2

17

55

50

13

7

50

45

18

18

50

46

16

8

57

52

8

19

60

54

3

9

52

48

20

20

55

50

12

10

42

40

14

21

56

56

10

11

57

60

5

22

54

53

21

a. Was the test reliable? b. Explain your answer, indicating the type of reliability found and the kinds of problems associated with this technique. For example, were their afternoon scores in any way different and, if so, why do you suspect they changed? c. During the next few weeks a panel of four judges, trained in child psychology, watched all the children while on the school premises, both in the classroom and outside during recess. They ranked all the children according to 393

How adventurous and daring they appeared to be and assigned a median rank to each child. Compare the child’s Test score with the ranks given by the judges.

Is the test statistically valid?

17-C2. A group of 15 college students was randomly selected from a population of students who had never flown an airplane and given the Pilot's Aptitude Test, a test designed by a manufacturer of aircraft for persons interested in general aviation. The test contained a total of 40 items. To control for practice and fatigue, the researcher decided to split the test into two halves, odd items versus even items. Calculate the split-half reliability of the test. What would the reliability have been if the whole test were used? Later, all the subjects were given flying lessons for a total of three months, and at the end of the training an independent evaluator (an instructor who had never seen any of the subjects previously and who did not know their scores on the aptitude test) took each student pilot for a short flight and ranked each student on his or her ability to fly the plane. Assess the validity of the total score on the aptitude test.

Subj#

Odd X

Even Y

TotalX+Y)

1

19

18

37

4

2

14

16

30

1

3

4

5

9

14

4

12

13

25

2

5

11

9

20

12

6

13

10

23

6

7

3

2

5

13

8

16

15

31

3

Scores Continue 394

Ranks

9

10

7

17

10

10

8

11

19

15

11

9

12

21

7

12

13

8

21

11

13

11

13

24

8

14

12

14

26

5

15

12

11

23

9

17-C3. A researcher has constructed a new substance-abuse test consisting of ten items scored on a 5-point scale from strongly agree to strongly disagree. A random sample of 15 prison inmates is selected from the population of a large house of correction. The data follow: Subj#Item1Item2 Item3 item4 Item5 Item6 Item7 Item8 Item9 Item10 1

5

4

5

4

5

4

5

5

5

5

2

1

2

2

3

2

3

2

3

3

2

3

5

4

5

3

4

3

3

5

5

4

4

3

3

3

4

4

3

2

4

3

3

5

5

4

3

4

4

4

5

5

5

5

6

2

2

1

3

2

2

2

1

1

2

7

4

3

2

5

5

5

1

3

4

3

8

4

3

3

3

3

3

2

4

3

3

9

3

2

3

4

4

4

3

5

5

2

10

3

1

2

2

2

3

2

3

3

1

11

5

1

2

5

3

5

3

4

5

5

12

4

3

3

3

5

5

1

3

4

3

The data continue 395

13

3

2

1

5

4

5

2

5

3

2

14

4

2

2

4

4

3

3

4

4

2

15

4

5

4

5

4

5

5

4

4

5

The researcher is interested in the internal consistency reliability. Compute Coefficient Alpha and indicate whether the results prove the test to be reliable.

396

ANSWERS TO COMPUTER PROBLEMS CHAPTER 2 C-1.a. The range is 27750. The mean is 49702.76. The median is 47207. The estimated standard deviation is (s in SPSS = 6707.689. The true standard deviation is 6641.602. b. Since the mean is somewhat higher than the median, this distribution, as is the case with most salary distributions, is skewed to the right, Sk+(.605). Remember, however, that this is a distribution of means, not raw scores. Thus, we should expect that the variability (both range and standard deviation) in this distribution will be less than in the underlying distribution of individual salaries. If we were to examine the underlying distribution, we would have found a range of well over 60000.

C-2.The Mean in this distribution is 45924, and yet the median is only 1340. This distribution is definitely Sk+ (skewed to the right at 3.147).

397

CHAPTER 3 C-1. The mean is 100.000. The median is 100.000. The range is 90.000. The estimated standard deviation is 15.3107. The true standard deviation of this sample is 15.001. For this distribution the mean, median and mode fall at exactly the same point, so the distribution cannot be skewed. (Working out the Skew value from the formula shown in Appendix, part A., Sk = -.030, which in effect means no skew).

C-2. The mean is 50.000. The median is 50.000. The range is 50.000.

The estimated standard deviation is 9.5426. The true standard deviation of this sample is 9.447.

398

CHAPTER 8 C-1. a. t = 1.963, and t at .05 with 24 df = 2.064. Accept Ho: Difference is not significant. This sample of 25 persons probably could be a representative sample. b. With infinite df, however, t at .05 = 1.96; Reject Ho: Difference is significant at p < .05. If the sample had been of infinite size, it probably would not be representative of the population.

C-2. a. SE of mean = .5859 b. t = 2.048 b. t at .05 with 24 df = 2.064. Accept Ho: Difference not significant. The sample probably could represent such a population.

C-3. The single sample t test yields a value of -0.647. With 24 degrees of freedom, the table indicates that t ratios of 2.064 at the .05 level and 2.797 at the .01 level are needed for significance. Therefore we accept Ho and conclude that the sample probably does represent the known population whose mean is 200 mg/dl.

C-4. The single sample t test yields a value of 3.337. With 24 degrees of freedom, the table indicates that t ratios of 2.064 at the .05 level and 2.797 at the .01 level are needed for significance. Therefore we reject Ho at p < .01. The sample probably does not represent the known population whose mean is 80 mm Hg.

C-5. The single sample t test yields a value of -0.461. With 19 degrees of freedom, the table indicates that t ratios of 2.093 at the .05 level and 2.861 at the .01 level are needed for significance. We, thus, accept Ho and conclude that this sample, despite the special services being provided,(or perhaps because of the special services being provided), is not significantly different from the known population and could, in fact, be representative of the non-LD population.

399

CHAPTER 10 C-1. The value of the t ratio is 2.313. Reject Ho at p<.05. Non-smokers had significantly lower pulse rates.

C-2. The value of the t ratio is 1.017. Accept Ho. There was no significant difference between the two body-type groups.

C-3. t = 0.738 Accept Ho. Not significant. VOT scores represent a common population.

C-4. t = 1.701 Accept Ho. Although the difference favors the boys, the difference in this study is not significant.

C-5. t = -4.500. Reject Ho at p <.01. The difference in favor of the sleep deprived group is significant.

C-6. a. t = 27.534. b. df = 31+31-2 = 60. c. The table value of t for 60df, with alpha at .01, is 2.660. Since our calculated value of t is greater than the table value, we reject the null hypothesis and conclude that there is a statistically significant difference in the sleep times between the two groups (with Group 2, the melatonin group, getting to sleep in significantly less time).

C-7. a. t = 3.766, significant at p <.01. The difference is statistically significant. b. The confidence interval at .95 is from 2.3583 to 7.8417.

400

CHAPTER 11

C-1. The Pearson r correlation is a positive 0.659 and with df = 28 the r is significant at p <.01. On the basis of these results we conclude that the two variables are related, but we can't tell if dogmatism caused the anxiety, anxiety caused the dogmatism, or a third variable, such as a person's past child-rearing environment or even the person's genetic predispositions may have caused both the dogmatic and anxious personalities.

C-2. r = 0.790. With df at 18 the correlation is significant at p <.01 and indicates a significant positive relationship between shyness and illegal drug use. This result, however, does not imply a direct causal relationship.

C-3. r = 0.585. Reject Ho at p <.01. There is a modest, but significant, relationship between the college president's income and that of the faculty.

C-4. a. The Pearson r correlation = -0.591 and is significant at p <.01. The negative sign is due to the fact that in general the higher birth-order numbers, such as 6 or 7, are linking with lower IQ values, and the lower birth-order numbers, such as 1 or 2 are linking with higher IQs. b. There are a large number of other variables lurking in the background on this one. For example, perhaps with larger families, there aren't as many financial resources or as much parental time to spread around among the siblings. Perhaps the children in larger families don't get as much parental attention, or perhaps lower IQ parents tend to have more children. c. The only solid conclusion is that there is a signficant correlation between birth order and IQ. Keep in mind that high birth-order numbers can ONLY come from large families. Nobody has a birth-order of 7 in a 2 child family.

C-5. a. r= -.048, p > .05, ns. b. (with the added countries) r = .871, p <.01.

401

CHAPTER 12 C-1. F = 44.921, significant at p <.01. Anger increased significantly as a function of number of incarcerations.

C-2. F for rows = 0.142. F for columns = 217.503. F for the interaction = 37.095. Accept Ho for rows. Reject Ho at p <.01 for both columns and interaction. Since Ho was accepted for rows, we can conclude that method of teaching had no significant effect between the three grade levels. The F ratio for columns was clearly a reject of Ho, indicating that grade level did have an effect. The interaction of grade and method of teaching was significant, and an examination of the means showed that whereas phonics became less effective for children in Grade 2, the sight method became more effective.

Perhaps, as children gain more reading experience they are better able to take advantage of the sight method and less apt to slow down in order to sound out the words.

C-3. F for rows = 26.292. F for columns = 13.467. For the interaction = 1.319. Reject Ho for rows and columns at p <.01. Accept Ho for interaction. The F ratio of 26.292 for rows, behavior mod, was significant, since with 2 and 54 df, a table value of 5.01 was needed at .01 and 3.18 was needed at .05. The F ratio of 13.467 for columns, ritalin, was also significant, since again with 2 and 54 df we had to reject Ho. The F ratio of 1.319, however, was not significant, since with 4 and 54 df we needed an .05 value of 2.56. By re-tabling the data according to the means, we are better able to see that Drug Dose ________________________________________________________________________ ________ 20 mgs.

10 mgs.

Placebo ________________________________________________________________________ ________ No Behavior Mod Mean = 16.429 Mean = 22.000 Mean = 29.143 Behavior Mod every other day Mean = 17.143 Mean = 19.000 Mean = 22.429 Behavior Mod every day Mean = 8.714 Mean = 12.286 Mean = 14.429 The more behavior mod, the lower the scores, and the higher the drug dosage the lower the scores. However, behavior mod and drug did not interact to produce a significant difference.

402

C-4. F rows (Time) = 15.174 (Significant at p < .01) F columns (Drug) = 0.697 (Not significant) F interaction = 6.271 (Significant at p < .05) The F ratio for rows shows that immediate testing (time) produced a significant advantage over the delayed condition. The F ratio for drug, however, showed no significant difference. The interaction of time and drug indicated significance and showed that in the delayed condition the drug produced more interference with the retention than did the placebo.

403

CHAPTER 13 C-1. Chi Square = 118.4194 (significant at p <.01) C-2. Chi square = 22.1340 (significant at p <.01) C-3. Chi Square = 4.0257 (significant at p <.05) C-4. Chi Square = 1227.083 (significant at p <.01). Assaults are a function of time of day.

C-5. Chi Square = 10.232 (not significant). Homicide rates do not differ significantly as a function of the day of the week.

C-6. a. Chi Square = 25.480 (significant at p < .01). b. Juveniles who are adjudicated delinquent on their first referral to court are less likely to later go on to adult prisons.

C-7. Chi Square = 9.350 ( significant at p < .01). Notice that both groups showed some improvement (again showing the power of the placebo effect), but the drug group improved significantly more.

404

CHAPTER 14 C-1. a. The correlation is 0.706 b. With 45 degrees of freedom, an r of .372 is needed at .01, so therefore the correlation is significant. c. The regression equation is Ypred= .609X + 241.236 Predictions are as follows: X Y pred 370 466.566 610 612.726 510 551.826 700 667.536 C-2. a. The r between alcohol use and drug use is a significant .616. b. The r between alcohol use and anger is a significant .456 c. The r between drug use and anger is a significant .452 d. R = .505 e. Predicts an anger score of 240.7304 C-3.a. For the SAT scores, the mean is 500 and the median is 500. Our rough approximation of skewness, based on the comparison of the mean and median indicates no skew. (Working out the exact Skew value from the formula shown in Appendix, part A., Sk = -.244, which in effect means very little skew). b. For the SAT scores, with a high score of 750 and a low score of 250, the range is 500 and the estimated sd = 102.5978. The sample SD is 100. c. For the GPA scores, the mean is 2.70 and the median is 2.70. (Working out the exact Skew value from the formula shown in Appendix, part A., Sk = 0.00, which, of course, means absolutely no skew). d. For the GPA scores, with a high score of 4.00 and a low score of 1.40, the range is 2.60 and the estimated sd = .4318. The sample SD is 0.421. e. The Pearson r between SAT and GPA is .508 and is significant at .05. f. The regression line is Ypred = .002X+1.632 g. SAT of 470 predicted 2.572, 560 predicted 2.752 and 590 predicted 2.812.

C-4. a. For the IQ distribution, M = 112.00, Mdn = 110.00, R = 50 b. r between IQ and SAT = .690 and is significant at p < .01. c. r between IQ and GPA = .622 and is significant at p < .01. d. Partial correlation between SAT and GPA, with IQ as the variable being partialled out is .1383, p > .05, not significant.

405

CHAPTER 15 C-1. The Pearson r correlation is 0.770. The table value for r at .01 with 29 degrees of freedom equals 0.456. Thus the null hypothesis is rejected, and the correlation is found to be significant. This shows that the matching process is effective. The groups have been proven to be dependent. A paired t is performed, with df allocated on the basis of the number of paired scores minus 1. The paired t = -3.758, t at .01 with 30 df = 2.750. The null hypothesis is rejected at p <.01, and we may conclude that the discovery teaching technique led to higher scores.

C-2. F = 19.868. Reject the null hypothesis, since F at .01 with 2 and 28 degrees of freedom = 5.45. The means for the three groups showed a successive decline in the scores as the crowding became more severe.

C-3. The Pearson r, calculated between the two sets of change scores, was as follows: the Pearson correlation is 0.661, and r at .01 with 18 df = .561. Reject Ho; significant at p < .01. We have now established statistical proof that the matching process was effective, and we move on to determine if the differences between the groups was significant. The value of the t-statistic for the paired t-test is -2.222, and is significant at p < .05. with 19 df = 2.09. Reject Ho; significant at p < .05. Since the mean change for Group A was 4.900 and the mean change for Group B was -3.500, we conclude that Group A changed more than did Group B. In this case, negative change scores indicated that for both groups impulsivity was reduced. However, the Social Skills and Perception Training was shown to be more effective in reducing impulsivity more that was the mere passage of time.

C-4. a. t = 4.727, and with 15 degrees of freedom this is significant at the .01 level. b. r = 0.932, which with 14 degrees of freedom is significant at the .01 level. The matching was effective.

C-5. With an r of .737, significant at p < .05, the t ratio = 4.186, significant at p < .01.

C-6. With an r of .991, significant at p < .01, the t ratio = -.396, which is not significant.

It appears that coaching has no effect.

C-7. With an r of .996, significant at p <.01, the t ratio is –2.862, which is significant at p <.05. For those scoring 500 or better on the pre test, the coaching does appear to work.

Alas, the rich get richer and the poor stay poor.

406

CHAPTER 16 C1. U = 10.000, and with a Z of -1.283, p>.05. ns. Accept Ho. There was no significant difference between the two groups.

C2. H (as Chi Square) = 6.860, p<.05. Reject Ho. The differences are significant, with those arrested between ages 12 to 15 showing the most problems (highest ranks for problems), those arrested between the ages of 16-19 having the next highest problem ranking and those arrested at age 20 or above the least.

C3. a. U = 8.000, and the Z = -2.521 the p < .05. The difference is significant with those without felony convictions being perceived as better looking . b. This is post-facto. The researcher would not be able to manipulate convictions c. The IV is prior convictions and the DV is “looks”.

407

CHAPTER 17 C-1. The Pearson r between the test and retest was a significant (at .01) value of 0.919.

No Spearman-Brown was needed for correction since the whole test was used in both situations. The Spearman rs of -0.868 between test scores and ranks was also significant at p < 01. Remember to change the sign of this correlation since it was pairing interval with ordinal data. See Sprinthall, 2009, “SPSS From A to Z”.

Thus, the test was both reliable and valid.

C-2. The Pearson r between the odd and even items was a significant (at .01) value of 0.8327. With the Spearman-Brown correction, the reliability would be reported as 0.908.

The Spearman rs between the total test scores and the ranks was a significant (at p <.01) value of -.886. Again, the sign must be changed because the computer was comparing interval with ordinal data. Thus, the test was both reliable and valid.

C-3. Alpha = .9019, indicating a high degree of reliability.

408

ANSWERS TO EVEN-NUMBERED TEXT PROBLEMS

The answers given here are those for all the even-numbered problems found at the end of each chapter..

CHAPTER 1 1-2. The ad is misleading, since we are not given the yearly production figures. Perhaps 90% of the cars built during the past 15 years, were, in fact, built only last year, so of course they are still on the road.

1-4. No. We must first examine the company profits of a year ago. An increase of 150% may, in fact, still be only a matter of pennies. Suppose that last year the company had earned 2 cents. The impressive-sounding increase of 150% would really only amount to 3 cents.

1-6. It may have been that the troopers in unmarked cars encountered at least 37% more speeding cars than did the troopers in marked cars. Perhaps the visibility of the marked cars had a deterring effect on potential speeders.

Also, we should be told how many unmarked versus marked cars were on the road.

If more troopers in unmarked vehicles had been working, then of course they gave out more tickets.

1-8. Not necessarily, since it depends totally on the definition of that key word, "parts". Suppose, for example, that we identify four parts, (1) a valve cap, (2) a needle for the speedometer, (3) a knob to turn on the headlights, and (4) the car's engine. Assume further that of the 4 listed parts, Japan only builds the engine. Now, although 75% of the "parts" have indeed been manufactured in the U.S., the hourly-wages needed to produce these parts tilts heavily toward Japan.

1-10. It is possible that the researchers have their cause-and-effects backwards. That is, it may be that some wives stay at home because their husbands earn enough to allow it. In some families, where the husband is not earning at a high enough level, the wife has no choice but to join the workforce (especially if the couple wants to continue making the mortgage and car payments). Another possible explanation is that having a wife at home to take care of the children and cover the house-hold responsibilities allows the husband to focus more fully on his job, to work later, and get research and other help at home etc. In short, having a wife at home may be like having two people working at the same career.

1-12. You should find out how many auto deaths occur on any similar four-day time period. Interestingly, you will discover the numbers are just about equal, with or without the holiday.

409

1-14. Certainly the author's conclusion may be valid. However, other explanations are possible. For example, there are some developmental disabilities that impact both a child's height and IQ and those with lower IQs might not be progressing through the grades at a normal rate. Also, perhaps the teacher's perceptions are accurate and that many of the shorter children are in fact late maturers, both physically and cognitively.

1-16. Other issues may have intruded. For example, perhaps teens who are on teams have less free time, are more tired and have less time for unhealthy activities. Also they may have parents who promote healthy diets and behaviors.

1-18. It might be that the victims are more prone to neurotic symptoms to begin with and that's one of the reasons they are picked on. In fact research has shown (Olweus, 1991) that victims have high levels of anxiety, low levels of self esteem and a pattern of physical weakness. Ironically, the bullies scored higher than their victims on a variety of measures of mental health.

410

CHAPTER 2 2-2.

a. 50%

2-4.

a. 150% b. 25% c. -50% d. -70%

2-6.

Mean = 39.50, Median = 40.00 and the Mode =40.

2-8.

b. 16%

a. Mean = 10.615, b. Median = 11.000, c. the Mode =11.

2-10. a. Mean = 63.222, Median = 63.000 and the Modes = 62 and 65. b. Use the modes, since the distribution is bimodal.

2-12. a. Mean, b. Mode 2-14. Sk+ 2-16. Median 2-18 Mode, since the distribution is bimodal.

2- 20. T 2- 22. F (To prove this, look at the set of scores in problem 9. Halfway between 3, the lowest score, and 100, the highest score, is 51.5. Yet the median is only 12.00.

2-24. F 2-26. F

411

CHAPTER 3 3-2.

Mean = 10.375 , Range = 19 and SD = 7.26

3-4. a. Mean = 22.857, b. Range = 35, c. SD = 12.20 3-6. a. Range = 30, b. Median = 118, c. Interquartile range = 8, d. Interdecile range = 22.

3-8. M= 6.000, Mdn= 6.000, Mo = 8, R = 7, SD = 2.191 (or as estimated by SPSS std.dev. =2.309) 3-10. M = 8.846, Mdn = 9.000, Mo =10, R = 12, SD = 3.159 (or as estimated by SPSS std.dev. =3.288) 3-12. A negative SD can never be calculated (if the formula is correctly followed).

3-14. a. leptokurtic,Ku+, b. probably (but not guaranteed to be) mesokurtic, or Ku=0, c. platykurtic,Ku-.

3-16. Range and Standard Deviation 3-18. Variance should approximate 225 (the square of the SD).

3-20. F 3-22. F 3-24. F 3-26. T

412

CHAPTER 4 4-2. a. 91.77%, b. 55.17%, c. 19.49%, d. 2.50% 4-4. a. 49.53%, b. 94.43%, c. 37.63%, d. 22.17% 4-6. 21.77% 4-8. a. 99.87%, or percentile rank of 100 b. .62%, or percentile rank of 1 c 56.75%, or percentile rank of 57 d. 99.38%, or percentile rank of 99 4-10. a. 23.28% b. 23.28% 4-12. a. 50, b. 50, c. 50 4-14. a. 50%, b. 68%, c. 95% 4-16. F 4-18. T 4-20. T 4-22. T 4-24. T

413

CHAPTER 5 5-2. a. 338.64, b. 593.55, c. 306.45, d. 438.69.

5-4. 76.40 5-6. a. 11.48, b. 68.85, c. 154.43, d. 85.57 5-8. 33.38 5-10. a. 55.56, b. 34.72, c. 63.89, d. 48.61.

5-12. a. -1.00 and + 1.00, b. 40 and 60.

5-14. T 5-16. T 5-18. T 5-20. T 5-22. 94.60 5-24. 490.00 5-26. F Either the measures of central tendency are too high, or the high value of 25 is incorrect.

5-28. F Either the measures of central tendency are incorrect OR the distribution is not normal.

5-30. F If the scores are really normally distributed, then the SD is too high.

It should be R/6 = 15.

414

CHAPTER 6 6-2. .65 6-4. a. .006 = .01, b. .309 = .31, c. .406 = .41.

6-6. .62 6-8. p = .07 6-10. p = .284 6-12. a. .125 = .13, b. .633 = .63, c. .497 = .50 6-14. .0014 6-16. a. .067 b. .933 c. .063 6-18. a. .091 b. .023 c. .886 6-20. 1.00 6-22. 50% 6-24. T 6-26. F 6-28. T 6-30 T

415

CHAPTER 7 7-2. p = 0.159, or .16 7-4. a. Standard error of the mean = 0.566 a. .566 (.57) b. z = 3.534 (3.51); reject Ho at P <.01. Since the calculated z of 3.534 is greater than the z score of 2.58 (which is known to exclude the extreme 1% of the distribution), the sample probably does NOT represent the known population.

7-6. 493 men.

7-8

855 men

7-10. Population.

7-12. Parameter 7-14. Sampling error 7-16. Sample mean 7-18. Standard error of the mean 7-20. 1. all samples selected randomly.

2. from a single population 3. and of sufficient size 7-22. F 7-24. F 7-26. T 7-28. Parameter 7-30. Parameter 7-32. Statistic

416

CHAPTER 8 8-2. t = 7.246. Reject Ho at p < .01. The difference is significant.

The sample mean is probably not representative of the population.

8-4. t = 2.144. Reject Ho at p < .05. The difference is significant.

The sample is probably not representative of the population.

8-6. a. At .95 the interval would be from 33.383 to 36.617. b. At .99 the interval would be from 32.837 to 37.163.

8-8. a t = 22.411 b. Significant at p<.01. This sample does not represent a population whose mean is .

21.

8-10 t = 6.708. Reject Ho; p < .05 and < .01. The illusion occurred.

8-12 a. t = 6.139 p<.01 Sample does not represent the known population. b. On a probability basis this sample represents a population that is higher in psychopathy than is true for the normal population. This should not be too surprising since this is a sample of prison inmates. .8-14 The normal distribution 8-16 The null hypothesis 8-18 Alpha error or the Type 1 error 8-20 3.106 8-22 Confidence Interval 8-24 Sample 8-26 T 8-28 . T 8-30. T

417

CHAPTER 9 9-2. Hypothesis of difference: experimental 9-4. Hypothesis of difference: experimental 9-6. a. experimental (although because of the shifting audience, different subjects are used in the pre and post conditions.) b. repeated-measures (before-after design) c. other variables which might confound are such things as: (1) weather (warm weather might increase Coke sales, and while buying Coke it might be easy to add popcorn), (2) the content of the movie, since scenes depicting eating behavior might induce hunger, (3) any change in the location or display of the Coke or popcorn booths could produce sales changes.

9-8. Whether the subjects received fluoride; manipulated 9-10. Whether the subjects received coordination training; manipulated 9-12. The "subliminal" message; manipulated 9-14. Problem #4 (as is usually the case with one-group before-after designs).

9-16. Obtaining true matches on variables which are indeed known to be related to the dependent variable.

9-18. a. IV is the times of the year (the major holidays) b DV is the heart attack c. Over indulging in food and/or drink, general excitement during holidays, being away from the security of one’s home 9-20. a. IV whether married b Earned income c. Perhaps the married twin works harder because he’s married and has a family to support. Perhaps the married twin is more generally appealing, both to women and to the work-place management 9-22

Post-facto. The IV is whether the Red Sox play-off games were on TV. The DV was based on the number of emergency room visits.

9-24

T

9-26 . T 418

9-28 . T 9-30 . T 9-32 . T 9-34 . T 9-36

T

9-38

Interval

9-40 . Ordinal 9-42

Nominal

419

CHAPTER 10 ANSWERS 10-2 a. t = -1.594. t at .05 with 18 df = 2.101; Accept Ho. b. No, the difference is not significant. c. Post Facto d. IV = college major and the DV = need dependence 10-4. a. t = -2.191. t at .05 with 60 df = 2.000; Reject Ho. b. Yes, the Difference is significant at p<.05. c. Experimental d. IV = whether they saw the movie and the DV = level of prejudice 10-6. a. t = 5.388. Reject Ho. b. Yes, the difference is significant at p< .01. c. Experimental d. IV = the type of light and the DV = depression scores 10-8 a. t = .190. Accept Ho. b. No, there is no significant difference c. Experimental d. IV = food supplement and the DV = weight e. Not needed since t is not significant 10-10. a. t = 2.928, b. Yes, significant at p <.01. Inmates who had committed violent crimes were greater risks than those who had not. c. Post facto d.. The IV (a subject variable) is whether they had committed violent crimes and the DV is their Level of Security Risk.

10-12 t=17.777, p<.01, Yes, the scores are significantly different.

10-14. Zero 10-16. Decrease. Less likely to reject Ho. 10-18. Distribution of differences.

10-20. Alpha 10-22. Ho 10-24. F

420

10-26. F 10-28. F 10-30. T

421

CHAPTER 11 11-2. r = -.976. r at .01 with 6 df = .834. Reject Ho, p < .01. The correlation is significant. There is a significant association between height and running speed among elementary school children. Taller children tend to have lower scores (faster speed), than do shorter children. Since a key third factor, age, has not been included, it may be that the association is more due to the age of the child than to the child's height.

11-4. a. rs = .519 b. Accept Ho. Correlation is not significant.

11-6. a. r = .649. Reject at p<.05. The correlation is significant. b. This correlation probably does exist in the population from which the samples were selected.

11-8. rs = -.865. Reject Ho, at p < .01; the correlation is significant.

Both these hypotheses may be true but neither is proven by this correlational analysis. It shows as a negative correlation because ranks 1,2 and 3 for example are high scores, but the actual numbers are low.

11-10.a. Pearson r = -.612 b. Reject Ho at p<.05. The correlation is significant.

11-12. Z = 1.369. Accept Ho, and conclude that the two correlations are not significantly different and do not represent separate populations.

11-14. A positive correlation, since the high scores are linking with high scores and the low scores are linking with low scores.

11-16. The central tendency of BOTH distributions.

11-18. Spearman rs 11-20. With equal sample sizes, the Pearson r is more powerful in detecting a correlation in the population when there really is such a correlation.

11-22. T 11-24. T

422

11-26. T 11-28. F 11-30. T

423

CHAPTER 12 12-2.HSD at .05 = 1.810 HSD at .01 = 2.320 At .05, all mean differences are significant except between groups 2 and 3.

At .01, significant mean differences can be found between groups 1 and 3, 1 and 4, and 3 and 4.

12-4a. F = 24.667 b. Sig at p <.01 c. HSD at .05 = 2.793, and HSD at .01 = 3.840 d All the mean differences are significant at p<.05 12-6. a. F = 2.400 b. Accept Ho, not significant. c. Tukey's not appropriate 12-8. a. The drug, Factor A, was in the columns. F for columns = 17.064; Reject Ho, Significant at p < .01 b. Psychotherapy, Factor B, was in the rows. F for rows = 2.217; Accept Ho, not significant. c. F for the interaction was 0.473; Accept Ho, not significant.

10-10 a. Three levels b. With 2 and 18 df, an F ratio of 3.55 is needed to reject Ho at .05.

The row F of 65.000 was significant at .05. c. With 2 and 18 df, an F ratio of 6.01 is needed to reject Ho at .01.

The row F of 65.000 was significant at .01. d. Three levels e. As shown above, with 2 and 18 df, an F ratio of 3.55 is needed to reject Ho at .05. The column F of 27.000 was significant at .05.

With 2 and 18 df, an F ratio of 6.01 is needed to reject Ho at .01.

The column F of 27.000 was significant at .01. g. With 4 and 18 df, an F ratio of 2.93 is needed to reject Ho at .05.

The interaction F was 2.285 and was therefore NOT significant at .05. h. Since the interaction F was not significant at .05 it could not be significant at the stricter level of .01. i. There was a total of 27 subjects in all the groups combined. j. There were 9 groups of 3 subjects each.

12-12. Total Mean 12-14. Larger 12-16. Between SS and Within SS 424

12-18. More Leptokurtic 12-20. Between 12-22. F 12-24. T 12-26. T 12-28. T

425

CHAPTER 13 13-2. Chi Square = 6.000; Reject Ho. Significant at p < .05.

13-4. Chi Square = 138.131; Reject Ho. Significant at p< .01.

13-6. Chi Square = 8.163; Reject Ho. Significant at p < .01.

13-8. Chi Square = 75.556; Reject Ho. Significant at p < .01.

13-10. Chi Square = 9.890; Reject Ho. Significant at p < .01.

13-12. Chi Square = 17.766. Reject Ho C = 0.236. Significant at p<.01.

13-14. Chi Square = 6.618. Reject Ho. Significant at p <.05. C =0.249 13-16. a. Chi Square = 3.838. Yes there is a significant difference when using a rounded, two-place chi Square. b. Perhaps the gender of the driver should be considered. Also, perhaps the helping scene should be varied so that the helper is a woman half the time. Other variables to consider might be the age of the driver, weather conditions, time of day and the amount of traffic. Finally, perhaps another control could have been added where a new scene was created in which the driver would have seen in succession two scenes of women alone attempting to change their tires. In this way the researcher would have been better able to establish whether the helping behavior in scene 1 provoked the driver to stop, or whether just seeing a woman in distress in a new scene would produce similar results through the sensitizing process.

13-18. Chi Square = 0.448.. Accept Ho, ns.

13-20 Chi Square = 22.289 Reject Ho Significant at <.01 13-22 fo = fe 13-24 Repeated Measures or Matched Subjects 13-26. T. 13-28. T 13-30. F

426

CHAPTER 14 14-2. a. 85.585 b. 103.575 c. As correlation decreases, predictions regress toward the mean.

14-4. a. Digit Span and Block design r = .787 b. Sig at p<.05 c. Digit Span and Total IQ r = .954 d. Sig at p < .01 e. Block Design and Total IQ r = .778 f. Sig at p < .05 14-6 a. 76.500 b. 97.505 14-8. a. 69.608 b. 71.546 c. 75.908 14-10. a. r = .667 Sig at p< .05 b. r = .800 Sig at p< .01 c. r = .695 Sig at p <.05 14-12. Multiple R = .901 14-14. Partial r = .140 14-16 Partial r = .583 14-18 Multiple R =.606 14-20 F = 2.899. Accept Ho. Multiple R is no longer significant 14-22 13.122 14-24 Not significant 14-26 Partial correlation of .375 and 15 df is NOT significant 14-28 F =2.857 Multiple R is NOT significant 14-30. Regression Line 14-32. Partial Correlation 427

14-34. The mean of Y 14-36. A pair of scores 14-38. + 1.00 14-40. T 14-42. T 14-44. T 14-46. T

428

CHAPTER 15 15-2. t = 4.311; Reject Ho at p< .01.

15-4. t = 2.236 Accept Ho ns.

15-6. t = 2.236 Accept Ho ns .

15-8 a. Experimental b. IV is diet pill vs placebo, and DV is weight change. c. t = -1.627 Accept Ho ns.

15-10 For this question assume that for both tests all their assumptions are met/ The paired t has more power for equal numbers of scores. The correlation between the scores will reduce the standard error of difference and thereby increase the size of the t ratio ( increasing the likelihood of rejecting Ho).

15-12.The correlation will act to reduce the value of the estimated standard error of difference.

15-14 .Since the correlation will act to reduce the value of the estimated standard error of difference, it will therefore INCREASE the value of the t ratio.

15-16. Since the paired t has fewer degrees of freedom than does its independent counterpart, that fact alone tends to make it less likely to achieve statistical significance. This loss of power, however, is typically compensated for by the higher value of the resulting t ratio.

15-18. More able.

15-20. F. 15-22. F. 15-24. T. 15-26. T. 15-28. T. 15-30. T. 15-32 T 429

CHAPTER 16 16-2. H = 3.285; Accept Ho. Difference is not significant.

16-4. Friedman = 10.400; Reject Ho. Difference is significant at p< .01.

16-6. Friedman = 20.000; Reject Ho. Difference is significant at p< .01.

16-8. Zu = -2.495. Difference is significant at p < .05.

16-10. H =8.540, Reject Ho. Difference is significant at p<.05.

16-12. Wilcoxon T. 16-14. Kruskal-Wallis H 16-16. T 16-18. F. 16-20. Wilcoxon T. 16-22. Friedman ANOVA by ranks 16-24. Kruskal-Wallis H 16-26 Friedman ANOVA .by ranks

430

CHAPTER 17

17-2. rtt = .943 and with a p < .01.

The test is reliable.

17-4. a. .907 b. .864 c. .958 d. .667 e. r of .40 is not significant for an N of 20, so the whole-test reliability could not be calculated. Had the correlation been significant, the whole-test reliability would have been .571.

7-6. a. DI = .20 b. DI = .25 c. DI = .40 d. DI = .30 e. DI = -.20 17-8. a. rs = .709 and with a p of less than.05, the test is valid. b. This would be an example of predictive validity.

17-10. a. 256.096 to 343.904 b. 456.096 to 543.904 c. 506.096 to 593.904 d. 606.096 to 693.904 17-12. a. 212.388 to 387.612 b. 412.388 to 587.612 c. 462.388 to 637.612 d. 562.388 to 737.612 17-14. rtt = .640 17-16. SEM = 3.013 17-18. rtc = .714 17-20. F = 40.779, Reject Ho at p < .01. The F ratio is significant and rtt = .975. The test is highly reliable.

17-22. a .844 b. Increase of 30% 431

17-24. T 17-26. T 17-28. F 17-30. T 17-32. T 17-34 Reliability 17-36. Reliability 17-38. Reliability 17-40. Validity

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