Test Bank For Intermediate Statistical Investigations 1st Edition. Nathan Tintle, Beth L. Chance, Ka

Test Bank For Intermediate Statistical I nvestigations 1e Nathan Tintle, Beth L. Chance, Karen Mc Gaughey, Soma Roy, Todd Swa nson, Jill VanderStoep

Chapter 1-6 Chapter 1 Intermediate Statistical Investigations Test Bank Question types:

FIB = Fill in the blank Ma = Matching MC = Multiple choice

Calc = Calculation MS = Multiple select TF = True-false

CHAPTER 1 TERMINAL LEARNING OUTCOMES TLO1-1: Apply the six-step investigative process in the context of a well-designed experiment.

TLO1-2: Partitioning variation in the response variable into variation explained by the model and unexplained variation, and measuring and reporting the percentage of variation explained TLO1-3: Assess the statistical significance of the difference between two groups on a quantitative response variable using both simulation and theory-based approaches TLO1-4: Compare more than two treatments on a quantitative response using both simulation and theory-based approaches TLO1-5: Apply Post-hoc analysis after significant F-test (pairwise differences, as well as confidence and prediction intervals for single means) TLO1-6: Understand statistical power and how it is impacted by sample size, variability within groups, number of groups, and significance level

Section 1.1: Sources of Variation in an Experiment LO1.1-1: Apply the six-step investigative process.

LO1.1-2: Distinguish experiments and observational studies.

LO1.1-3: Review basic study design principles such as inclusion criteria and random assignment.

LO1.1-4: Define terminology specific to an experimental study (e.g., treatments).

LO1.1-5: Produce a Sources of Variation diagram for an experiment.

Questions 1 through 3: A study published in Psychological Science in 2007 examined a possible link between mindset and health. The following is an excerpt from the abstract of the article: ―84 female room attendants working in seven different hotels were measured on physiological health variables affected by exercise. Those in the informed condition were told that the work they do (cleaning hotel rooms) is good exercise and satisfies the Surgeon General's recommendations for an active lifestyle. Examples of how their work was exercise were provided. Subjects in the control group were not given this information.‖ 1. Identify the experimental units in this study.

A. The eighty-four room attendants FOR INSTRUCTOR USE ONLY

1-2 B. The seven different hotels C. The physiological health variables D. The two groups (informed and control) Ans: A; LO: 1.1-4; Difficulty: Easy; Type: MC 2. The researchers chose to include room attendants from seven different hotels (as opposed to using stricter inclusion criteria that would limit the study to room attendants at one particular hotel). Describe the consequences of this decision.

Using broader inclusion criteria may ______ (increase/decrease) the amount of variation in the observed health outcomes. However, this decision also __________ (supports/limits) generalizability to a larger population of room attendants.

Ans: increase, supports; LO: 1.1-3; Difficulty: Medium; Type: FIB 3. Room attendants were randomly assigned to either the informed condition or the control group. What is the most important reason for the random assignment?

A. Random assignment ensures that the study is double-blind.

B. Random assignment reduces the impact of outliers.

C. Random assignment creates two groups of room attendants that are as similar as possible, which supports cause-and-effect conclusions.

D. Random assignment makes it possible to generalize the results to the population.

Ans: C; LO: 1.1-3; Difficulty: Medium; Type: MC Questions 4 through 6: An online retailer is using an experiment to decide whether to modify their website. When visitors type in the web address or click a link to the site, they are randomly re-directed to one of two versions of the website: the version that has been in use for the last year (version A) or an updated version (version B). The retailer’s goal is to maximize the amount of time (in minutes) visitors stay on the site.

4. Identify the experimental units and variables. Note: One of the answer choices will not be used.

Experimental units: Explanatory variable: Response variable:

A. Version of the website (A and B) B: Online retailers C: Visitors to the website D: Time spent on the website (in minutes)

Ans: Experimental units: C, Explanatory variable: A, Response variable: D; LO: 1.1-4; Difficulty: Easy; Type: Ma 5. Consider two possible models for analyzing time spent on this retailer’s website.

Single-mean model: A. Predicted time spent on site 12.33, SE of residuals  4.64 Separate-means model: FOR INSTRUCTOR USE ONLY

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 9.9 forVersion A , SE of residuals  3.98 14.7 forVersion B

B. Predicted time spent on site  

Does the version of the website appear to explain any of the variation in time spent on the site? Note: If more than one of these justifications is appropriate, select multiple answers.

A. Yes, because the mean time spent on the site is higher for Version B than for Version A. B. Yes, because the SE of the residuals is smaller for the separate-means model than for the single-mean model.

C. No, because the mean time spent on the site is not the same for Version A and for Version B. D. No, because the SE of the residuals is smaller for the separate-means model than for the single-mean model.

Ans: A, B; LO: 1.1-1; Difficulty: Medium; Type: MS 6. The researcher decides that the difference between Version A and Version B in this study is meaningful. Is it reasonable to generalize these results to all customers of this retailer?

A. Yes, because visitors to the website were randomly assigned to either Version A or Version B. B. Yes, because the study’s inclusion criteria would exclude potential subjects who are not customers.

C. It depends whether visitors to the website knew about the research question being investigated. The study may not be double-blind.

D. It depends who visited the website during the study period. The sample may not be representative.

Ans: D; LO: 1.1-1; Difficulty: Medium; Type: MC Questions 7 through 8: Researchers at a university were interested in the effectiveness of a calculus workshop program for students who fail Calculus I and need to retake the course.

As part of the study, students who were retaking Calculus I were allowed to enroll in a calculus workshop at their own discretion. At the end of the grading term, all students (even those with different instructors) took the same final exam. The researchers then compared the scores for those who enrolled in the workshop while re-taking calculus to those who re-took calculus without enrolling in the workshop.

7. Is this an experiment? Justify your answer.

A. Yes, this is an experiment, because there was a treatment group who enrolled in the calculus workshop and a control group that did not.

B. Yes, this is an experiment, because the study was double-blind (as long as the calculus teachers did not know which students enrolled in the workshop).

C. No, this is an observational study, because it does not take place in a laboratory or other tightly controlled research environment.

D. No, this is an observational study, because the choice of whether to participate in the workshop was made by the students not the researchers.

Ans: D; LO: 1.1-2; Difficulty: Easy; Type: MC FOR INSTRUCTOR USE ONLY

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8. Which of the following are sources of unexplained variation in this study? Select all that apply.

A. Whether or not students enrolled in the workshop B. Whether or not students had failed a calculus class in the past C. Student attendance in class (number of absences) D. Student motivation to study calculus E. Calculus instructor F. Difficulty of the final exam Ans: C, D, E; LO: 1.1-5; Difficulty: Easy; Type: MS 9. A study published in Athletic Training examined the effects of three different types of knee stabilizing braces on agility test speed. College football players from all different positions (running back, wide receiver, linebacker, lineman, etc.) were recruited to participate in the study. All players in the study had torn their ACL (anterior cruciate ligament) in the past, and needed to wear a knee brace to play football. Agility tests were administered in an outdoor football stadium, and the time to complete the test was recorded by a Lafayette photoelectric Cell and Light Time Unit (in seconds).

Put the components of the study into the correct boxes in the Sources of Variation Diagram. Note: Some boxes will include more than one answer.

Observed variation in:

Sources of explained variation

Sources of unexplained variation

Inclusion criteria: Design:

A. College football players from all different positions B. Type of knee brace C. Players’ current condition (health, mood, motivation, etc.)

D. Time to complete agility test (in seconds) E. Measurement error F. History of torn ACL and need for a knee brace G. Details of the agility test H. Players’ natural speed and agility I. Environmental factors (weather, wind, etc.)

Ans: Observed variation in: D; Inclusion criteria: A, F; Design: G; Sources of explained variation: B; Sources of unexplained variation: C, E, H, I; LO: 1.1-5; Difficulty: Medium; Type: Ma 10. In a separate-means model, the standard error of the residuals can be thought of as the typical deviation of an observed response from: A. The residuals (prediction errors) B. The response predicted by the model (group mean) C. The overall mean of the response variable D. The overall mean of the explanatory variable FOR INSTRUCTOR USE ONLY

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Ans: B; LO: 1.1-1; Difficulty: Easy; Type: MC 11. A study published in the Journal of Sports Science & Medicine tested the effectiveness of the Power Balance © bracelet, which has been marketed as a way to improve balance, flexibility, strength, and power through the use of hologram technology. Subjects, who were all college athletes, completed tests of their athletic performance while wearing either a Power Balance © bracelet or a plain rubber placebo bracelet. The bracelets were covered with a wristband, so the athletes and those measuring their performance could not see which bracelet was being worn. Only the researcher who analyzed the data knew which measurements corresponded to the Power Balance © bracelet and which to the placebo bracelet. Classify this study.

A. This study is not blinded.

B. This is a single-blind study.

C. This is a double-blind study.

D. There is not enough information to classify this study.

Ans: C; LO: 1.1-4; Difficulty: Easy; Type: MC 12. Which of the following is an experiment? Select all that apply.

A. Executives at a large department store chain selected 100 stores and randomly assigned 50 of them to reduce their hours, opening an hour later than before; hours for the other 50 stores were not changed. After six months, the executives compared revenue for the two groups of stores.

B. A researcher recruited a group of American adults whose demographics were similar to the American population. The researcher measured each subject’s forced expiratory volume, an indicator of lung function. Then each subject was asked whether or not they smoke cigarettes.

C. A survey was administered to a large group of high school students. The survey asked whether the students were employed outside of school (in a paying job) and how much sleep they got the night before (in hours).

Ans: A; LO: 1.1-2; Difficulty: Medium; Type: MS 13. A university professor teaches two sections of introductory statistics: one section meets at 8:00 am and the other meets at 11:00 am. She wants to evaluate the effectiveness of a new method of teaching statistics compared to the standard method she has used in the past. She flips a coin to assign her sections to teaching methods and determines that she will use the standard method at 8:00 am and the new method at 11:00 am. At the end of the term, she compares the final exam scores for the two sections. Which of the following best describes the potential for confounding in this scenario?

A. Different students have different levels of talent and motivation, so it is impossible to attribute differences in final exam scores to the teaching method.

B. The two sections may not be exactly the same size, which would lead to inappropriate comparisons between the treatment and control groups.

C. Students may find it difficult to pay attention at 8:00 am, which may negatively impact the exam scores of those taught with the standard method.

D. Confounding is not a concern in this scenario, because the professor used random assignment as part of the study design.

Ans: C; LO: 1.1-1; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

1-6 14. True or False: The standard error of the residuals is a way to measure the amount of variation in the response variable that remains unexplained after applying the model.

Ans: True; LO: 1.1-1; Difficulty: Easy; Type: TF 15. True or False: Because of the possibility of confounding, you should always avoid using causal language (action verbs like ―affect‖ and ―lead to‖) in your conclusions.

Ans: False; LO: 1.1-1; Difficulty: Easy Type: TF

Section 1.2: Quantifying Sources of Variation LO1.2-1: Partitioning variation in the response variable into variation explained by the model and unexplained variation.

LO1.2-2: Measuring percentage of variation explained.

LO1.2-3: Understanding effect size and practical significance.

Questions 1 through 3: Dog agility is a sport where trainers guide their dogs through an obstacle course as quickly as possible. Two trainers, Abby and Lauren, have dogs participating in agility competitions. Each dog completes the same agility course, and they measure the time it takes each dog to complete the course (in seconds). The times for Abby’s three dogs were 30, 40, and 50. The times for Lauren’s three dogs were 50, 60, and 70.

1. Calculate the Sum of Squares Total (SSTotal).

Solution:  30  50    40  50    50  50    50  50    60  50    70  50  1000 2

2

2

2

2

2

Ans: 1000  0; LO: 1.2-1; Difficulty: Medium; Type: Calc 2. Calculate the Sums of Squared Errors (SSError).

Solution:  30  40    40  40    50  40    50  60    60  60    70  60   400 2

2

2

2

2

2

Ans: 400  0; LO: 1.2-1; Difficulty: Medium; Type: Calc 3. Calculate the Sum of Squares for the Model (SSModel).

Solution:  40  50    40  50    40  50    60  50    60  50    60  50   600 2

2

2

2

2

2

Ans: 600  0; LO: 1.2-1; Difficulty: Medium; Type: Calc Questions 4 and 5: Dog agility is a sport where trainers guide their dogs through an obstacle FOR INSTRUCTOR USE ONLY

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course as quickly as possible. Two trainers, Abby and Lauren, have dogs participating in agility competitions. Each dog completes the same agility course, and they measure the time it takes each dog to complete the course in seconds. Compare the sums of squares for two possible datasets that could occur in this context.

Dataset 1: Times for Abby’s dogs: 30, 40, 50 Times for Lauren’s dogs: 50, 60, 70

Dataset 2: Times for Abby’s dogs: 20, 40, 60 Times for Lauren’s dogs: 40, 60, 80

4. SSTotal for Dataset 1 __________ (<, >, or =) SSTotal for Dataset 2 SSModel for Dataset 1 __________ (<, >, or =) SSModel for Dataset 2 SSError for Dataset 1 __________ (<, >, or =) SSError for Dataset 2 Ans: <, =, <; LO: 1.2-1; Difficulty: Medium; Type: FIB 5. The R2 value for Dataset 1 __________ (<, >, or =) the R2 value for Dataset 2 Ans: >; LO: 1.2-2; Difficulty: Medium; Type: FIB Questions 6 and 7: An office designer claims that a new ergonomic desk chair makes typing at a computer terminal faster and easier. A client company plans to test it by asking 30 employees who do a lot of typing to take part in an experiment. They will randomly assign 15 employees to use the new ergonomic chair and 15 to use a regular chair. The 30 employees will then type a selected passage for 5 minutes, recording the total number of words that are typed correctly.

FOR INSTRUCTOR USE ONLY

1-8 Consider two hypothetical data sets that could result from this experiment:

6. Which of the datasets would result in a larger R2 value?

A. Dataset 1 would result in a larger R2 value, because SSModel for Dataset 1 is larger than SSModel for Dataset 2.

B. Dataset 1 would result in a larger R2 value, because SSError for Dataset 1 is smaller than SSError for Dataset 2.

C. Dataset 2 would result in a larger R2 value, because SSModel for Dataset 2 is larger than SSModel for Dataset 1.

D. Dataset 2 would result in a larger R2 value, because SSError for Dataset 2 is smaller than SSError for Dataset 1.

Ans: C; LO: 1.2-2; Difficulty: Medium; Type: MC 7. Compare the value of the effects for the two datasets.

A. The effects for Dataset 1 would be larger (in absolute value), because in Dataset 1 there is a smaller difference between the group means.

B. The effects for Dataset 2 would be larger (in absolute value), because in Dataset 2 there is a larger difference between the group means.

C. The effects for Dataset 1 would be the same size as the effects for Dataset 2, because the standard deviations are the same for both datasets.

D. The effects for Dataset 1 would be the same size as the effects for Dataset 2, because the sample sizes are the same for both datasets.

Ans: B; LO: 1.2-2; Difficulty: Medium; Type: MC

Questions 8 through 11: The graphs below display the outcomes of three different experiments FOR INSTRUCTOR USE ONLY

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to compare a treatment group with a control group.

8. For which of the experiments does SSModel = 0? Select one or more than one.

A. Experiment A B. Experiment B C. Experiment C Ans: A, C; LO: 1.2-1; Difficulty: Easy; Type: MS 9. For which of the experiments does SSError = 0? Select one or more than one.

A. Experiment A B. Experiment B C. Experiment C Ans: A, B; LO: 1.2-1; Difficulty: Easy; Type: MS 10. The R2 value for Experiment B is _______ (0, 0.5, 1), because ________ (none, half, all) FOR INSTRUCTOR USE ONLY

1-10 of the variability in outcomes is explained by the treatment group model.

The R2 value for Experiment C is ________ (0, 0.5, 1), because ________ (none, half, all) of the variability in outcomes is explained by the treatment group model.

Ans: 1, all, 0, none; LO: 1.2-2; Difficulty: Medium; Type: FIB 11. The value of the effect for the treatment group in Experiment B is _____ (-4, -2, 0, 2, or 4).

The value of the effect for the treatment group in Experiment C is _____ (-4, -2, 0, 2, or 4).

Ans: -2, 0; LO: 1.2-3; Difficulty: Easy; Type: FIB Questions 12 and 13: A statistics class conducted an experiment to investigate whether standing heart rates tend to be higher than sitting heart rates. Students were randomly assigned to either sit or stand, then the students measured their heart rates (in beats per minute). They used software to calculate the sums of squares and found that SSModel = 614.8 and SSTotal = 13232.1 12. Calculate SSError.

Solution: 13232.1 – 614.8 12617.3 Ans: 12617.3  0; LO: 1.2-1; Difficulty: Easy; Type: Calc 13. Calculate the R2 value. Give your answer as a proportion.

Solution: 614.8 /13232.1 0.046 Ans: 0.046  0.006; LO: 1.2-2; Difficulty: Easy; Type: Calc 14. An online retailer is using an experiment to decide whether to modify their website. When visitors type in the web address or click a link to the site, they are randomly re-directed to one of two versions of the website: the version that has been in use for the last year (version A) or an updated version (version B). The retailer’s goal is to maximize the amount of time (in minutes) visitors stay on the site.

Single-mean model:

Predicted time spent on site 12.3, SE of residuals  4.64 Separate-means model:

 9.9 forVersion A Predicted time spent on site   , SE of residuals  3.98 14.7 forVersion B Calculate the effect for Version A. Solution: 9.9 –12.3 2.4 Ans: -2.4  0; LO: 1.2-3; Difficulty: Medium; Type: Calc Questions 15 and 16: The following output displays the amount (in dollars) that a sample of male and female college students spent on their most recent haircuts.

FOR INSTRUCTOR USE ONLY

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15. Calculate SSModel. Note that the sample sizes are the same for the two groups.

Solution: 20 19.10 – 30.18   20  41.25  30.18   4906.23 2

2

Ans: 4906.23  25 LO: 1.2-1; Difficulty: Medium; Type: Calc 16. Calculate the standard error of the residuals. Note that the sample sizes are the same for the two groups.

Solution:

11.052  32.902  24.54 2

Ans: 24.54  0.1; LO: 1.2-1; Difficulty: Medium; Type: Calc 17. Match each sum of squares (SS) with its description.

SSModel:

A. Measures the amount of variability in the response variable without accounting for groups SSError: B. Measures the variability within groups, the variability unexplained by the model SSTotal: C. Measures the variability between groups, the variability explained by the model Ans: SSModel: C; SSError: B; SSTotal: A; LO: 1.2-1; Difficulty: Easy Type: Ma 18. In general, study results are considered practically significant when the R2 value is ______ (large/small) and the effect sizes are ______ (large/small).

Ans: large, large; LO: 1.2-3; Difficulty: Easy; Type: FIB

19. What is the cutoff for determining whether study results are practically significant?

A. Results are practically significant when the R2 value is less than 0.05.

B. Results are practically significant when the R2 value is less than 0.5.

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1-12 C. Results are practically significant when the R2 value is greater than 0.5.

D. Results are practically significant when the R2 value is greater than 0.95.

E. There is no set cut-off for practical significance. It differs based on context.

Ans: E; LO: 1.2-3; Difficulty: Easy; Type: MC 20. If a researcher were unhappy with his/her effect size or R2 value, what steps could they take when planning a follow-up study?

A. They could try to improve the model by adding new explanatory variables.

B. They could try to reduce unexplained variation in the response through experimental controls or stricter inclusion criteria.

C. Both of these strategies are reasonable.

D. Neither of these strategies would impact the effect size or R2 value.

Ans: C; LO: 1.2-3; Difficulty: Medium; Type: MC

Section 1.3: Is the Variation Explained Statistically Significant LO1.3-1: Carry out and evaluate a randomization test comparing two groups on a quantitative response variable.

LO1.3-2: Assess the statistical significance of a two-group comparison.

LO1.3-3: Apply two-sample t-procedures for tests of significance and confidence intervals.

Questions 1 through 3: The following output displays the amount (in dollars) that a sample of male and female college students spent on their most recent haircuts. You may assume that the sample is representative of a larger population.

1. Does this data provide strong evidence that female students spend more per haircut than male students, on average? Choose the appropriate statement of the null hypothesis.

A. H 0 :  F  M  0 B. H 0 :  F  M  0 C. H 0 : xF  xM  0 D. H 0 : xF  xM  0 Ans: A; LO: 1.3-3; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

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2. Interpret the 95% confidence interval.

A. We are 95% confident that the sample mean for females is between $6.10 and $38.20 higher than the sample mean for males.

B. We are 95% confident that the population mean for females is between $6.10 and $38.20 higher than the population mean for males.

C. If we randomly select one male student and one female student from this sample, we are 95% confident that the haircut value for the female student will be higher.

D. If we randomly select one male student and one female student from the population, we are 95% confident that the haircut value for the female student will be higher.

Ans: B; LO: 1.3-3; Difficulty: Medium; Type: MC

3. Based on the 95% confidence interval, we would expect the two-sided p-value to be _______ (>, <, or =) 0.05.

Ans: <; LO: 1.1-2; Difficulty: Medium; Type: FIB Questions 4 through 6: A study published in Psychological Science in 2007 examined a possible link between mindset and health. The following is an excerpt from the abstract of the article: ―84 female room attendants working in seven different hotels were measured on physiological health variables affected by exercise. Those in the informed condition were told that the work they do (cleaning hotel rooms) is good exercise and satisfies the Surgeon General's recommendations for an active lifestyle. Examples of how their work was exercise were provided. Subjects in the control group were not given this information.‖ Over the course of four weeks, the informed group lost an average of 1.79 lbs and the uninformed group lost an average of 0.20 lbs. Are these results statistically significant? To decide, we can use a randomization test. The dotplot below shows 1000 simulated differences in mean weight loss: yinformed  yuninformed .

4. Suppose we want to use the 3S Strategy to investigate whether being informed about how their work qualifies as exercise affects room attendants’ weight loss. How would we design the simulation?

A. Write the weight loss amounts on 84 cards. Shuffle and deal them into two groups to represent the informed and uninformed groups. Calculate the difference of means. Repeat.

B. Write the group labels (informed or uninformed) on 84 cards. Shuffle and deal them into groups to represent weight loss. Calculate the difference of means. Repeat.

C. Both of these designs are appropriate in this context.

D. Neither of these designs is appropriate in this context.

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1-14 Ans: A; LO: 1.3-1; Difficulty: Medium; Type: MC 5. Why is the distribution of simulated statistics centered at 0?

A. Because some of the room attendants in the sample gained weight and others lost weight, but the average is close to 0 B. Because if we repeated this study again, some of the results would be positive and some of the results would be negative, but the average is close to 0 C. Because the statistics were simulated under the assumption that being informed about how their work qualifies as exercise doesn’t affect room attendants’ weight loss D. Because the data in this study do not provide sufficient evidence to conclude that being informed about how their work qualifies as exercise affects room attendants’ weight loss Ans: C; LO: 1.3-1; Difficulty: Medium; Type: MC 6. Estimate the p-value. Include three decimal places in your answer.

Solution: Approximately 8 out of 1000 simulated differences were greater than or equal to 1.59, so we estimate that the p-value is 0.008 Ans: 0.008  0.004; LO: 1.3-1; Difficulty: Medium; Type: Calc Questions 7 through 9: Researchers used an experiment to investigate whether cell phone use impairs drivers’ reaction times. 64 students who volunteered to participate in the study were assigned to one of two driving conditions: cell phone use (n1=32) or no distractions (n2=32). The students then participated in a simulation of driving situations, pressing a brake button as soon as they saw a red light. A device recorded their reaction times (in milliseconds).

7. The standardized statistic for testing whether Cell  Control  0 is t  2.72 . Interpret.

A. The standard deviation of the reaction times is 2.72 milliseconds.

B. The standard deviation of the reaction times is 2.72 milliseconds higher than we would have expected based on the null hypothesis.

C. The sample mean for the cell phone group is 2.72 milliseconds above the sample mean for the control group.

D. The sample mean for the cell phone group is 2.72 standard errors above the sample mean for the control group.

Ans: D; LO: 1.3-3; Difficulty: Medium; Type: MC 8. You want to use a theory-based pooled t-test to assess the statistical significance of the difference between these two groups, so you would use a t-distribution with ____ degrees of freedom.

Ans: 62  0; LO: 1.1-3; Difficulty: Easy; Type: FIB 9. The graph below shows the appropriate t-distribution for assessing the statistical significance of the difference between these two groups. Which of the following statements includes a reasonable p-value and conclusion?

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A. The p-value = 0. This study provides only weak evidence to suggest that cell phone use impairs drivers’ reaction times.

B. The p-value = 0.0042. This study provides strong evidence to suggest that cell phone use impairs drivers’ reaction times.

C. The p-value = 0.2495. This study provides only weak evidence to suggest that cell phone use impairs drivers’ reaction times.

D. The p-value = 0.4856. This study provides strong evidence to suggest that cell phone use impairs drivers’ reaction times.

Ans: B; LO: 1.3-3; Difficulty: Hard; Type: MC Questions 10 and 11: Anchoring is the common human tendency to rely too heavily, or ―anchor‖, on one trait or piece of information when making decisions. A group of statistics students from California were asked to guess the population of Milwaukee, Wisconsin. Some of the students were randomly chosen to be told that the nearby city of Chicago, Illinois, has a population of about 3 million people, while the rest of the students were told that the nearby city of Green Bay, Wisconsin, has a population of about 100,000.

City Chicago Green Bay

N 35 34

Mean 1357.34 271.38

SD 802.21 370.96

10. Based on the information given above, is the effect of anchoring statistically significant in this context?

A. Yes, because the difference of means observed in this sample would be unlikely to occur if anchoring really had no effect.

B. Yes, because the sample means in this study are different and the sample sizes are both larger than 20.

C. No, because the shuffled differences in means are centered at 0, so it is reasonaFOR INSTRUCTOR USE ONLY

1-16 ble to conclude that anchoring has no effect.

D. No, because shuffled differences in means generally fall between -500 and 500.

That means the results of this experiment were an unlikely fluke.

Ans: A; LO: 1.3-2; Difficulty: Medium; Type: MC 11. Are the validity conditions met for a theory-based pooled t-test?

A. No, because the samples are not independent of each other.

B. No, because the sample sizes are not the same.

C. No, because the sample standard deviations for the two groups are very different.

D. Yes. The only potential violation is the skewness in the sample distributions, but this is not a problem, because the sample sizes are both larger than 20.

Ans: C; LO: 1.3-3; Difficulty: Easy; Type: MC Questions 12 through 14: A psychology study (Rutchick, Slepian, and Ferris, 2010) investigated whether using a red pen causes people to assign lower scores than using a blue pen. A group of 128 students in an undergraduate psychology class were asked to grade 128 different eighth graders’ essays on a scale of 0—100. Half of the students were randomly assigned a red pen while grading, and the other half were given blue. The results are given in the table below: Pen Color Red Blue

N 64 64

Mean Score 76.20 80.00

Standard Deviation 12.29 9.36

12. State the alternative hypothesis.

A. H A :  Red  Blue B. H A : xRed  xBlue C. H A :  Red   Blue D. H A : xRed  xBlue Ans: C; LO: 1.3-1; Difficulty: Medium; Type: MC 13. Is it appropriate to use a pooled t-test to compare these groups?

A. Yes, a pooled t-test is appropriate, because the group means are fairly similar.

B. Yes, a pooled t-test is appropriate, because the group standard deviations are fairly similar.

C. No, an unpooled two-sample t-test is more appropriate, because it is not reasonable to assume that the group means are equal in the population.

D. No, an unpooled two-sample t-test is more appropriate, because it is not reasonable to assume that the group standard deviations are equal in the population.

Ans: B; LO: 1.3-3; Difficulty: Medium; Type: MC 14. Calculate the t-statistic. Note that the sample sizes are equal.

12.29  9.36  10.924;t  Solution: SE of residuals = 2

2

2

80.00 – 76.20 1.97 1 1 10.924*  64 64

Ans: 1.97 ± 0.05; LO: 1.3-3; Difficulty: Medium; Type: Calc FOR INSTRUCTOR USE ONLY

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15. In the context of a pooled t-test for comparing two groups, what does it mean to say that the study results are statistically significant? Select all that apply.

A. It means there is large difference between the means of the two groups.

B. It means there is small difference between the means of the two groups.

C. It means the observed sample difference in means would be unlikely to occur if there were really no difference between the two groups in the population.

D. It means the observed sample difference in means would be likely to occur if there were really no difference between the two groups in the population.

Ans: C; LO: 1.3-3; Difficulty: Medium; Type: MS 16. Which of the following statistics reflect both the effect size and the sample size? In other words, which of the following statistics can be used to assess statistical significance?

Select all that apply.

A. R2 B. Difference in means, x1  x2 C. Standardized statistic, t D. p-value Ans: C, D; LO: 1.3-2; Difficulty: Medium; Type: MS 17. True or False: The p-value is the probability that the null hypothesis is true.

Ans: False; LO: 1.3-2; Difficulty: Easy; Type: TF 18. True or False: A small p-value indicates strong evidence against the null hypothesis.

Ans: True; LO: 1.3-2; Difficulty: Easy; Type: TF 19. True or False: A confidence interval is a range of plausible values for a parameter calculated using sample statistics.

Ans: True; LO: 1.3-3; Difficulty: Easy; Type: TF

FOR INSTRUCTOR USE ONLY

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Section 1.4: Comparing Several Groups LO1.4-1: Compare more than two treatments using randomization tests.

LO1.4-2: Calculate an F-statistic and use the F-distribution to find theory-based p-values.

LO1.4-3: Assess the validity of an F-test.

LO1.4-4: Complete an Analysis of Variance table.

Questions 1 through 4: Does seeing a picture have any effect on college students’ understanding of ambiguous prose? 57 students were randomly assigned to three groups: 19 saw a picture before reading a difficult passage of text, 19 saw the picture after reading the passage, and 19 were shown no picture at all. The groups were then tested on their reading comprehension and assigned a quantitative score.

Does this data provide convincing evidence that seeing a picture has an effect on reading comprehension scores?

1. Which of the following is an appropriate statement of the null hypothesis? Select all that apply.

A. H0 : None  Before   After B. H0 :  After  None  Before C. H 0 : At least one  differs from the others D. H 0 : There is an association between seeing a picture (before, after, or not at all) and reading comprehension scores.

E. H 0 : There is no association between seeing a picture (before, after, or not at all) and reading comprehension scores.

Ans: A, E; LO: 1.4-1; Difficulty: Medium; Type: MS

FOR INSTRUCTOR USE ONLY

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2. Which of the following is an appropriate statement of the alternative hypothesis? Select all that apply.

A. H A : None  Before   After B. H A :  After   None  Before C. H A : At least one  differs from the others D. H A : There is an association between seeing a picture (before, after, or not at all) and reading comprehension scores.

E. H A : There is no association between seeing a picture (before, after, or not at all) and reading comprehension scores.

Ans: C, D; LO: 1.4-1; Difficulty: Medium; Type: MS 3. What does the graph of shuffled R-squared statistics represent?

A. The values of R2 for all 57 students who participated in this study B. The values of R2 that would occur if the null hypothesis were really true C. The values of R2 that would occur if the alternative hypothesis were really true D. The values of R2 that would occur if we repeated this study many times in the real world Ans: C; LO: 1.4-1; Difficulty: Medium; Type: MC 4. Does this study provide strong evidence that seeing a picture affects reading comprehension?

A. Yes, because an R2 value of 0.271 does not appear in the graph of shuffled Rsquared statistics, which suggests it would be unlikely to occur by chance alone.

B. Yes, because the mean comprehension scores are different in each group, and all the validity conditions for the significance test are satisfied.

C. No, because the graph of shuffled R-squared statistics is centered at 0.036, which suggests that very little variability is explained by the model.

D. No, because an R2 value of 0.271 does not appear in the graph of shuffled Rsquared statistics, which suggests the data from this experiment was a fluke that occurred by chance alone.

Ans: A; LO: 1.4-1; Difficulty: Hard; Type: MC Questions 5 and 6: A study was carried out to investigate whether the type of message on the back of customer checks at a restaurant would affect tips (recorded as a percentage of the total bill). Sixty tables were selected to participate over a weekend at a restaurant in Philadelphia. Each table was randomly assigned to receive either (1) a picture of a happy face, (2) the words ―Thank you!‖ written out, or (3) no message.

Does this study provide convincing evidence that the message written on the back of the check affects tip percentage? You can use a randomization test to decide.

5. How would you design the physical simulation?

Write the tip percentages on cards. Shuffle and deal the cards into ____ groups.

A. 2 B. 3 C. 20 D. 60 FOR INSTRUCTOR USE ONLY

1-20 Ans: B; LO: 1.4-1; Difficulty: Medium; Type: MC 6. Which of the following statistics could be used to summarize each simulated sample? Select all that apply.

A. Difference in means B. R2 C. t-statistic D. F-statistic Ans: B, D; LO: 1.4-1; Difficulty: Medium Type: MS 7. A food company was interested in how texture might affect the palatability of a particular food. They set up an experiment in which they looked at whether the ―coarseness‖ of the final product (coarse or fine) affected the palatability scores given by 50 people. A partially filled in ANOVA table is given below. Calculate the F-statistic.

Source Model Error Total

df

SS

MS

F ?

6113 16722

Solution: SSModel 16722 – 6113 10609; F 

10609 /  2  1

6113 /  50  2 

 83.30

Ans: 83.30  0.5 LO: 1.4-4; Difficulty: Medium; Type: Calc Questions 8 through 11: Multiple researchers have conducted studies to examine the time it takes for three different medications to register in a patient’s blood system (in minutes). Each researcher wants to test whether the type of medication affects time.

8. Which researcher would obtain a larger F-statistic based on their results?

A. Researcher 1 would obtain a larger F-statistic.

B. Researcher 2 would obtain a larger F-statistic.

C. Researchers 1 and 2 would obtain very similar F-statistics.

D. There is not enough information to determine which F-statistic would be larger.

Ans: B; LO: 1.4-2; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

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9. Compare the results for Researcher 1 and Researcher 3.

Which researcher would obtain a larger F-statistic based on their results?

A. Researcher 1 would obtain a larger F-statistic.

B. Researcher 3 would obtain a larger F-statistic.

C. Researchers 1 and 3 would obtain very similar F-statistics.

D. There is not enough information to determine which F-statistic would be larger.

Ans: A; LO: 1.4-2; Difficulty: Medium; Type: MC

10. Researcher 4 had a total sample size of 90, with 30 patients per treatment group, and a standardized statistic of F = 12.72. She intends to use the F-distribution to find a theorybased p-value. Which F-distribution should she use?

She should use an F-distribution with Model df = _______ and Error df = ________.

Ans: 2  0, 87  0; LO: 1.4-2; Difficulty: Easy; Type: FIB

FOR INSTRUCTOR USE ONLY

1-22 11. Researcher 4 had a total sample size of 90, with 30 patients per treatment group, and a standardized statistic of F = 12.72. The graph below shows the appropriate F-distribution for assessing the statistical significance of the association between type of medication and time. Which of the following statements includes a reasonable p-value and conclusion?

A. The p-value is close to 0. This study provides only weak evidence to suggest that type of medication has an effect on the time it takes for the medication to register.

B. The p-value is close to 1. This study provides only weak evidence to suggest that type of medication has an effect on the time it takes for the medication to register.

C. The p-value is close to 0. This study provides strong evidence to suggest that type of medication has an effect on the time it takes for the medication to register.

D. The p-value is close to 1. This study provides strong evidence to suggest that type of medication has an effect on the time it takes for the medication to register.

Ans: C; LO: 1.4-2; Difficulty: Hard; Type: MC 12. The table below summarizes the results of an experiment to compare yields (as measured by the dried weight of plants) obtained under a control and two different treatment conditions. Calculate SSModel.

Full sample

Sample Size 30

Mean 5.07

SD 0.701

Treatment 1 Treatment 2 Treatment 3

10 10 10

5.03 4.66 5.53

0.583 0.794 0.443

Solution: 10  5.03  5.07   10  4.66  5.07   10  5.53  5.07   3.81 2

2

Ans: 3.81  0.01; LO: 1.4-4; Difficulty: Medium; Type: Calc

FOR INSTRUCTOR USE ONLY

2

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13. The following output displays the amount (in dollars) that a sample of male and female college students spent on their most recent haircuts. You may assume that the sample is representative of a larger population. Calculate the F-statistic.

Solution: 2.85  8.12 Ans: 8.12  0.01; LO: 1.4-2; Difficulty: Easy; Type: Calc 2

14. A randomized experiment was conducted exploring the effectiveness of acupuncture in treating chronic lower back pain. Patients in the study were randomly assigned to one of three treatment groups: Verum acupuncture (traditional Chinese medicine), Sham acupuncture (placebo), and nonacupuncture therapy (drugs, physical therapy, etc.). After six months, each patient’s pain reduction was measured on a quantitative scale. Which inference procedure(s) could you use to test for an association between type of treatment and pain reduction?

A. Two sample t-test (pooled or unpooled) B. ANOVA F-test C. Both of these tests are appropriate in this scenario.

D. Neither of these tests is appropriate in this scenario.

Ans: B; LO: 1.4-2; Difficulty: Medium; Type: MC

FOR INSTRUCTOR USE ONLY

1-24 15. A random sample of 1450 birth records was selected from the state of North Carolina in the year 2001. One question of interest is whether the distribution of birth weights (in ounces) differs based on the race/ethnicity of the mother (White, Black, Hispanic, or other).

Would an ANOVA F-test be valid for these data?

A. No, because the samples are not independent of each other.

B. No, because the sample sizes for the groups are not similar enough.

C. No, because distribution of weights is slightly skewed left for two of the groups.

D. Yes, because all of the validity conditions are met.

Ans: D; LO: 1.4-3; Difficulty: Medium; Type: MC

16. True or False: The validity conditions for the ANOVA F-test are the same as the validity conditions for the two-sample pooled t-test.

Ans: True; LO: 1.4-3; Difficulty: Easy; Type: TF

FOR INSTRUCTOR USE ONLY

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Section 1.5: Confidence and Prediction Intervals LO1.5-1: Apply post-hoc analysis after significant F-test (pairwise differences).

LO1.5-2: Calculate and interpret confidence intervals on single means and differences in two means.

LO1.5-3: Calculate and interpret prediction intervals on quantitative variables.

LO1.5-4: Identify factors that impact widths of confidence intervals and prediction intervals.

Questions 1 through 3: Does seeing a picture have any effect on college students’ understanding of ambiguous prose? 57 students were randomly assigned to three groups: 19 saw a picture before reading a difficult passage of text, 19 saw the picture after reading the passage, and 19 were shown no picture at all. The groups were then tested on their reading comprehension and assigned a quantitative score. The pairwise confidence intervals for the difference in mean comprehension scores are given below.

95% confidence interval for  After  Before : (-2.60, -0.88) 95% confidence interval for  After   None : (-1.02, 0.70) 95% confidence interval for Before  None : (0.72, 2.44) 1. Based on the confidence intervals, which conditions are significantly different from each other? Select all that apply.

A. After is significantly different from Before.

B. After is significantly different from None.

C. Before is significantly different from None.

Ans: A, C; LO: 1.5-1; Difficulty: Medium; Type: MS 2. Which of the following letters tables is consistent with the confidence intervals given above?

A. Group Letters Before A After A None A B. Group Before After None

Letters A B B

C. Group Before After None

Letters A B C

D. The confidence intervals do not provide enough information to construct a letters table.

Ans: B; LO: 1.5-1; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

1-26 3. How could you use a confidence interval to decide whether the difference in comprehension scores for the Before group and the None group is important in a practical sense?

A. Subtract the endpoints to find the width of the interval. If the interval is wide, then the true difference is practically important.

B. Subtract the endpoints to find the width of the interval. If the interval is narrow, then the true difference is practically important.

C. Look to see whether the interval includes 0. If the interval does not include 0, then the true difference is practically important.

D. Look to see whether the endpoints of the interval are close to 0 or far from 0. If the endpoints are far from 0, then the true difference may be practically important.

Ans: D; LO: 1.5-2; Difficulty: Medium; Type: MC Questions 4 and 5: An experiment was conducted to compare yields (as measured by the dried weight of plants in kg) obtained under a control and two different treatment conditions.

The pairwise confidence intervals for the difference in mean yields are given below.

95% confidence interval for Treatmen1  Treatment 2 : (-1.44, -0.29) 95% confidence interval for Treatment1  Control : (-0.94, 0.20) 95% confidence interval for Treatment 2  Control : (-0.08, -1.07) 4. Based on the confidence intervals, which of the treatments are significantly different from each other?

A. All three treatments are significantly different, because none of the confidence intervals have similar endpoints.

B. All three treatments are significantly different, because none of the confidence intervals have midpoints that are equal to 0.

C. None of the treatments are significantly different, because the widths of the confidence intervals are all very similar to each other.

D. Treatment 1 is significantly different from Treatment 2, because the confidence interval for this comparison does not include 0.

Ans: D; LO: 1.5-1; Difficulty: Hard; Type: MC 5. The grower’s goal is to find a treatment that produces yields of at least 1.4 kg, on average.

Which of the treatments meets this goal?

A. None of the treatments meet the grower’s goal.

B. Only Treatment 2 meets the grower’s goal.

C. Both Treatment 1 and Treatment 2 meet the grower’s goal.

D. The pairwise confidence intervals for the difference in mean yields do not provide enough information to decide if the treatments meet the grower’s goal.

Ans: D; LO: 1.5-2; Difficulty: Hard; Type: MC Question 6 through 8: In 2018, a sample of academic faculty were surveyed about their salaries (in US dollars). The results were classified according to academic rank: instructor, assistant professor, associate professor, and full professor. The table below shows 95% confidence intervals for the population mean of each rank.

FOR INSTRUCTOR USE ONLY

Introduction to Financial Statements Rank

Sample size

Group Mean

95% CI for μi

Instructor Assistant Associate Full Professor

75 175 145 234

63680 92029 105133 154509

(54583, 72776) (86073, 97984) (98591, 111676) (149359, 159659)

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6. Which of the following statements is an appropriate interpretation based on the confidence intervals? You may assume that this sample is representative of a larger population of academic faculty.

A. The average salaries of all four ranks are significantly different from each other, because none of the confidence intervals include 0.

B. We are 95% confident that the population mean salary for instructors is between $54,583 and $72,776.

C. Roughly 95% of instructors in the population earn between $54,583 and $72,776 per year.

D. More than one of these statements is an appropriate interpretation of the confidence intervals.

Ans: B; LO: 1.5-2; Difficulty: Medium; Type: MC 7. Why is the 95% confidence interval for  full narrower than the other intervals?

A. The sample size for full professors is largest, and as sample size increases, the width of the confidence interval tends to decrease.

B. The group mean for professors is largest, and as group mean increases, the width of the confidence interval tends to decrease.

C. We can predict with a high level of certainty that full professors make more than other ranks, so the confidence interval provides a precise estimate.

D. None of the justifications above are reasonable, so professors’ salaries must be less variable (lower SD) compared to the other ranks.

Ans: A; LO: 1.5-4; Difficulty: Medium; Type: MC 8. If we changed the confidence level from 95% to 99% (holding everything else constant), would the width of the confidence intervals change?

A. The width of the confidence intervals would decrease.

B. The width of the confidence intervals would increase.

C. The width of some intervals would increase and the width of the other intervals would decrease.

D. The width of all the confidence intervals would stay the same.

Ans: B; LO: 1.5-4; Difficulty: Medium; Type: MC 9. An online retailer is using an experiment to decide whether to modify their website. When visitors type in the web address or click a link to the site, they are randomly re-directed to one of three versions of the website. The retailer’s goal is to maximize the amount of time (in minutes) visitors stay on the site.

Version Letters 1 A 2 AB 3 B True or False: In this study, there is a statistically significant difference in mean time spent FOR INSTRUCTOR USE ONLY

1-28 on the site for Version 1 and Version 2.

Ans: F; LO: 1.5-1; Difficulty: Easy; Type: TF 10. Body temperature measurements (in Fahrenheit) were taken from 65 healthy female volunteers aged 18 to 40 that were participating in vaccine trials. Based on this data, researchers calculated a 95% prediction interval: (96.90, 99.89). Interpret the interval. You may assume that the sample is representative of a larger population.

A. Roughly 95% of healthy females in this population would have body temperatures between 96.90 and 99.89.

B. We are 95% confident that the sample mean body temperature is between 96.90 and 99.89.

C. We are 95% confident that the population mean body temperature is between 96.90 and 99.89.

D. If we were to collect another sample of size 65, we are 95% confident that the sample mean body temperature would be between 96.90 and 99.89.

Ans: A; LO: 1.5-3; Difficulty: Medium; Type: MC 11. In a context with a quantitative response variable and a multi-level categorical explanatory variable, which of the following best describes the purpose of the ANOVA F-test?

A. The F-test helps us assess whether or not there is convincing evidence of an association between the variables.

B. The F-test helps us measure the strength of the association between the variables by indicating how much the groups differ in terms of the mean response.

C. The F-test helps us determine the direction of the association between the variables by indicating which group means are higher than others.

D. The F-test serves all three of the purposes listed above.

Ans: A; LO: 1.5-1; Difficulty: Medium; Type: MC 12. A researcher has conducted an experiment to study four different treatments, and they decide to analyze the data by comparing each treatment group mean to every other treatment group mean. This involves tests for six pairwise comparisons. If the researcher uses a significance level of   0.05 for each test, then the probability of making at least one Type I error is ________ (>, <, =) 0.05.

Ans: >; LO: 1.5-1; Difficulty: Easy; Type: FIB 13. What should you do to protect against an inflated experiment-wise Type I error rate?

A. Conduct pairwise comparisons using t-procedures first and only conduct an F-test if the p-values for the t-tests are all large.

B. Conduct pairwise comparisons using t-procedures first and only conduct an F-test if the p-values for the t-tests are all small.

C. Conduct an F-test first and only conduct pairwise comparisons using t-procedures if the p-value for the F-test is large.

D. Conduct an F-test first and only conduct pairwise comparisons using t-procedures if the p-value for the F-test is small.

Ans: D; LO: 1.5-1; Difficulty: Medium; Type: MC 14. How are prediction intervals different from confidence intervals?

A. Prediction intervals predict the population mean for a particular group, thus they FOR INSTRUCTOR USE ONLY

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are wider than confidence intervals.

B. Prediction intervals predict the population mean for a particular group, thus they are narrower than confidence intervals.

C. Prediction intervals predict the response of a new individual observation, thus they are wider than confidence intervals.

D. Prediction intervals predict the response of a new individual observation, thus they are narrower than confidence intervals.

Ans: C; LO: 1.5-3; Difficulty: Medium; Type: MC 15. The validity conditions for confidence intervals and prediction intervals on means require that the data distribution be reasonably bell-shaped and symmetric. Is this condition always important, even when the sample size is large?

A. This condition is not very important for confidence intervals or prediction intervals, as long as the sample size is large.

B. This condition is very important for prediction intervals. It is less of a concern for confidence intervals, as long as the sample size is large.

C. This condition is very important for confidence intervals. It is less of a concern for prediction intervals, as long as the sample size is large.

D. This condition is very important for both confidence intervals and prediction intervals, regardless of sample size.

Ans: B; LO: 1.5-4; Difficulty: Medium; Type: MC 16. Which of the following is the best way to reduce the width of a prediction interval?

A. Increase the confidence level B. Increase the sample size C. Reduce the unexplained variation within groups D. Use a pooled estimate of the total variation Ans: C; LO: 1.5-3; Difficulty: Medium; Type: MC 17. True or False: Suppose we test H0 : 1  2  3  4 using an ANOVA F-test. This is preferable to using 6 pairwise t-tests, because testing all parameters at once controls the probability of Type II error.

Ans: False; LO: 1.5-1; Difficulty: Medium; Type: TF

Section 1.6: More Study Design Considerations LO1.6-1: Understand statistical power and how it is impacted by sample size, variability within groups, number of groups, and significance level.

LO1.6-2: Use statistical power analysis to plan the sample size of a study.

1. A Type II error occurs when researchers _________ (do/don’t) find convincing evidence against the null hypothesis, when the null hypothesis is actually __________ (true/false).

Ans: don’t, false; LO: 1.6-1; Difficulty: Easy; Type: FIB 2. Suppose that you analyzed data from an experiment and obtained a large p-value. Which type of error is possible in this case?

FOR INSTRUCTOR USE ONLY

1-30 A. This result could be due to a Type I error.

B. This result could be due to a Type II error.

C. This result could be due to either a Type I error or a Type II error.

D. As long as the validity conditions were met, the large p-value is not due to an error.

Ans: B; LO: 1.6-1; Difficulty: Easy; Type: MC 3. The statistical power of a study is the probability that the researchers _______ (will/won’t) find convincing evidence against the null hypothesis, when the null hypothesis is actually __________ (true/false).

Ans: will, false; LO: 1.6-1; Difficulty: Easy; Type: FIB 4. Suppose researchers design a study such that the Type I error rate is 5% and the Type II error rate is 20%. Calculate the power of the study. Include the % sign in your answer.

Solution: 100%  20%  80% Ans: 80% ± 0; LO: 1.6-1; Difficulty: Easy; Type: Calc 5. True or False: The aspects of a study that impact the strength of evidence (sample size, unexplained variation, number of groups, etc.) are the same ones that impact a study’s power.

Ans: True; LO: 1.6-1; Difficulty: Easy; Type: TF 6. How are the probabilities of Type I and Type II error affected by sample size? Assume that variability within groups, number of groups, and significance level remain unchanged.

As the sample size increases, the probability of making a Type I error ________ (increases / decreases / stays the same), and the probability of making a Type II error ________ (increases / decreases / stays the same).

Ans: stays the same, decreases; LO: 1.6-1; Difficulty: Medium; Type: FIB

7. How are the probabilities of Type I and Type II error affected by the significance level,  ?

Assume that sample size, variability within groups, and number of groups remain unchanged.

As the significance level increases, the probability of making a Type I error ________ (increases / decreases / stays the same), and the probability of making a Type II error ________ (increases / decreases / stays the same).

Ans: increases, decreases; LO: 1.6-1; Difficulty: Medium; Type: FIB 8. True or False: When comparing groups with a quantitative response, using a smaller number of groups (fewer levels of the categorical variable) always increases the statistical power of the test.

Ans: False; LO: 1.6-1; Difficulty: Medium; Type: TF FOR INSTRUCTOR USE ONLY

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9. Suppose a researcher wants to design an experiment with a high level of statistical power.

What should they do?

Include a _________ (large/small) number of experimental units in the study.

Choose a number of groups that is as_________ (large/small) as possible without compromising the amount of variability explained.

Take steps during study design and data collection to __________ (increase/decrease) the amount of variability within groups.

Ans: large, small, decrease; LO: 1.6-2; Difficulty: Medium; Type: FIB Questions 10 and 11: For a class project, a statistics student plans to conduct an experiment to investigate whether standing heart rates tend to be higher than sitting heart rates. They will randomly assign their participants to either sit or stand, then the participants’ heart rates will be measured (in beats per minute).

The student considers a difference of 5 beats per minute to be practically important, so if the difference is 5 beats per minute or larger, they want to be able to detect it. The student expects the standard deviation of each group to be about 12 bpm and plans to use a significance level of   0.01 and a sample size of 20 in each group.

10. Which of the graphs above would the student use to find the rejection region?

A. Graph A B. Graph B FOR INSTRUCTOR USE ONLY

1-32 C. Either of these two graphs could be used to find the rejection region.

D. Neither of these two graphs could be used to find the rejection region.

Ans: A; LO: 1.6-2; Difficulty: Medium; Type: MC 11. The rejection region for this study is a difference of means ( xStand  xSit ) of 9.1 or higher.

Which of the values below is closest to the power of the test, given that the difference in standing and sitting heart rates is really 5 beats per minute?

A. 1% B. 15% C. 50% D. 90% Ans: B; LO: 1.6-2; Difficulty: Hard; Type: MC Questions 12 and 13: Do older adults (ages 65+) have lower body temperatures than younger adults (ages 18-64), on average? Researchers decide to conduct a test of H0 : older   younger vs. H A : older   younger at the   0.01 significance level with sample sizes of 25 in each group.

Suppose that the true mean body temperature for older adults is 97.5º F, the true mean body temperature for younger adults is 98.6º F, and both groups have a standard deviation of 0.75º F.

12. Match each term to its representation in the graphs above.

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Introduction to Financial Statements

Power: Prob(Type I error): Prob(Type II error):

1-33

A. The area to the left of the red line in Graph 1 B. The area to the left of the red line in Graph 2 C. The area to the right of the red line in Graph 2

Ans: Power: B, P(Type I): A, P(Type II): C; LO: 1.6-1; Difficulty: Medium; Type: Ma 13. The rejection region for this study is a difference of means ( xOlder  xYounger ) of -0.51 or lower. Suppose the significance level was changed from   0.01 to   0.05 . The boundary of the rejection region (the red line) would shift to the _____ (left/right), and the power would ______ (increase/decrease).

Ans: right, increase; LO: 1.6-1; Difficulty: Medium; Type: FIB Question 14 and 15: Olestra was approved by the FDA for use in snack foods as a fat substitute in the 1990s. Because there were anecdotal reports of stomach (GI) problems associated with Olestra consumption, researchers planned to carry out an experiment to compare GI symptoms after consuming Olestra potato chips or regular potato chips.

The researchers consider a difference of proportions of 0.05 to be practically significant.

That is, if 20% of people experience GI problems when eating Olestra and 15% experience GI problems while eating regular potato chips, they want to be able to detect the difference between the two. They use software to conduct a power analysis to decide how large the sample size needs to be in order for the study to have 80% power with a significance level of   0.05 and a one-sided alternative hypothesis.

14. Statistical power is the probability of concluding that the risk of GI problems for those eating chips with Olestra is ___________ (higher than / the same as) the risk for those eating regular potato chips, given that the difference in proportions who experience GI problems is really equal to ____ (0 / 0.05).

Ans: higher than, 0.05; LO: 1.6-2; Difficulty: Medium; Type: FIB 15. The power analysis suggests a sample size of at least 714. One of the researchers’ colleagues is surprised that such a large sample size is necessary. Which of the following is the best explanation?

A. The difference between 15% and 20% is small, and small effect sizes are more difficult to detect, so a large sample size is required.

B. The significance level is fairly high. If the researchers changed the significance level from   0.05 to   0.01, they wouldn’t need such a large sample size.

C. 80% power is an unusually high value that demands an unusually large sample.

Lowering the power would make their plan more acceptable to funding agencies.

D. One-sided tests always require larger samples. If they used a two-sided test, the necessary sample size would be roughly half as large.

Ans: A; LO: 1.6-2; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

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Chapter 2 Intermediate Statistical Investigations Test Bank Question types:

FIB = Fill in the blank Ma = Matching MC = Multiple choice

Calc = Calculation MS = Multiple select TF = True-false

CHAPTER 2 TERMINAL LEARNING OUTCOMES TLO2-1: Analyze paired data as a way to control observational unit variation appropriately.

TLO2-2: Analyze data on quantitative response from a randomized complete block design using both simulation and theory-based approaches.

TLO2-3: Analyze an observational study with two sources of explained variation understanding adjustment for another variable in the analysis and covariation.

Section 2.1: Paired Data LO2.1-1: Use pairing to potentially reduce unexplained variation and increase the power of a study.

LO2.1-2: Explain how to analyze paired data appropriately.

Questions 1 through 3: Statisticians recommend randomizing the order of treatments in paired experiments, but is order really important? To investigate, 22 students in a statistics class each completed a memory puzzle twice, recording the amount of time (in seconds) it took to complete the puzzle on each attempt. They also calculate the difference between their two attempts (first – second). The results of the experiment are shown below.

First Attempts (n=22) Mean = 83.755 SD = 18.318

Second Attempts (n=22) Mean = 75.963 SD = 18.008

1. What is the value of the t-statistic for this study?

Sol: Ans: 4.412

0.05; LO2.1-2; Difficulty: Medium; Type: Calc

2. What is the value of the F-statistic for this study?

Sol: ( Ans: 19.469

) 0.1; LO2.1-2; Difficulty: Medium; Type: Calc

FOR INSTRUCTOR USE ONLY

Differences (n=22) Mean = 7.792 SD = 8.283

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3. The 95% confidence interval for the mean difference between first and second attempts is (4.120, 11.465). Does this interval provide strong evidence of an order effect in this context? You may assume 0.05.

A. Yes, there is strong evidence of an order effect, because the sample mean (7.792) is inside the interval.

B. Yes, there is strong evidence of an order effect, because 0 is outside the interval.

C. No, there is not strong evidence of an order effect, because the sample mean (7.792) is inside the interval.

D. No, there is not strong evidence of an order effect, because 0 is outside the interval.

Ans: B; LO2.1-2; Difficulty: Medium; Type: MC Questions 4 and 5: An economist at Vanderbilt University designed a study to compare different types of online auctions. In one experiment, he compared a Dutch auction to a first-price sealed bid auction. In the Dutch auction, the item for sale starts at a very high price and is lowered gradually until someone finds the price low enough to buy. In the first-price sealed bid auction, each bidder submits a single sealed bid, and the highest bid wins.

The researcher auctioned off collectible trading cards from the game Magic: The Gathering. He placed pairs of identical cards up for auction; one would go into the Dutch auction and other to the first-price sealed bid auction. He then looked at the difference in the prices he received on each pair. He repeated this for a total of 88 pairs.

4. Which of the following are appropriate ways to state the null hypothesis in this scenario?

Select all that apply.

A. B. C. There is an association between type of auction and selling price.

D. There is no association between type of auction and selling price.

Ans: A, D; LO2.1-2; Difficulty: Medium; Type: MS 5. Are the validity conditions met for a one-sample t-test for paired data?

A. No, because the distribution of selling prices are skewed right for both types of auctions.

B. No, because the spread of the differences is less than half the spread of the original selling prices.

C. Yes, because the distribution of the differences is unimodal and approximately symmetric.

D. Yes, because the spread of the selling prices is roughly the same for both types of auctions.

Ans: C; LO2.1-2; Difficulty: Medium; Type: MS Questions 6 through 10: Is there a difference between standing and sitting heart rates? To deFOR INSTRUCTOR USE ONLY

1-36 cide, 52 students measured their heart rates (in beats per minute) in two positions: once while standing and once while sitting. Each student then calculated the difference in their measurements: standing heart rate – sitting heart rate. Data are shown below.

6. Which of the following inference methods is appropriate for this scenario? Select all that apply.

A. Two-sample t-test B. One-way ANOVA F-test C. t-test for paired data Ans: C; LO2.1-2; Difficulty: Medium; Type: MS 7. In this study, it is helpful to control for person-to-person variability, removing it as a source of unexplained variation. Which features of the graph above indicate that pairing will lead to a more powerful test in this scenario? Select all that apply.

A. The connected lines show that those with high standing heart rates also tend to have high sitting heart rates and vice versa.

B. On average, standing heart rates tend to be higher than sitting heart rates (based on sample means).

C. The variability of the differences is less than the variability within treatment groups (based on sample standard deviations).

D. The distribution of difference is unimodal and roughly symmetrical, so it is reasonable to assume the residuals are Normally distributed.

Ans: A, C; LO2.1-1; Difficulty: Hard; Type: MS

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8. The following statistical model could be used to predict heart rates.

Predicted heart rate = overall mean + position effect + person effect One participant had a standing heart of 88 and a sitting heart rate of 82. Using this information and the data shown above, calculate the person effect for this participant.

Sol: Person mean = (88+82)/2 = 85; Overall mean = (77.0+70.0)/2 = 73.5; Person effect = 85–73.5 = 11.5 Ans: 11.5 0.5; LO2.1-2; Difficulty: Medium; Type: Calc 9. Based on the ANOVA table below, what proportion of the variability in heart rates is due to participants (person-to-person variability)? Give your answer as a decimal.

Source DF Sum of Squares Mean Squares p-value F Position 1 1281.0 1281.0 43.4243 <0.001 Participant 51 8154.5 8154.5 5.4201 <0.001 Residuals 51 1504.5 29.50 Total 103 10940.0 Sol: 8154.5/10940.0 = 0.74 Ans: 0.74 0.06; LO2.1-2; Difficulty: Easy; Type: Calc 10. As shown in the ANOVA table below, there is strong evidence of a position effect – a difference between standing and sitting heart rates. Suppose you had used only position as a predictor of heart rate, ignoring person-to-person variability in your model. How would the strength of evidence change?

The evidence of a position effect would be ______ (stronger/weaker). Specifically SSError would be ______ (larger/smaller), which would lead to a ______ (larger/smaller) Fstatistic for testing the effect of position.

Source DF Sum of Squares Mean Squares p-value F Position 1 1281.0 1281.0 43.4243 <0.001 Participant 51 8154.5 8154.5 5.4201 <0.001 Residuals 51 1504.5 29.50 Total 103 10940.0 Ans: weaker, larger, smaller; LO2.1-2; Difficulty: Medium; Type: FIB

FOR INSTRUCTOR USE ONLY

1-38 11. In a paired experiment to compare sleep aids, participants all try two different treatments.

The order of the treatments is randomized, and sleep quality is measured for each treatment. A partially completed ANOVA table is given below.

Source DF Sum of Squares Mean Squares Treatment 1 322 Person 19 561 Error 19 Total 39 1211

F

Calculate the standard error of the residuals.

Sol: SSError = – – ; SE of residuals = √ Ans: 4.155 0.05; LO2.1-2; Difficulty: Medium; Type: Calc 12. Which of the following are examples of paired designs? Select all that apply.

A. A teacher gives each of her students two cookies: one made with eggs and butter and one made with a vegan recipe. The students rate the taste of each cookie on a scale of 1-10.

B. The managing partner of a consulting firm wants to compare the effectiveness of two computer skills training programs. Employees are randomly assigned to complete one of the two programs. Afterward, both groups take the same computer skills test.

C. A college student is trying to decide which of two grocery stores – Publix or Kroger – offers better prices. The student selects 50 items that are available in both stores, and for each of these items, records both the Publix price and the Kroger price.

Ans: A,C; LO2.1-1; Difficulty: Medium; Type: MS 13. Which of the following are benefits of a paired design compared to an independent groups design?

A. Paired designs allow us to generalize results to a larger population.

B. Paired designs always support cause-and-effect conclusions.

C. Paired designs are always practical, whereas independent groups designs do not make sense in some contexts.

D. Paired designs often lead to tests with more statistical power to detect an effect.

Ans: D; LO2.1-1; Difficulty: Medium; Type: MC 14. In a paired design where each participant is observed under two conditions (treatment and control), why is it inappropriate to conduct a two-sample t-test?

A. Because it is not reasonable to consider multiple observations on the same individual as independent B. Because it is not reasonable to consider multiple observations on the same individual as Normally distributed C. Because this design leads to large differences in standard deviations for the treatment and control groups D. Because this design leads to large differences in means for the treatment and control group Ans: A; LO2.1-2; Difficulty: Medium; Type: MC 15. True or False: In a paired design where each participant is observed under two conditions FOR INSTRUCTOR USE ONLY

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(treatment and control), the sum of the person effects across all participants will be zero.

Ans: True; LO2.1-2; Difficulty: Easy; Type: TF

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Section 2.2: Randomized Complete Block Designs LO2.2-1: Analyze a randomized complete block design using simulation-based methods.

LO2.2-2: Analyze a randomized complete block design using theory-based methods.

1. True or False: In a randomized complete block design, each block should be a diverse group of observational units, roughly mirroring the population of interest.

Ans: False; LO2.2; Difficulty: Easy; Type: TF 2. A matched pairs design is a specific kind of block design. When can the term ―matched pairs‖ be used to describe a block design?

A. When there are only two blocks of experimental units B. When there are only two observations in each block C. When there are two independent groups of experimental units D. When the alternative hypothesis in the test for a treatment effect is two-sided Ans: B; LO2.2; Difficulty: Easy; Type: MC 3. A blocking variable is a variable that the researcher believes is ________ (related to/independent of) the response variable; this variable ________ (is/is not) manipulated by the researcher.

Ans: related to, is not; LO2.2; Difficulty: Easy; Type: FIB 4. The table below shows the means and standard deviations for each block in a randomized block design.

Block 1 2 3 4

Mean Response 500 501 498 497

Standard Deviation of Responses 12 14 9 12

Was the choice of blocking variable effective in this study?

A. Yes, because the mean responses are very similar for the four blocks.

B. No, because the mean responses are very similar for the four blocks.

C. Yes, because the standard deviations are fairly similar for the four blocks.

D. No, because the standard deviations are not similar enough for the four blocks.

Ans: B; LO2.2; Difficulty: Medium; Type: MC Questions 5 through 7: A farmer has a field with three rows, where each row contains nine plants. The farmer suspects that the row may affect plants’ yield, because of variations in sunlight across his property, so he decides to use a randomized block design to test the effectiveness of three different fertilizers.

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5. Describe an appropriate randomized complete block design.

A. Randomly assign the plants to one of the three fertilizers, which results in nine plants per treatment. Record both the yield and the row for each plant.

B. Randomly sample one plant from each of the three rows, then randomly assign the three selected plants to the three different fertilizers.

C. Randomly assign each of the three rows to one of the three fertilizers, so that all plants in a given row will receive the same treatment.

D. For each row, randomly assign the nine plants to the three fertilizers, which results in three plants per treatment in each row.

Ans: D; LO2.2; Difficulty: Medium; Type: MC 6. The null distribution of F-statistics was produced by randomly re-assigning the plant yields within blocks and calculating the F-statistic for each shuffle. The observed F-statistic for the real study was 0.92. What do you conclude?

A. After adjusting for block effects, this data provides strong evidence that all three fertilizers are the same in terms of their effect on yield.

B. After adjusting for block effects, this data provides strong evidence that all three fertilizers are different in terms of their effect on yield.

C. After adjusting for block effects, this data provides strong evidence that at least one of the fertilizers are different in terms of its effect on yield.

D. Even after adjusting for block effects, this data provides only weak evidence of a fertilizer effect.

Ans: C; LO2.2-1; Difficulty: Medium; Type: MC

FOR INSTRUCTOR USE ONLY

1-42 7. The null distribution of F-statistics was produced by randomly re-assigning the plant yields to treatments within blocks and calculating the F-statistic for each shuffle. How would the null distribution change if the plant yields were randomly re-assigned to treatments completely at random ignoring blocks?

A. The null distribution of F-statistics would have a larger spread.

B. The null distribution of F-statistics would have a smaller spread.

C. The null distribution of F-statistics would be skewed left instead of skewed right.

D. The null distribution of F-statistics would be approximately Normal.

Ans: A; LO2.2-1; Difficulty: Medium; Type: MC Questions 8 through 12: Concentration is a one-person memory game in which cards are laid face down on a surface and two cards are flipped face up at a time. The object of the game is to turn over matching pairs of cards.

An online version of this game includes three different sets of cards: one has images of animals on the cards, one has images of babies, and one has images of holiday scenes.

Are these three variations equally difficult? To investigate, eight students tried all three versions of the game in random order. They recorded the amount of time (in seconds) it took to complete the game. The overall mean time to complete the game was 79.31 seconds. The graphs below display the raw data and the data after adjusting for block effects.

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8. To find adjusted data values, it is first necessary to calculate block effects for each student in the sample. One student completed the animal version of the game in 65 seconds, the baby version in 73 seconds, and the holiday version in 59 seconds. Calculate the block effect for this student.

Sol: Student average = (65+73+59)/3 = 65.67; Student effect = 65.67–79.31 = -13.64 Ans: -13.64 0.1; LO2.2; Difficulty: Medium; Type: Calc 9. Compare the variability in the adjusted data to the variability in the raw data.

In the adjusted data, the between-groups variability, as measured by SSModel, is ______ (>, <, =) the between-groups variability in the raw data.

In the adjusted data, the within-groups variability, as measured by SSError, is ______ (>, <, =) the within-groups variability in the raw data.

Ans: =, <; LO2.2; Difficulty: Medium; Type: FIB 10. In a test to decide whether the three versions of the game are equally difficult, which dataset would produce a larger F-statistic?

A. The raw dataset (not accounting for blocks) would produce a larger F-statistic.

B. The adjusted dataset (accounting for blocks) would produce a larger F-statistic.

C. The F-statistics for the two datasets would be equal.

Ans: B; LO2.2; Difficulty: Medium; Type: MC

FOR INSTRUCTOR USE ONLY

1-44 11. You want to use simulation to test whether the three versions of the game are equally difficult. Which of the following is an appropriate design for the simulation. Select all that apply.

A. Write the times from the raw dataset on cards. Shuffle and deal them into three piles representing the three versions of the game ignoring blocks. Calculate the Fstatistic based on a one-variable ANOVA. Repeat.

B. Write the times from the raw dataset on cards. Randomly re-assign the times to versions within blocks. Calculate the F-statistic based on a one-variable ANOVA. Repeat.

C. Write the times from the adjusted dataset on cards. Shuffle and deal them into three piles representing the three versions of the game ignoring blocks. Calculate the F-statistic based on a one-variable ANOVA. Repeat.

Ans: B,C; LO2.2-1; Difficulty: Medium; Type: MS 12. True or False: The validity conditions for ANOVA with a blocking variable are the same as the validity conditions for ANOVA without a blocking variable, except that the conditions are checked based on the adjusted data not the raw data.

Ans: True; LO2.2-2; Difficulty: Easy; Type: TF Questions 13 through 15: Sixty-eight allergy sufferers were recruited to participate in a study comparing the effectiveness of different allergy treatments. During allergy season, each participant tried three treatments with active ingredients plus one placebo treatment, with the order of the treatments randomized and adequate time between treatments. Participants rated the severity of their allergy symptoms while taking each treatment. A partially filled in ANOVA table is given below.

Source Treatment Participant Error Total

DF

Sum of Squares 2174.4 3913.6

201 271

Mean Squares

F

8697.4

13. In this study, what are the blocks?

A. The 68 allergy sufferers B. The 3 allergy treatments with active ingredients C. The 4 allergy treatments (including the placebo) D. The 272 ratings recorded across participants Ans: A; LO2.2; Difficulty: Medium; Type: MC 14. Calculate the F-statistic that could be used to test for a treatment effect.

Sol: Ans: 55.83

;

;

0.1; LO2.2-2; Difficulty: Medium; Type: Calc

15. What if this analysis had failed to account for person-to-person variability in allergy symptoms? How would the ANOVA table change?

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The row for Participant would be removed, and the amount of variation due to individual differences in allergy symptoms (SSParticipants = 3913.6) would be added to _________ (SSTreatment/SSError), which would make the F-statistic _________ (larger/smaller).

Ans: SSError, smaller; LO2.2-2; Difficulty: Medium; Type: FIB

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Section 2.3: Observational Study with Two Explanatory Variables LO2.3-1: Analyze an observational study with two sources of explained variation.

LO2.3-2: Understand what it means to adjust for another variable in the analysis.

LO2.3-3: Explore effects of covariation in explanatory variables on the analysis.

Questions 1 through 4: In 2018, a sample of 628 academic faculty from universities across the country were surveyed about their salaries (in US dollars). The results were classified according to each faculty member’s academic rank (instructor, assistant professor, associate professor, and full professor) and gender (male, female).

1. Calculate the effect of ―female‖ on wages. Note that the sample sizes for male and female faculty members are not equal.

Full sample (n = 628) Male (n = 428) Female (n = 200)

Salary Mean $114,946 $123,953 $95,671

SD $51,681 $55,281 $55,281

Sol: Ans: -14141

; 0; LO2.3-1; Difficulty: Medium; Type: Calc

2. In the sample, male faculty members are paid more than female faculty members. Does this analysis provide evidence that gender causes differences in salary?

A. It depends whether the results are statistically significant. If this difference results in a small p-value, then the analysis supports causal conclusions.

B. It depends how the sample was selected. If the sample is representative of the population, then the analysis supports causal conclusions.

C. No, because this sample is not large enough. There are over a million academic faculty in the U.S., so it is not appropriate to draw conclusions based on only 628.

D. No, because data come from an observational study not a randomized experiment, so there may be confounding variables present.

Ans D; LO2.3-1; Difficulty: Medium; Type: MC 3. The ANOVA table below summarizes a model that uses both gender and academic rank to predict a faculty member’s salary (in $1000s).

Source Gender Rank Error Total

DF 1 3 623 627

Adj SS 15,167 575,905 989,757 1,674,691

Adj MS 15,167 191,968 1,589

F 9.54 120.81

p-value 0.0021 <0.0001

Calculate the sum of squares for the covariation - the amount of variation in salaries that is attributed jointly to gender and rank.

A. SSCovariation = SSGender + SSRank = 591,072 FOR INSTRUCTOR USE ONLY

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B. SSCovariation = SSGender + SSRank + SSError = 1,580,829 C. SSCovariation = SSTotal – SSGender – SSRank = 1,083,619 D. SSCovariation = SSTotal – SSGender – SSRank – SSError = 93,862 Ans D; LO2.3-3; Difficulty: Medium; Type: MC 4. Statistical software was used to calculate the adjusted effects of gender and rank. The statistical model is given below.

{

{

with According to this model, if we compare two faculty members of the same rank, on average, the male faculty member earns _______ dollars more per year than the female faculty member.

Ans: 10978; LO2.3-2; Difficulty: Medium; Type: FIB Questions 5 through 8: A real estate agent collected information on 100 recent home sales in their town. In addition to selling prices (in $1000s), the agent recorded information about the home’s location (north side of town or south side of town) and the number of bedrooms (at least 3 or less than 3). A partially filled in ANOVA table for the two-variable model is given below.

Source Location Bedrooms Error Total

DF 1 1 97 99

Adj SS 19,843 22,180 278,799 314,433

Adj MS

F

p-value 0.0100 0.0066

R2 0.0631 0.0705

5. What does the adjusted sum of squares for Location represent?

The variation in prices that is explained by Location, _________ (excluding/including) the variation explained jointly by Location and Bedrooms.

Ans excluding; LO2.3-2; Difficulty: Easy; Type: FIB 6. For a one-variable model that uses only Location as a predictor of price, R2 = 0.09. Is there an association between Location and Bedrooms?

A. Yes, there is an association between Location and Bedrooms.

B. No, there is not an association between Location and Bedrooms.

C. We do not have enough information to decide if there is an association between Location and Bedrooms.

Ans A; LO2.3-3; Difficulty: Medium; Type: MC

FOR INSTRUCTOR USE ONLY

1-48 7. Calculate the F-statistic for Location.

Sol: Ans: 6.90

0.1; LO2.3-1; Difficulty: Medium; Type: Calc

8. After adjusting for the number of bedrooms, does this data provide strong evidence of a difference in the population mean prices for houses on the north side and houses on the south side? You may assume that the sample is representative of all houses in this town.

A. No, because the p-value for Location is small.

B. No, because the R2 value for Location is small.

C. Yes, because the p-value for Location is small.

D. Yes, because the R2 value for Location is small.

Ans: C; LO2.3-1; Difficulty: Medium; Type: MC Questions 9 and 10: Sixty-eight allergy sufferers were recruited to participate in a study comparing the effectiveness of different allergy medications. During allergy season, all 68 participants tried three medications with active ingredients plus one placebo medication, with the order of the treatments randomized and adequate time in between. Participants rated the severity of their allergy symptoms while taking each treatment.

9. True or False: The mean of the four treatment means will be the same as the overall mean rating.

Ans: True; LO2.3-1; Difficulty: Medium; Type: TF 10. True or False: The design of this study ensures there is no relationship between the treatments and the blocks; thus there is no covariation.

Ans: True; LO2.3-3; Difficulty: Medium; Type: TF Questions 11 and 12: In an observational study, two categorical explanatory variables are used to predict a response. You may assume that there is a moderate amount of covariation.

Source Variable 1 Variable 2 Error Total

DF

Adj SS

F

p-value

11. How do you calculate SSModel for this study?

A. SSModel = SSVariable1 + SSVariable2 B. SSModel = SSTotal – SSError C. Both (A) and (B) would produce the correct value for SSModel.

D. Neither (A) nor (B) would produce the correct value for SSModel.

Ans: B; LO2.3-3; Difficulty: Medium; Type: MC

12. Suppose Variable 2 were removed from the model. Would the adjusted sums of squares FOR INSTRUCTOR USE ONLY

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and p-value for Variable 1 change?

A. Removing Variable 2 would not affect the sum of squares or p-value for Variable 1.

B. Removing Variable 2 would not affect the sum of squares for Variable 1, but the pvalue for Variable 1 may change.

C. Removing Variable 2 would not affect the p-value for Variable 1, but the sum of squares for Variable 1 may change.

D. Removing Variable 2 may change both the sum of squares and the p-value for Variable 1.

Ans: D; LO2.3-2; Difficulty: Medium; Type: MC Questions 13 through 15: Students in a statistics class wanted to know how customers rate one of their favorite coffee shops. The students administered surveys and asked customers to rate their experience in the shop on a scale of 1-10. They also asked whether the survey respondent was a college student and how often they visit coffee shops (at least once a week or less than once a week).

13. The students decide to use a two-variable model that predicts ratings based on whether the customer is a student and the frequency of their coffee shop visits. Based on the graph below, do you expect covariation – variation in ratings that is attributed jointly to Student and Frequency?

A. Yes, because more than half of the customers in the sample are students.

B. Yes, because students tend to give higher ratings than non-students.

C. Yes, because students are more likely to visit coffee shops at least once per week.

D. No, because the sample is evenly divided between customers who visit coffee shops at least once per week and those who visit less than once per week.

Ans: C; LO2.3-3; Difficulty: Medium; Type: MC 14. Consider two possible models for predicting ratings. A one-variable model that predicts ratings based on frequency of coffee shop visits has an R2 value of 0.06. A one-variable model that predicts ratings based on whether the customer is a student has an R2 value of 0.03. What is the R2 value of a two-variable model that uses both Frequency and Student to predict ratings?

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1-50 A. R2 = 0.06 + 0.03 = 0.09 B. R2 = 0.06 - 0.03 = 0.03 C. R2 = (0.06 + 0.03) 2 = 0.045 D. There is not enough information to calculate R2 for the two-variable model.

Ans: D; LO2.3-3; Difficulty: Medium; Type: MC 15. Consider two possible models to predict ratings: Model 1: {

, with

{

{

Model 2:

with Compare the adjusted sum of squares for student in each of these two models.

The adjusted sum of squares for student in Model 1 ________ (>, <, =) the adjusted sum of squares for student in Model 2.

Ans: >; LO2.3-2; Difficulty: Medium; Type: FIB

Chapter 3 Intermediate Statistical Investigations Test Bank Question types:

FIB = Fill in the blank Ma = Matching MC = Multiple choice

Calc = Calculation MS = Multiple select TF = True-false

CHAPTER 3 TERMINAL LEARNING OUTCOMES TLO3-1: Design and analyze a multi-factor experiment realizing the difference between blocking variables and experimental factors.

TLO3-2: Understand the concept of a statistical interaction, and calculate and interpret interaction effects.

TLO3-3: Explain the benefits and challenges of replication particularly as they relate to generalized block designs and within block factorial designs.

TLO3-4: Run a two-variable ANOVA including an interaction term for an observational study with 2 two-level variables.

Section 3.1: Multifactor Experiments LO3.1-1: Design an experiment with more than one variable of interest.

LO3.1-2: Explore the benefits of a two-variable study where the levels of both variables are asFOR INSTRUCTOR USE ONLY

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signed by the researcher.

1. A researcher is interested in the relative difficulty of two sets of mathematical tasks and how student success in completing the tasks may be affected by distractions. Forty-eight college students have been recruited to participate in a study. Which of the following is the most appropriate study design?

A. Randomly assign the students into two groups of size 24. One group completes Task Set 1 with distractions, and the other completes Task Set 2 with no distractions.

B. Randomly assign the students into four groups of size 12. One group completes Task Set 1 with distractions, one completes Task Set 1 without distractions, one completes Task Set 2 with distractions, and the last completes Task Set 2 without distractions.

C. Divide the students into two groups of 24. Within Group 1, assign 12 students to complete Task Set 1 and 12 students to complete Task Set 2. Within Group 2, assign 12 students to work with distractions and 12 to work without distractions.

D. All three study designs above are equally appropriate ways to investigate the researcher’s statistical questions.

Ans: B; LO3.1-1; Difficulty: Medium; Type: MC

FOR INSTRUCTOR USE ONLY

1-52 Questions 2 through 5: In a balanced, full-factorial experiment, 40 volunteers were randomly assigned to receive one of two pain medications (A, B) at one of two dosages (high, low).

After an hour, their response to pain was recorded on a scale of 0 to 20 (with higher numbers indicating more severe pain). The ANOVA table below corresponds to a twovariable additive model using drug and dose as predictors of pain response.

Source Drug Dose Error Total

DF 1 1 37 39

SS 16.9 313.6 157.5 488.0

MS 16.9 22.22 4.26

F 3.97 73.67

p-value 0.0537 <0.0001

2. Calculate SSModel.

A. SSModel = 16.9 + 313.6 = 330.5 B. SSModel = (16.9 + 313.6) / 2 = 165.25 C. SSModel = 16.9 + 313.6 – 157.5 = 173.0 D. There is not enough information to calculate SSModel, because an association between Drug and Dosage may lead to covariation.

Ans: A; LO3.1-2; Difficulty: Medium; Type: MC 3. True or False: The 95% confidence interval for estimating the difference in mean pain response for low and high doses includes 0.

Ans: False; LO3.1-2; Difficulty: Medium; Type: TF 4. Suppose the researchers had ignored dosage in the analysis. What values would you expect in the ANOVA table for a one-variable model using only drug as a predictor of pain response?

Source Drug Error Total

DF

SS

MS

F

p-value

SSDrug in the one-variable ANOVA table would be _________ (>, <, =) 16.9 The p-value for Drug in the one-variable ANOVA table would _________ (>, <, =) 0.0537, indicating _________ (stronger/weaker/the same) evidence of a difference between Drug A and Drug B. Ans: =, >, weaker; LO3.1-2; Difficulty: Hard; Type: FIB 5. Suppose the researchers had ignored dosage in the analysis. How would this affect the width of the 95% confidence interval for estimating ?

The 95% confidence interval for estimating would be ________________ (wider/narrower) than the confidence interval based on the two-variable model, because the ________________ (treatment means/residual standard error) would be different.

Ans: wider, residual standard error; LO3.1-2; Difficulty: Medium; Type: FIB

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Questions 6 through 9: A large field was divided into 24 equal-sized plots to be planted with the same number of potato plants. Each plot was randomly assigned a type of fertilizer (A, B, C) and a manure level (high, low) based on a balanced, full-factorial design. The table below shows the mean yield (by weight) for each combination of fertilizer and manure.

Low Manure High Manure

Fertilizer A 6 7

Fertilizer B 8 11

Fertilizer C 10 12

6. True or False: By using a balanced design, the researchers have ensured that there is no association between the type of fertilizer and the level of manure.

Ans: True; LO3.1-1; Difficulty: Medium; Type: TF 7. Fill in the blanks using numerical values.

In this experiment, there are ________ explanatory variables (or factors) and there are ________ treatments with ________ replications.

Ans: 2, 6, 4; LO3.1-1; Difficulty: Easy; Type: FIB 8. Calculate the main effects of fertilizer and manure to fill in the additive statistical model. {

{

Ans: First column: 9; Second column: -2.5, 0.5, 2; Third column: -1, 1; LO3.1-2; Difficulty: Medium; Type: FIB 9. For the additive model that uses fertilizer type and manure level to predict yield, the standard error of the residuals is 4.5. What does the standard error of the residuals measure?

A. The variability of individual plot yields around the observed mean yields given in the table B. The variability of individual plot yields around the predicted mean yields calculated based on the additive model C. The variability between the observed mean yields given in the table and the predicted mean yields calculated based on the model D. The variability between the observed mean yields given in the table, the predicted mean yields calculated based on the model, and the overall mean yield Ans: B; LO3.1-2; Difficulty: Medium; Type: MC

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1-54 Questions 10 through 13: Researchers want to reduce potato rot while potatoes are being stored for future use. In an experiment, potatoes were injected with a bacteria known to cause rot and then stored under a variety of conditions. Two of the experimental factors were temperature during storage and amount of oxygen during storage. The response variable was the diameter of the rotted area (in millimeters). A partially filled in twovariable ANOVA table is given below.

Source Temperature Oxygen Error Total

DF 1 2 14 17

SS 600.89 44.44 365.77 1011.11

MS 600.89 22.22 26.14

F ?

p-value 0.0003 0.4483

10. Fill in the blanks using numerical values.

Based on the ANOVA table, there were ________ levels of temperature and ________ levels of oxygen tested in this experiment. There were ________ replications for each combination of temperature and oxygen.

Ans: 2, 3, 3; LO3.1-1; Difficulty: Medium; Type: FIB 11. Calculate the F-statistic for testing whether temperature has an effect on rot.

Solution: Ans: 22.99 0.1; LO3.1-2; Difficulty: Medium; Type: Calc 12. What conclusions would you draw based on the p-values in this ANOVA table?

After adjusting for oxygen, this study provides _______ (strong/weak) evidence that temperature has an effect on rot.

After adjusting for temperature, this study provides _______ (strong/weak) evidence that oxygen level has an effect on rot.

Ans: strong, weak; LO3.1-2; Difficulty: Medium; Type: FIB 13. Initially, the researchers had planned to use a one-variable model with separate means for each combination of temperature and oxygen. Which of the following are advantages of a two-variable additive model over a one-variable separate means model? Select all that apply.

A. The two-variable model allows us to estimate the main effects of temperature and oxygen separately.

B. The two-variable model explains more of the variability (higher SSModel) compared to the separate means model.

C. The two-variable model has fewer degrees of freedom for treatment and more degrees of freedom for error.

D. The two-variable model has more degrees of freedom for treatment and fewer degrees of freedom for error.

Ans: A, C; LO3.1-2; Difficulty: Hard; Type: MS

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14. An experiment was conducted to compare two different applications for text messaging on cell phones, each with its own interface and software. The experiment included participants in two age groups: 15-44 years old and 45+ years old. Within each age group, half of the participants were randomly assigned to each application. The researchers then asked each participant to type the words from a given passage of text and recorded the number of words typed correctly in a fixed period of time. Describe the explanatory variables in this study.

A. This study has two explanatory variables: one blocking variable and one experimental factor.

B. This study has two explanatory variables, both of which are experimental factors.

C. This study has four explanatory variables: two blocking variables and two experimental factors.

D. This study has four explanatory variables, all of which are experimental factors.

Ans: A; LO3.1-2; Difficulty: Easy; Type: MC

15. The plots below display the residuals for a two-factor model with six treatments. Match each validity condition to the description of how that condition should be checked. Two of the descriptions will not be used.

Independence:

A. Check that fitted values are spaced fairly evenly along the x-axis.

Equal variance:

B. Check that the vertical spread of the residuals at each of the fitted values is reasonably similar.

Normality:

C. Check that the histogram of the residuals is reasonably symmetric and bell-shaped.

D. Check that the mean of the residuals is 0.

E. Check that experimental units were randomly assigned to treatments with no repeated measures.

Ans: Independence: E; Equal variance: B; Normality: C; LO3.1-2; Difficulty: Medium; Type: Ma

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Section 3.2: Statistical Interactions LO3.2-1: Understand the concept of a statistical interaction.

LO3.2-2: Interpret an interaction plot.

LO3.2-3: Calculate interaction effects.

LO3.2-4: Use simulation- and theory-based p-values to assess the significance of an interaction.

Questions 1 through 6: A balanced, full-factorial experiment was used to investigate the effect of protein source (beef, cereal) and protein level (high, low) on weight gain (in grams) for male rats. Ten rats were assigned to each combination of source and level. A table of mean weight gains and an interaction plot are shown below.

Level Source

Low

High

Beef

79.2

100

Cereal

83.9

85.9

1. Suppose the researchers used an additive model to predict weight gain, as shown below. {

{

According to the interaction plot, which treatment is expected to cause the smallest amount of weight gain?

A. Beef/Low B. Beef/High

According to the additive model, which treatment is expected to cause the smallest amount of weight gain?

C. Cereal/Low D. Cereal/High

Ans: A, C; LO3.2-1; Difficulty: Easy; Type: Ma 2. Calculate the interaction effect for the Beef/Low treatment.

{

Sol: Answer: -4.7

{

{

; 0; LO3.2-3; Difficulty: Medium; Type: Calc

3. Calculate SSInteraction.

Sol: ; Answer: 883.6 0.5; LO3.2-3; Difficulty: Medium; Type: Calc 4. Which of the following statements are appropriate descriptions of the statistical interaction FOR INSTRUCTOR USE ONLY

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in this sample? Select all that apply.

A. Weight gain increases when rats are fed a high protein diet instead of a low protein diet.

B. When beef is the source, higher protein levels lead to higher weight gains, whereas when cereal is the source, higher protein levels do not have much impact.

C. Beef has higher protein levels than cereal, and thus, beef has a positive effect on weight gain.

D. Beef and cereal sources are very different in terms of weight gain when the protein level is high, but they are similar when the protein level is low.

Ans: B, D; LO3.2-2; Difficulty: Medium; Type: MS 5. Use the 3S Strategy to test the significance of the interaction based on the difference in the differences statistic. The histogram to the right shows the results of simulating 10,000 trials under the assumption of no interaction between source and level.

Based on the null distribution and difference in the differences for the sample data, what do you conclude?

If there were really ______ (an interaction/no interaction), we’d get a difference in the differences as or more extreme as our observed statistic about ________ (0.5, 7, or 50)% of the time. This provides ___________ (moderately/very) strong evidence of an interaction between source and level.

Ans: no interaction, 7, moderately; LO3.2-4; Difficulty: Hard; Type: FIB 6. The table below shows all possible pairwise confidence intervals for the four treatments.

Groups Compared Beef/High – Beef/Low Beef/High – Cereal/Low Beef/High – Cereal/High Cereal/High – Beef/Low Cereal/Low – Beef/Low Cereal/High – Cereal/Low

95% Confidence Intervals (7.23, 34.36) (2.53, 29.66) (0.54, 27.66) (-6.86, 20.26) (-8.86, 18.26) (-11.56, 15.56)

What conclusion can you draw from the table? Note that the term “significant” in these statements refers to statistical significance not practical importance.

A. Beef/High is significantly different from every other treatment in this study.

B. Cereal/Low is significantly different from every other treatment in this study.

C. Beef/High and Beef/Low are the only two treatments that are significantly different from each other.

D. Cereal/High and Cereal/Low are the only two treatments that are significantly different from each other.

Ans: A; LO3.2-4; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

1-58 Questions 7 through 11: In a balanced, full-factorial experiment, 60 volunteers were randomly assigned to receive one of three pain medications (A, B, C) at one of two dosages (high, low).

After an hour, their response to pain was recorded on a scale of 0 to 20 (higher numbers indicate more severe pain).

7. Suppose there were no interaction between pain medication and dosage. How would you complete the table of means below?

Low Dose High Dose

Drug A 10.4 5.3

Drug B 12.2 ?

Drug C 8.8 ?

Ans: 7.1, 3.1; LO3.2-1; Difficulty: Medium; Type: FIB 8. Which of the following statements are appropriate descriptions of the statistical interaction in this sample?

A. The mean pain response for low dosages is always higher than the mean pain response for high dosages.

B. Drug A and Drug B are usually prescribed at low dosages. Drug C is prescribed at a high dosage nearly half the time.

C. The dosage effect is not the same for all three drugs. The dosage effect for Drug C is much smaller than for Drug A or Drug B. D. All of the statements above are appropriate descriptions of the statistical interaction.

Ans: C; LO3.2-2; Difficulty: Medium; Type: MC 9. The researchers decided to use a two-variable model with an interaction.

{

{ {

True or False: The predicted pain responses based on this model will be equal to the FOR INSTRUCTOR USE ONLY

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observed treatment means for each of the six drug-dose combinations Ans: True; LO3.2-3; Difficulty: Medium; Type: TF 10. The partially filled in ANOVA table below corresponds to the two factor model with interaction.

Source Drug Dose Interaction Error Total

DF

SS 21.70 281.67 50.63 258.60 612.60

MS 10.85 281.67

F 2.27 58.82

p-value 0.1135 <0.0001 0.008

Calculate the F-statistic for the interaction.

Sol: Ans: 5.29

0.05; LO3.2-4; Difficulty: Medium; Type: Calc

11. The p-value for testing the interaction between drug and dosage is 0.008. What do you conclude? You may assume .

This study provides ___________ (strong/weak) evidence that there ________ (is/is not) an interaction between drug and dosage.

Ans: strong, is; LO3.2-4; Difficulty: Medium; Type: FIB

12. You plan to use a theory-based p-value to test the interaction in a 2x2 full-factorial design.

Does the residual plot below indicate a violation of the validity conditions?

A. Yes, because the residuals are centered at 0, so the independence condition is violated.

B. Yes, because data points appear in four stacks, so the normality condition is violated.

C. Yes, because the points are not evenly distributed along the x-axis, so the equal variance condition is violated.

D. No, this graph does not indicate any major violation of the validity conditions.

Ans: D; LO3.2-4; Difficulty: Medium; Type: MC

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1-60 13. Below are two interaction plots for two factors (A and B) and a quantitative response variable.

Assuming the sample sizes are the same, which graph corresponds to a smaller p-value for the interaction?

A. Graph 1 would correspond to a smaller p-value for the interaction.

B. Graph 2 would correspond to a smaller p-value for the interaction.

C. Both samples would result in the same p-value for the interaction.

D. These graphs to not provide enough information to decide which sample would correspond to a smaller p-value for the interaction.

Ans: A; LO3.2-2; Difficulty: Medium; Type: MC 14. A large field was divided into 24 equal-sized plots to be planted with the same number of potato plants. Each plot was randomly assigned a type of fertilizer (A, B, C) and a manure level (high, low) based on a balanced, full-factorial design.

True or False: By using a balanced design, the researchers have ensured that there is no interaction between the type of fertilizer and the level of manure.

Ans: False; LO3.2-1; Difficulty: Medium; Type: TF 15. True or False: When there is a substantial interaction between Factor A and Factor B in a two-factor design, the researcher should interpret the main effects of Factor A and Factor B separately.

Ans: False; LO3.2-1; Difficulty: Medium; Type: TF

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Section 3.3: Replication LO3.3-1: Explain the benefits and challenges of replication.

LO3.3-2: Define and describe advantages of a generalized block design.

LO3.3-3: Define and describe advantages of within-blocks factorial designs.

Questions 1 through 3: Concentration is a one-person memory game in which cards are laid face down on a surface and two cards are flipped face up at a time. The object of the game is to turn over matching pairs of cards.

An online version of this game includes three different sets of cards: one has images of animals on the cards, one has images of babies, and one has images of holiday scenes.

Are these three variations equally difficult? To investigate, eight students tried all three versions of the game in random order. They recorded the amount of time (in seconds) it took to complete the game.

1. How would you describe this design?

A. Randomized complete block design with repeated measures B. Generalized block design with replication C. Full factorial design with independent groups D. Within-block factorial design with two experimental factors Ans: A; LO3.3; Difficulty: Medium; Type: MC 2. Suppose you want to test for a person-version interaction as part of the analysis. Fill in the degrees of freedom in the ANOVA table below.

Source Version Person Version Error Total

Person

DF ? ? ? ? ?

SS

MS

F

Ans: 2, 7, 14, 0, 23; LO3.3-1; Difficulty: Medium; Type: FIB 3. True or False: In this scenario, the person-version interaction is completely confounded with error, so the only way to carry out statistical tests of the version effect and the person effect is to exclude the interaction term from the model.

Ans: True; LO3.3-1; Difficulty: Medium; Type: TF Questions 4 and 5: A researcher is interested in how distractions affect students as they work on mathematical tasks, so she plans to assign 150 students to work under different conditions (with or without distractions) for a fixed period of time, recording the number of math problems that each student answers correctly. The study will be carried out over the course of a week, and the researcher worries that student motivation may vary day to day, so she decides to use day as a blocking variable. Each day (Monday through Friday), she will assign 15 students to work with distractions and 15 to work without distractions.

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1-62 4. Suppose the researcher wants to test for a condition-day interaction as part of the analysis. Fill in the degrees of freedom in the ANOVA table below.

Source DF SS MS Distraction Condition ?

Day ? ?

Condition Day Error ?

Total ?

Ans: 1, 4, 4, 140, 149; LO3.3-1; Difficulty: Medium; Type: FIB

F

5. True or False: In this scenario, the condition-day interaction is completely confounded with error, so the only way to carry out statistical tests of the condition effect and the day effect is to exclude the interaction term from the model.

Ans: False; LO3.3-1; Difficulty: Medium; Type: TF 6. Researchers used a paired design to investigate whether cell phone use impairs drivers’ reaction times. 64 students participated in a simulation of driving situations, pressing a brake button as soon as they saw a red light. A device recorded their reaction times (in milliseconds). Each student completed the simulation under two different conditions: once while talking on a cell phone and once while listening to music.

The study design described above requires researchers to assume that the effect of the driving condition (cell phone or music) is the same for every student. How could you modify the study design in order to estimate and test for a statistically significant interaction?

A. Increase the sample size by recruiting more students to participate.

B. Ask each student to complete the simulation more than once under each condition.

C. Add a control condition where students are not distracted by cell phones or music.

D. Collect data on a potential confounding variable (how confident students feel as drivers, how much sleep they got last night, etc.).

Ans: B; LO3.3-1; Difficulty: Medium; Type: MC 7. Which of the following is the best description of replication?

A. Replication means there are at least 20 observational units in the study (or more if the data distribution is not symmetrical).

B. Replication means the study employs random assignment and a placebo control group for comparison.

C. Replication means each observational unit is measured more than once under different experimental conditions.

D. Replication means each set of experimental conditions (or factor-block combination) occurs more than once in the design.

Ans: D; LO3.3-1; Difficulty: Easy; Type: MC

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8. Which of the following is an advantage of replication in a study design?

A. Replication reduces unexplained variation within groups, which leads to more powerful study designs.

B. Replication makes more efficient use of resources, which reduces the cost and/or time required to complete a study.

C. Replication makes it possible to estimate treatment and interaction effects separately from random error.

D. Replication ensures that there is no association between experimental factors, which prevents confounding.

Ans: C; LO3.3-1; Difficulty: Medium; Type: MC Questions 9 and 10: Ten male and ten female personnel officers were shown a front view photograph of a job applicant’s face and asked to rate the likely job success of the applicant on a scale of 0 to 20. Half of the officers in each gender were chosen at random to receive a version of the photograph in which the applicant made eye contact with the camera.

9. How would you describe this design?

A. Randomized complete block design with repeated measures B. Generalized block design with replication C. Full factorial design with independent groups D. Within-block factorial design with two experimental factors Ans: B; LO3.3-2; Difficulty: Medium; Type: MC 10. The plot below shows the mean success rating for each combination of gender and eye contact. Is there a substantial interaction between gender and eye contact in this sample?

A. Yes, because the means for all four combinations of gender and eye contact are different from each other.

B. Yes, because female offers tend to give higher ratings and officers who see the photo with eye contact tend to give higher ratings.

C. No, because the effect of eye contact is roughly the same regardless of officer gender.

D. No, because male and female officers are equally likely to see a photo with eye contact, since this is a balanced design.

Ans: C; LO3.3-2; Difficulty: Medium; Type: MC

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1-64 Questions 11 through 14: A waitress works part-time at two different restaurants: one is a casual deli and the other is a more upscale Italian restaurant. She decides to conduct an experiment to investigate whether having a conversation with her customers or writing ―Thank you!‖ on the check will affect her tip percentages. At each restaurant, she will assign 32 tables of customers to one of four treatments: conversation and message on check, no conversation and message on check, conversation and no message on check, or no conversation and no message on check.

11. Identify the components of this study.

Observational unit(s): Blocking variable(s): Experimental factor(s): Response variable(s):

A. Conversation B. Message on check C. Restaurant D. Tables of customers E. Tip percentage Ans: Observational units: D; Blocking variable: C; Experimental factors: A, B; Response variable: E; LO3.3-3; Difficulty: Medium; Type: Ma

12. The graphs below show the interaction between conversation and message for each of the two restaurants. Do these graphs indicate a three-way interaction between restaurant, conversation, and message in this sample?

A. Yes, because although the shapes of the graphs are similar, the tips at the Italian restaurant tend to be higher than tips at the deli.

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B. Yes, because having a conversation with the customers and writing a message on the check tend to have a positive effect on the tip percentage at both restaurants.

C. No, the message effect is larger when the waitress has a conversation with the customers, and this interaction is roughly the same at both restaurants.

D. No, because the lines do not intersect on either of these graphs, and we are not given a graph that shows all three variables.

Ans: C; LO3.3-3; Difficulty: Hard; Type: MC 13. The p-value for the conversation-message interaction is 0.0273. After adjusting for restaurant, does this data provide sufficient evidence of an interaction between conversation and message? You may use .

A. No, at the significance level, this study provides insufficient evidence of an interaction between conversation and message.

B. Yes, at the significance level, this study provides sufficient evidence of an interaction between conversation and message.

C. It is not appropriate to interpret the theory-based p-value for testing interaction, because the study design lacks replication.

D. It is not appropriate to interpret the theory-based p-value for testing interaction, because the customers were not randomly assigned to restaurants.

Ans: B; LO3.3-3; Difficulty: Medium; Type: MC 14. The p-value for the conversation-message interaction is 0.0273. Is it appropriate to interpret the main effects of conversation and message separately?

A. Yes, when the p-value for an interaction is small, it is appropriate to interpret the main effects separately.

B. No, when the p-value for an interaction is small, it is not appropriate to interpret the main effects separately.

C. It depends on the p-value for the three-way interaction. As long as the p-value for the three-way interaction is small, it is appropriate to interpret the main effect separately.

D. It depends on the p-value for the three-way interaction. As long as the p-value for the three-way interaction is large, it is appropriate to interpret the main effect separately.

Ans: B; LO3.3-3; Difficulty: Hard; Type: MC 15. True or False: A matched pairs design is a special case of a within-blocks factorial design with one binary factor.

Ans: True; LO3.3-3; Difficulty: Medium; Type: TF

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Section 3.4: Interactions in Observational Studies LO3.4-1: Interpret interactions with observational data.

LO3.4-2: Sketch and interpret interaction plots.

LO3.4-3: Run a two-variable ANOVA including an interaction term for an observational study with 2 two-level variables.

Questions 1 through 6: In 2018, a sample of 628 academic faculty from universities across the country were surveyed about their salaries (in US dollars). The results were classified according to each faculty member’s academic rank (instructor, assistant professor, associate professor, and full professor) and gender (male, female).

1. The plot below shows the interaction between rank and gender. Which of the following are statements about main effects and which are statements about interaction?

Statements about main effects: Statements about interaction: A. At each academic rank, male faculty earn higher salaries than female faculty, on average.

B. The salary ―gap‖ between male and female faculty is highest at the academic rank of professor.

C. Gender modifies the association between rank and salary.

Ans: Main effects: A; Interaction: B, C; LO3.4-2; Difficulty: Medium; Type: Ma 2. Which of the following is the appropriate null hypothesis for testing the statistical significance of the interaction between rank and gender?

A. There is no interaction between rank and gender on salaries in this population.

B. There is an interaction between rank and gender on salaries in this population.

C. Neither rank nor gender has an interaction with salary in this population.

D. Both rank and gender have an interaction with salary in this population.

Ans: A; LO3.4-1; Difficulty: Medium; Type: MC

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3. The partially filled-in ANOVA table below can be used to test the statistical significance of the interaction between rank and gender. Calculate the F-statistic for the interaction.

Source Rank Gender Rank Gender Error Total

DF

SS 390,047 11,767 13,133 976,624 1,674,691

MS

F J;

p-value <0.0001 0.0065 0.0404

Sol: Ans: 2.78 0.1; LO3.4-3; Difficulty: Medium; Type: Calc 4. What would happen if the rank-gender interaction term were removed from the model?

Source Rank Gender Rank Gender Error Total

DF

SS 390,047 11,767 13,133 976,624 1,674,691

MS

F J;

p-value <0.0001 0.0065 0.0404

A. SSTotal would increase to 1,674,691 + 13,133 = 1,687,824.

B. SSTotal would increase, but we can’t say what the new value would be, because of potential covariation between the interaction and the main effects.

C. SSError would increase to 976,624 + 13,133 = 989,757.

D. SSError would increase, but we can’t say what the new value would be, because of potential covariation between the interaction and the main effects.

Ans: D; LO3.4-3; Difficulty: Medium; Type: MC 5. The table below shows 95% confidence intervals for the difference in mean salaries for male and female faculty members at each academic rank.

Male – Female Instructor Assistant professor Associate professor Professor

95% confidence interval (-13955, 23165) (-6734, 17991) (-10345, 16890) (14101, 40181)

At which academic rank(s) are the differences in mean salaries for male and female faculty members statistically significant? Select all that apply. Note: This question refers to statistical significance not practical importance.

A. Instructor B. Assistant professor C. Associate professor D. Professor Ans: D; LO3.4-1 Difficulty: Medium; Type: MS FOR INSTRUCTOR USE ONLY

1-68 6. Does the residual plot below indicate any violation of conditions that might cause us to question the validity of theory-based p-values or confidence intervals in this context?

A. Yes, the independence condition is violated.

B. Yes, the equal variance condition is violated.

C. Yes, the normality condition is violated.

D. No, this graph does not indicate any major violation of the validity conditions.

Ans: B; LO3.4-1 Difficulty: Medium; Type: MC Questions 7 through 11: Students in a statistics class wanted to know how customers rate one of their favorite coffee shops. The students administered surveys and asked customers to rate their experience in the shop on a scale of 1-10. They also asked whether the survey respondent was a college student and how often they visit coffee shops (at least once a week or less than once a week).

7. How would you describe this design?

A. Observational study with two explanatory variables B. Observational study with four explanatory variables C. Randomized complete block design with repeated measures D. Generalized block design with replication Ans: A; LO3.3; Difficulty: Easy; Type: MC 8. What can you say about the relationships among the variables based on the plot below?

Select all that apply.

A. There is a substantial association between student and frequency.

B. There is a substantial interaction between student and frequency.

C. The student variable has a main effect on ratings.

D. The frequency variable has a main effect on ratings.

Ans: C, D; LO3.4-2; Difficulty: Medium; Type: MS FOR INSTRUCTOR USE ONLY

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9. Fill in the blank with a numerical value: In an ANOVA table, the interaction term would have ___ degree(s) of freedom.

Ans: 1; LO3.4-1; Difficulty: Medium; Type: FIB 10. Using the adjusted sums of squares below, calculate

for this model.

Source Frequency Student Frequency Student Error Total

Adjusted SS 6.954 1.879 0.005 128.290 141.556

Sol: Ans: 0.094 0.06; LO3.4-3; Difficulty: Medium; Type: Calc 11. Suppose the interaction term were removed from the model. How would the R2 value change?

Source Frequency Student Frequency Student Error Total

Adjusted SS 6.954 1.879 0.005 128.290 141.556

A. Removing the interaction term would cause R2 to increase dramatically.

B. Removing the interaction term would cause R2 to decrease dramatically.

C. Removing the interaction term would cause R2 to increase slightly.

D. Removing the interaction term would cause R2 to decrease slightly.

Ans: D; LO3.4-3; Difficulty: Medium; Type: MC Questions 12 and 13: In an observational study of rheumatoid arthritis, researchers recorded several indicators of disease activity as well as information about treatment. The table below shows the mean CDAI value (an indicator of disease activity) for patients receiving different types of treatments.

N Steroids Biologics Mean CDAI 71 No No 10.51 76 No Yes 10.78 31 Yes No 15.94 28 Yes Yes 22.82

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1-70 12. Does this table of means suggest a substantial interaction between treatment with steroids and treatment with biologics?

A. Yes, because the sample sizes for the four groups are all different.

B. Yes, because treatment with steroids and treatment with biologics are both associated with higher CDAI values.

C. Yes, because the effect of biologics is much larger for patients who are taking steroid treatment than for those who are not taking steroid treatment.

D. No, because 64.5% of patients who take steroids take biologics and 62.2% of patients who don’t take steroids take biologics, so the relationship is weak.

Ans: C; LO3.4-1; Difficulty: Medium; Type: MC 13. Fill in the blanks: It is plausible that patients who are being treated more aggressively had more severe symptoms before treatment began; thus the higher CDAI scores among patients taking steroids and biologics could be due to _________ (confounding/covariation). In order to draw causal conclusions, we would need a study that involves random __________ (selection/assignment).

Ans: confounding, assignment; LO3.4-1; Difficulty: Medium; Type: FIB 14. Suppose a study is done where the response variable is fuel efficiency (miles per gallon) and the two explanatory variables are horsepower (low, medium, high) and weight (heavy, light). A table with two of the interaction effects filled in is given below.

Heavy Light

Low Horsepower 0.23

Medium Horsepower 2.97

High Horsepower

Fill in the missing interaction effects.

Ans: Heavy-High: -3.20, Light-Low: -0.23, Light-Medium: -2.97, Light-High: 3.20; LO3.4-1; Difficulty: Medium; Type: FIB 15. A real estate agent collected information on 100 recent home sales in their town. In addition to selling prices (in $1000s), the agent recorded information about the home’s location (north side of town or south side of town) and the number of bedrooms (at least 3 or less than 3). Match the terms with their descriptions in this context.

Covariation: Association: Interaction:

A. Houses on the south side of town are more likely to have at least three bedrooms than houses on the north side.

B. The relationship between number of bedrooms and price changes based on the home’s location.

C. Some of the variability in prices cannot be attributed exclusively to one source (location, bedrooms, or interaction).

Ans: Covariation: C, Association: A, Interaction: B; LO3.4-3; Difficulty: Medium; Type: Ma

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Intermediate Statistical Investigations Test Bank Question types:

FIB = Fill in the blank Ma = Matching MC = Multiple choice

Calc = Calculation MS = Multiple select TF = True-false

CHAPTER 4 TERMINAL LEARNING OUTCOMES TLO4-1: Represent the association between two quantitative variables with a linear regression model, and compare and contrast with a separate-means model.

TLO4-2: Assess the evidence of a linear association between two quantitative variables using both simulation and theory-based approaches.

TLO4-3: Use a regression model to represent the adjusted relationship between two quantitative variables based on a binary categorical variable.

TLO4-4: Include interaction between quantitative and categorical variables in a multiple regression model, and interpret the nature of the interaction.

TLO4-5: Interpret an interaction between a quantitative variable and a multi-level categorical variable in a multiple regression model.

Section 4.1: Quantitative Explanatory Variables LO4.1-1: Describe the association between two quantitative variables numerically and graphically.

LO4.1-2: Interpret least squares regression models between two quantitative variables.

LO4.1-3: Compare and contrast separate means vs. linear regression models.

Questions 1 through 6: Does vitamin C affect tooth growth in guinea pigs? Sixty guinea pigs were randomly assigned to one of three dose levels of vitamin C (0.5, 1, or 2 mg/day).

The response variable was the length of odontoblasts – cells responsible for tooth growth.

The table below summarizes three models that could be used to analyze this data.

Model Single-mean Separate-means Linear

Predicted Length at each Dosage 0.5 mg 1 mg 2 mg 18.813 18.813 18.813 10.605 19.735 26.100 12.304 17.186 26.950

SSError 3452.2 1025.8 ?

1. If one of these models were used to predict length for a dosage of 1.5 mg/day, that would be called ________ (extrapolation/interpolation).

If one of these models were used to predict length for a dosage of 5 mg/day, that would be called ________ (extrapolation/interpolation).

Ans: interpolation, extrapolation; LO4.1-1; Difficulty: Easy; Type: FIB 2. Which of the models could be used to predict length for a dosage of 1.5 mg/day. Select all that apply.

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1-72 A. Single-mean model B. Separate-means model C. Linear model Ans: C; LO4.1-3; Difficulty: Medium; Type: MS 3. Fill in the degrees of freedom for the separate-means model and the linear model.

Separate-means model Source DF Model Error Total

Linear model Source DF Model Error Total

Ans: Separate-means: 2, 57, 59; Linear: 1, 58, 59; LO4.1-3; Difficulty: Medium; Type: FIB 4. Calculate the y-intercept and slope for the linear regression model. Keep three decimal places in your answer. _______

_______

Ans: 7.422 0; 9.764 0, LO4.1-2; Difficulty: Medium; Type: FIB 5. How does SSError for the linear model compare to SSError for the other models?

A. SSError for the linear model is less than 1025.8.

B. SSError for the linear model is greater than 1025.8 but less than 3452.2.

C. SSError for the linear model is greater than 3452.2.

D. There is not enough information to determine how SSError for the linear model compares to SSError for the other models.

Ans: B; LO4.1-3; Difficulty: Medium; Type: MC 6. Does the table provide any indication that the linear model is not a good fit for the data?

A. Yes, because the separate-means model and the linear model lead to substantially different predicted values, especially for dosages of 0.5 and 1.

B. Yes, because 17.186 – 12.304 does not equal 26.950 – 17.186. If there were a linear relationship, we’d expect the predicted values to increase at a constant rate.

C. No, because the linear model shows that odontoblast lengths increase at a constant rate of 9.764 units per mg/day of Vitamin C. D. No, because length and dosage are both quantitative variables, so a linear regression model is the most appropriate analysis method.

Ans: A; LO4.1-3; Difficulty: Medium; Type: MC

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Questions 7 through 9: A dataset contains information about 325 books for sale at amazon.com. The dataset includes two prices for each book: List Price (the price set by the publisher in dollars) and the Amazon Price (in dollars). The linear model that uses a book’s list price to predict its Amazon price is given below.

7. Interpret the slope by filling in the blanks.

As the ______ (Amazon/list) price increases by 1 dollar, the ______ (Amazon/list) price is predicted to increase by ______ dollar(s).

Ans: list, Amazon, 0.83; LO4.1-2; Difficulty: Medium; Type: FIB 8. Does the y-intercept have a meaningful interpretation in this context?

A. Yes. The y-intercept describes how SSError for the linear model compares to SSError for the single-mean model.

B. Yes. The y-intercept describes how Amazon prices compare to list prices, on average.

C. No. $0 is not a reasonable value for the Amazon price, so the intercept does not have a meaningful interpretation.

D. No. $0 is not a reasonable value for the list price, so the intercept does not have a meaningful interpretation.

Ans: D; LO4.1-2; Difficulty: Medium; Type: MC 9. Calculate the residual for a book that has a list price of $12.95 and an Amazon price of $5.18.

Sol: Ans: -3.16 0.01; LO4.1-2; Difficulty: Medium; Type: Calc Questions 10 and 11: The graph below shows the relationship between life expectancy (in years) and fertility rate (babies per woman) in 1960. Each dot represents a country, and the size of the dot indicates the country’s population.

FOR INSTRUCTOR USE ONLY

1-74 10. If you fit a linear model to these data…

The slope would be _________ (positive/negative).

The y-intercept would be _________ (positive/negative).

Ans: negative, positive; LO4.1-1; Difficulty: Easy; Type: FIB 11. In 1960, China had a fertility rate of 3.99 babies per woman and life expectancy of 31.6 years.

The residual for China would be _________ (positive/negative), which means that the linear model would _________ (overestimate/underestimate) China’s life expectancy.

Ans: negative, overestimate; LO4.1-1; Difficulty: Medium; Type: FIB 12. The scatterplots below show two different datasets (A and B).

In Dataset A, R2 for the separate-means model _______ (>, <, =) R2 for the linear model.

In Dataset B, R2 for the separate-means model _______ (>, <, =) R2 for the linear model.

Ans: >, =; LO4.1-3; Difficulty: Medium; Type: FIB 13. Which sum of squares is minimized by the least squares regression line?

A. SSModel B. SSError C. SSTotal D. All three of these values are minimized by the least squares regression line.

Ans: B; LO4.1-2; Difficulty: Easy; Type: MC 14. True or False: When two models have similar SSError values, you should generally choose the one that uses more degrees of freedom (DF for Model is higher).

Ans: False; LO4.1-3; Difficulty: Medium; Type: TF

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15. You are considering two models to predict a quantitative response: a separate-means model and linear model. You find that SSError is smaller for the separate-means model, yet the SE of the residuals is smaller for the linear model. Does this indicate that you have made a calculation mistake?

A. Yes. When the data are quantitative, SSError for the linear model should always be smaller than SSError for the separate-means model.

B. Yes. SSError and the SE of the residuals both measure the amount of prediction error. As SSError increases, SE of the residuals should always increase.

C. No. The separate-means model and the linear model have different degrees of freedom for error, so SSError and the SE of the residuals don’t always ―agree.‖ D. No. Despite their similar names, SSError and the SE of the residuals measure completely different things. It is not surprising that these two values do not ―agree.‖ Ans: C; LO4.1-3; Difficulty: Medium; Type: MC

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Section 4.2: Inference for Simple Linear Regression LO4.2-1: Carry out simulation-based inference to assess the strength of evidence of a linear association between the quantitative explanatory and response variables.

LO4.2-2: Use a theory-based approach to assess the strength of evidence of a linear association between the quantitative explanatory and response variables.

LO4.2-3: Evaluate the validity of the theory-based approach using residual plots.

Questions 1 through 6: A teacher randomly assigned students to a seat in one of six rows during a particular unit of instruction. At the end of the unit, the teacher recorded each student’s score on the unit test. The scatterplot and least squares regression equation are given below. Do these data provide convincing evidence that sitting further away from the front of the room causes lower scores? (Higher row numbers are further from the front.)

1. Match used.

and

A. Sitting further from the front causes lower scores, on average.

B. Sitting further from the front causes higher scores, on average.

C. Sitting further from the front has an effect on average scores.

D. There is no association between seat location and score.

.

Ans:

to the appropriate statement in words. Two of the statements will not be

: D,

A; LO4.2-1; Difficulty: Medium; Type: Ma

2. You decide to use the 3S strategy to evaluate the evidence. Which of the following are statistics you could use to summarize the linear association? Select all that apply.

A. Sample mean B. Sample proportion C. Correlation D. Slope E. Ans: C, D, E; LO4.2-1; Difficulty: Medium; Type: MS

3. You decide to use the 3S strategy to evaluate the evidence. How would you conduct the FOR INSTRUCTOR USE ONLY

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simulation?

A. Write the scores on cards. Shuffle and deal the cards into two groups to represent seats close to the board and seats far from the board.

B. Write the scores on cards. Shuffle and randomly re-assign the scores to row numbers 1 through 6.

C. For each data point, randomly select an integer between 1 and 6 as the row number and randomly select an integer between 0 and 100 as the test score.

D. For each data point, randomly select an integer between 1 and 6 as the row number and randomly select an integer between 52 and 100 as the test score.

Ans: B; LO4.2-1; Difficulty: Medium; Type: MC 4. You decide to use the 3S strategy to evaluate the evidence. The dotplot below shows the shuffled slopes for 100 repetitions of the simulation.

Calculate the p-value based on this dotplot.

Sol: 10 of the 100 repetitions are less than -1.02, so we estimate the p-value to be 10/100.

Ans: 0.10 0.02; LO4.2-1; Difficulty: Medium; Type: Calc 5. One student in the 6th row made a 100 on the test. Suppose that this student’s score were removed from the dataset. Would this affect the p-value?

After removing this point, the p-value would be ________ (higher/lower/the same as) before.

Ans: lower; LO4.2-1; Difficulty: Hard; Type: FIB 6. The plots below display the residuals for this regression model. Match each validity condition to the description of how that condition should be checked. Two of the descriptions will not be used.

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1-78 Independence:

A. Check that the mean of the residuals is 0.

Equal variance:

B. Check that the students were randomly assigned to rows with no repeated measures.

Normality:

C. Check that the histogram of the residuals is reasonably symmetric and bell-shaped.

D. Check that predicted values are spaced fairly evenly along the x-axis.

E. Check that the vertical spread of the residuals at each of the predicted values is reasonably similar.

Ans: Independence: B; Equal variance: E; Normality: C; LO4.2-3; Difficulty: Medium; Type: Ma Questions 7 through 10: The scatterplot and software output below show the relationship between CEO salary and number of employees for 100 companies.

Term Coeff.

SE t-stat p-value Intercept 85378.8 5604.3 15.23 <0.0001 Employees 35.7438 14.8028 2.41 0.0176

7. Calculate a 95% confidence interval to estimate the population slope. Even using a rough estimate of t*, you should be able to choose the best option below.

A. (0.07, 71.41) B. (6.37, 65.12) C. (20.94, 50.55) D. (33.33, 38.15) Ans: B; LO4.2-2; Difficulty: Medium; Type: MC 8. State the hypotheses for testing the association between CEO salary and number of employees.

A. B. C. D. Ans: C; LO4.2-2; Difficulty: Medium; Type: MC 9. The theory-based p-value shown above was calculated using a t distribution with _____ degrees of freedom.

Ans: 98

; LO4.2-2; Difficulty: Medium; Type: FIB FOR INSTRUCTOR USE ONLY

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10. Which of the following is an appropriate conclusion based on the small p-value for the slope?

A. There is strong evidence of an association between CEO salary and Number of Employees in the population; the sample slope would be unlikely to occur by chance.

B. There is a strong association between CEO salary and Number of Employees in the sample; a large portion of the variability in salaries is explained by the model.

C. There is sufficient evidence to conclude that hiring more employees causes a CEO’s salary to increase slightly.

Ans: A; LO4.2-2; Difficulty: Medium; Type: MC Questions 11 through 13: The owner of a café suspects that coffee sales are related to the weather. For a period of 60 days in the spring, she records the number of coffees sold at her café as well as the minimum daily temperature (in degrees Fahrenheit).

11. Using the partially filled in ANOVA table below, calculate the F-statistic.

Source Model Error Total

DF 1 58 59

SS 8010 13575 21585

MS

F

Sol: Ans: 34.22

0.1; LO4.2-2; Difficulty: Medium; Type: Calc

12. The 95% confidence interval for the slope is (-1.37, -0.67). Interpret this interval.

We are 95% confident that…

A. The sample slope is between 0.67 and 1.37 standard errors below the null hypothesis value ( .

B. The true population slope is between 0.67 and 1.37 standard errors below the null hypothesis value ( .

C. The café sells between 0.67 and 1.37 fewer coffees, on average, for every one degree increase in minimum daily temperature.

D. The café sells between 0.67 and 1.37 fewer coffees, on average, for every one degree decrease in minimum daily temperature.

Ans: C; LO4.2-2; Difficulty: Medium; Type: MC 13. The 95% confidence interval for the slope is (-1.37, -0.67). Suppose the owner continued to collect data until she had a sample size of 300 days. What would you expect to happen to the width of the confidence interval?

A. The CI would get narrower, because the SE of would decrease.

B. The CI would get narrower, because the SE of the residuals would decrease.

C. The CI would get wider, because the critical value, t*, would increase.

D. The CI would get wider, because the confidence level would increase.

Ans: A; LO4.2-2; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

1-80 14. Describe how each of the following factors are related to the standard error of the slope.

As sample size increases, SEb tends to ________ (increase/decrease).

As SE of the residuals increases, SEb tends to ________ (increase/decrease).

As SD of the explanatory variable increases, SEb tends to ________ (increase/decrease).

Ans: decrease, increase, decrease; LO4.2-2; Difficulty: Medium; Type: FIB 15. The scatterplots below display two datasets of the same size with roughly the same SD of X. Which of the following would produce a larger F statistic?

A. Dataset 1 would have a larger F statistic, because SSModel is larger.

B. Dataset 1 would have a larger F statistic, because SSError is smaller.

C. Dataset 2 would have a larger F statistic, because SSModel is larger.

D. Dataset 2 would have a larger F statistic, because SSError is smaller.

Ans: D; LO4.2-2; Difficulty: Medium; Type: MC 16. Which of the datasets below would produce the smallest p-value?

A. Dataset A would produce the smallest p-value.

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Introduction to Financial Statements B. Dataset B would produce the smallest p-value.

C. Dataset C would produce the smallest p-value.

D. Two of these datasets are tied for the smallest p-value.

Ans: C; LO4.2-2; Difficulty: Medium; Type: MC

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Section 4.3: Quantitative and Categorical Explanatory Variables LO4.3-1: Adjust the relationship between two quantitative variables based on a categorical variable.

LO4.3-2: Evaluate the validity of the regression model.

LO4.3-3: Create indicator variables in order to include binary categorical variables in the regression model.

Questions 1 through 5: Consider a linear regression model that predicts body temperature (in degrees Fahrenheit) based on heart rate (in beats per minute, bpm), and sex (male, female).

The categorical variable, sex, was defined using effect coding: Term Intercept Heart Rate Sex [female]

Estimate 96.39 0.025 0.135

SE 0.649 0.009 0.616

t-stat 148.47 2.88 2.19

{ p-value <0.0001 0.0046 0.0307

1. True or False: Based on this model, the predicted body temperature for a male subject whose heart rate is 0 bpm is 96.39 degrees Fahrenheit. Note: Interpretations of the intercept often involve extrapolation.

Ans: False; LO4.3-1; Difficulty: Medium; Type: TF 2. True or False: Based on this model, body temperature is predicted to increase by 0.025 degrees for each 1 bpm increase in heart rate, holding sex constant.

Ans: False; LO4.3-1; Difficulty: Medium; Type: TF 3. Suppose there are two subjects, one male and one female, who both have the same heart rate. Based on this model, what is the predicted difference in their body temperatures, female – male?

Ans: 0.270 0; LO4.3-1; Difficulty: Medium; Type: Calc 4. Does this dataset provide strong evidence of a difference in body temperatures for males and females? You may assume that 0.05.

This dataset provides ________ (strong/weak) evidence of a difference in body temperature for males and females, ____________ (ignoring/adjusting for) heart rate.

Ans: strong, adjusting for; LO4.3-1; Difficulty: Medium; Type: FIB

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5. Are conclusions based on theory-based p-values valid for this regression model?

A. No, because the independence condition has been violated.

B. No, because the linearity condition has been violated.

C. No, because the equal variance condition has been violated.

D. No, because the normality condition has been violated.

E. Yes, because all the validity conditions are met in this scenario.

Ans: E; LO4.3-2; Difficulty: Medium; Type: MC Questions 6 through 8: A botanist collected data to measure the variation of Iris flowers of two related species. The data set consists of 50 flowers from two species of Iris – Iris setosa and Iris versicolor. Various features were measured including the length and width of the sepals, in centimeters. (Sepals are a part of the flower.)

6. A simple linear regression model used sepal length to predict sepal width. The plot below shows residuals from that model broken down by species.

The simple linear regression model tends to ____________ (overestimate/underestimate) the sepal widths for the setosa species and____________ (overestimate/underestimate) the sepal widths for the versicolor species.

This residual plot suggests that Species should be ____________ (included/excluded) in the model to predict sepal width.

Ans: underestimate, overestimate, included; LO4.3-3; Difficulty: Medium; Type: FIB 7. A multiple linear regression model used sepal length and species to predict sepal width. {

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1-84 Which of the following interpretations is correct?

A. Sepal width is predicted to be 0.517 cm after adjusting for sepal length and species.

B. As sepal length increases by 1 cm, sepal with is predicted to increase by 0.472 cm, holding species constant.

C. For both species, sepal width is predicted to increase by 0.548 cm for each one cm increase in sepal length.

D. For the setosa species, sepal width is predicted to increase by 0.548 cm for each one cm increase in sepal length.

Ans: B; LO4.3-1; Difficulty: Medium; Type: MC 8. A multiple linear regression model used sepal length and species to predict sepal width. {

Suppose the binary categorical variable species had been recorded using indicator coding instead of effect coding. Which of the numerical values would change? Select all that apply.

A. The intercept would change B. The slope corresponding to sepal length would change.

C. The slope(s) corresponding to species would change.

D. The SE of the residuals would change.

Ans: C; LO4.3-3; Difficulty: Medium; Type: MS Questions 9 through 11: The plot below represents a model that uses two explanatory variables, one quantitative and one categorical, to predict y. {

9. Why are the lines parallel?

A. On average, Y changes at the same constant rate relative to X for both Category A and Category B in the sample data.

B. The model assumes that Y changes at the same constant rate relative to X for both Category A and Category B. Ans: B; LO4.3-1; Difficulty: Medium; Type: MC

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{

Complete the model statement below: { Ans: 40 0, 2 0, -10 0, 10 0; LO4.3-3; Difficulty: Hard; Type: FIB

11. Suppose the categorical variable is defined by indicator coding.

{

Find the intercept and slopes for this model.

Term Intercept Variable Variable

Coefficient

Ans: 50 0, 2 0, -20 0; LO4.3-3; Difficulty: Hard; Type: FIB Questions 12 and 13: The scatterplots below display data collected from 27 automotive plants.

For each plant, three variables were recorded: Defects = number of assembly defects per 100 cars, Time = time (in hours) to assemble each vehicle, and Location = whether or not the plant is located in Japan. The graph on the left shows the relationship between Defects and Time for all 27 plants. The graph on the right shows the relationship between Defects and Time for each plant location (in Japan and not in Japan).

{ 12. The slope coefficient for Time is not given in the model above. What is the interpretation of the missing coefficient?

The predicted change in _________ (Location/Defects/Time) for a one-unit change in _________ (Location/Defects/Time), holding ________ (Location/Defects/Time) constant.

Ans: Defects, Time, Location; LO4.3-1; Difficulty: Medium; Type: FIB

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1-86 13. The slope coefficient for Time is not given in the model above. Based on the scatterplots, we expect the missing slope coefficient for time to be __________ (positive/negative).

Ans: negative; LO4.3-1; Difficulty: Medium; Type: FIB 14. When evaluating a regression model with multiple explanatory variables, which significance test(s) should you conduct first? Why? A. First conduct an F-test for the overall model. This protects against Type I error.

B. First conduct an F-test for the overall model. This protects against Type II error.

C. First conduct t-tests for the individual predictors. This protects against Type I error.

D. First conduct t-tests for the individual predictors. This protects against Type II error.

Ans: A; LO4.3-1; Difficulty: Medium; Type: MC 15. Match the terms below with their features. Each choice (A-D) will be used exactly once.

Effect coding: Indicator coding: .

A. The binary variable is coded (0, 1).

B. The binary variable is coded (-1, 1).

C. The slope coefficient is the difference in the group means.

D. The slope coefficient is the difference between the group means and the overall average (least squares mean).

Ans: Effect: B, D; Indicator: A, C; LO4.3-3; Difficulty: Easy; Type: Ma 16. A dataset contains information about a random sample of 200 house sales in a town in the Pacific Northwest. A regression model was fit for predicting the selling price of a house (in thousands of dollars) using house size (sq ft) and whether the house has a garage as predictors. The ANOVA table is given below.

Source House Size Garage Error Total

DF 1 1 197 199

Sum of Squares 1,380,568 58,076 1,263,472 2,668,870

F 215.26 9.06

p-value <0.0001 0.0030

Does SSHouseSize + SSGarage = SSModel?

A. No, because there is likely covariation between size and garage in this observational study.

B. No, because there is no covariation among explanatory variables when the data come from a random sample.

C. Yes, because there is likely covariation between size and garage in this observational study.

D. Yes, because there is no covariation among explanatory variables when the data come from a random sample.

Ans: A; LO4.3-1; Difficulty: Medium; Type: MC

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17. A simple linear regression model is being used to describe the relationship between two quantitative variables, and y, in an observational study. The p-value for testing vs. is 0.043.

Suppose the researchers added a new categorical explanatory variable, , to the model.

Would the p-value corresponding to the quantitative explanatory variable, , change?

A. No, adding a new variable, , would not affect the p-value corresponding to .

B. Yes, the p-value would be larger than 0.043 after adding the new variable, .

C. Yes, the p-value would be smaller than 0.043 after adding the new variable, .

D. The p-value is likely to change, but there is not enough information to say whether the p-value would get larger or smaller.

Ans: D; LO4.3-1; Difficulty: Hard; Type: MC

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Section 4.4: Quantitative/Categorical Interactions LO4.4-1: Include interaction between quantitative and categorical variables in a statistical model.

LO4.4-2: Interpret the nature of the interaction.

Questions 1 through 5: A dataset contains information about a random sample of 200 house sales in a town in the Pacific Northwest. A regression model was fit for predicting the selling price of a house (in thousands of dollars) using house size (sq ft), garage (garage=1 if the house has a garage and 0 otherwise), and the interaction between size and garage.

Term Intercept Size Garage_ind Size Garage_ind

Coeff.

141.835 0.039 -120.392 0.062

SE 18.130 0.006 24.163 0.008

t-stat.

7.82 7.07 -4.98 7.62

p-value <0.0001 <0.0001 <0.0001 <0.0001

1. True or False: In this town, there is strong evidence to suggest that the association between size and price differs based on whether the house has a garage.

Ans: True; LO4.4-2; Difficulty: Medium; Type: TF 2. True or False: In this town, there is strong evidence to suggest that houses with garages cost less, on average, than houses without garages.

Ans: False; LO4.4-1; Difficulty: Hard; Type: TF 3. Interpret the slope corresponding to size.

As _________ (price/size) increases by 1 unit, _________ (price/size) is predicted to increase by 0.039 units, for houses with _________ (a garage/no garage).

Ans: size, price, no garage; LO4.4-1; Difficulty: Medium; Type: FIB 4. Write an equation that predicts house price (in thousands of dollars) for houses with garages.

For houses with garages: Ans: 21.443 0, 0.101 0; LO4.4-1; Difficulty: Medium; Type: FIB 5. Suppose the binary variable garage had been recorded using effect coding instead of indicator coding. Which of the numerical values would change? Select all that apply.

A. The intercept would change.

B. The slope corresponding to house size would change.

C. The slope corresponding to garage would change.

D. The slope corresponding to the interaction would change.

E. The SE of the residuals would change.

Ans: A, B, C, D; LO4.4-1; Difficulty: Hard; Type: MS FOR INSTRUCTOR USE ONLY

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Questions 6 through 10: A botanist collected data to measure the variation of iris flowers of two related species. The data set consists of 50 flowers from two species of iris – iris setosa and iris versicolor. Various features were measured including the length and width of the sepals, in centimeters. (Sepals are a part of the flower.)

A multiple regression model was fit to predict sepal width based on sepal length, species, and the interaction between sepal length and species. Note: Effect coding was used.

{

{

6. Which of the following describes the relationship between length and width for the setosa species?

A. As length increases by 1 cm, width is predicted to increase by 0.559 cm.

B. As length increases by 1 cm, width is predicted to decrease by 0.721 cm.

C. As length increases by 1 cm, width is predicted to increase by 0.239 cm.

D. As length increases by 1 cm, width is predicted to increase by 0.798 cm.

Ans: D; LO4.4-1; Difficulty: Medium; Type: MC 7. Which of the following is the appropriate interpretation of the slope corresponding to species? Note: Interpretations may involve extrapolation.

A. The predicted sepal width of setosa is 0.721 lower than the predicted sepal width of versicolor for irises with a sepal length of 0 cm.

B. The predicted sepal width of setosa is 0.721 lower than the predicted sepal width of versicolor for irises of average sepal length.

C. The predicted sepal width of setosa is 1.442 lower than the predicted sepal width of versicolor for irises with a sepal length of 0 cm.

D. The predicted sepal width of setosa is 1.442 lower than the predicted sepal width of versicolor for irises of average sepal length.

Ans: C; LO4.4-1; Difficulty: Medium; Type: MC 8. Suppose the model used indicator coding for species instead of effect coding. { Calculate the coefficients for each term in the model.

Term Intercept Sepal length Species Length Species Ans: 0.872

Coefficient

0, -3.20

0, -1.442

0, 0.478

0; LO4.4-1; Difficulty: Hard; Type: Calc

FOR INSTRUCTOR USE ONLY

1-90 9. Which of the following statements is a description of the interaction between sepal length and species in this sample? Select all that apply.

A. Irises of the versicolor species tend to have longer sepal lengths than irises of the setosa species.

B. Irises of the setosa species tend to have wider sepals than the versicolor species, and sepal length and sepal width are positively associated for both species.

C. The relationship between sepal length and sepal width changes based on the species of iris.

D. The difference in sepal width for the two species of iris is smaller when the sepal length is small and larger when the sepal length is large.

Ans: C, D; LO4.4-2; Difficulty: Medium; Type: MS 10. The multiple linear regression model used sepal length, species, and the interaction between sepal length and species to predict sepal width. The scatterplot below shows the residuals of that model vs. the predicted values from that model, by species.

Does this residual plot indicate a problem with the regression model being used?

A. Yes. The residuals are not normally distributed.

B. Yes. There is no linear association between the residuals and predicted values.

C. Yes. The versicolor irises tend to have lower predicted values than setosa irises.

D. No. This residual plot does not indicate a problem with the regression model.

Ans: D; LO4.4-1; Difficulty: Medium; Type: MC Questions 11 and 12: Suppose you want to predict body temperature (in degrees Fahrenheit) based on heart rate (in beats per minute), and sex (male, female). You decide to exclude FOR INSTRUCTOR USE ONLY

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the interaction between heart rate and sex from your model, because the term was not ―statistically significant.‖ 11. How do you determine whether an interaction is statistically significant?

A. Fit a model that includes an interaction term. If the p-value for the interaction term is large, then the interaction is statistically significant.

B. Fit a model that includes an interaction term. If the p-value for the interaction term is small, then the interaction is statistically significant.

C. Fit a model that includes an interaction term. If the value for that model is large, then the interaction is statistically significant.

D. Fit a model that does not include an interaction term. If the value for that model is large, then the interaction is not statistically significant.

Ans: B; LO4.4-1; Difficulty: Medium; Type: MC 12. Explain what ―no interaction‖ means in the context of this question. Select all that apply.

A. The predicted difference in heart rates for men and women does not depend on body temperature.

B. The relationship between heart rate and body temperature is the same regardless of sex.

C. Neither sex nor heart rate is associated with body temperature, so neither of these are useful predictors in the model.

D. There is no association between sex and heart rate, so knowing a person’s sex does not help predict their heart rate.

Ans: A,B; LO4.4-2; Difficulty: Medium; Type: MS 13. What statistical term do we use when one explanatory variable modifies the association of another explanatory variable with the response?

A. Association B. Covariation C. Interaction D. Regression Ans: B; LO4.4-1; Difficulty: Easy; Type: MC 14. In which of the following study designs is it possible to have an interaction between two explanatory variables?

A. An observational study with substantial covariation between two predictors B. An observational study with minimal association between the predictors C. A designed experiment with factors that are independent of each other D. Interaction may occur in any of the three designs described above.

Ans: D; LO4.4-1; Difficulty: Medium; Type: MC 15. The plot below represents a model that uses two explanatory variables, one quantitative and one categorical, to predict a quantitative response. The model includes an interaction term. The categorical variable is defined using indicator coding (1 for category A, 0 for category B).

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Find the coefficients for this model.

Term Intercept Variable X1 Variable X2 Variable X1 Ans: 20

Coefficient

Variable X2

0, -1

0, -10

0, 3

0; LO4.4-1; Difficulty: Hard; Type: FIB

16. The plots below represent two different datasets. Each dataset has two explanatory variables, one quantitative and one categorical, that are being used to predict a quantitative response. The samples sizes and SE of the residuals are roughly the same for both datasets.

For each dataset, a significance test is used to decide whether the interaction is statistically significant. How do the p-values for the two tests compare?

A. Dataset 1 would produce a smaller p-value for testing the interaction.

B. Dataset 2 would produce a smaller p-value for testing the interaction.

C. Both datasets would produce the same p-value for testing the interaction.

D. There is not enough information to determine which test would result in a smaller p-value?

Ans: A; LO4.4-1; Difficulty: Medium; Type: MC

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Section 4.5: Multi-level Categorical Variables LO4.5-1: Include categorical variables with more than two categories in a linear model.

LO4.5-2: Interpret an interaction between a quantitative variable and a multi-level categorical variable.

Questions 1 through 5: Most universities ask students to fill out course evaluations at the end of the semester, but some students choose not to complete the survey. The statistics department wonders whether high response rates are associated with higher or lower average ratings. They also want to account for differences in ratings for courses at different levels: introductory, intermediate, and advanced.

A multiple regression model (with no interaction) was used to predict a section’s average rating using response rate and course level. Response rate was recorded as a proportion and course level was recorded using indicator coding. The table of coefficients is given below. Note: The observational units are course sections not individual students.

Term Intercept ResponseRate Level_intro Level_inter

Coeff.

3.8717 0.9559 -0.7026 -0.2585

SE 0.2404 0.3468 0.1111 0.1108

t-stat.

16.103 2.757 -6.321 -2.563

p-value <0.0001 0.0076 <0.0001 0.0128

1. The interaction term was not included in the final model for predicting average course ratings, because it was not statistically significant. Using the partially filled in ANOVA table below, calculate the F-statistic for testing the interaction between course level and response rate. Use three decimal places in your answer.

Source ResponseRate Level ResponseRate Error Total

DF

Level

SS 0.6888 8.9508 0.0307 5.6795

MS

F

p-value 0.0085 <0.0001 0.8483

66

Sol: Ans: 0.165

0.002; LO4.5-2; Difficulty: Medium; Type: Calc

2. Fill in the linear equations to predict the average rating based on the response rate for each course level. Do not round.

Introductory: Intermediate: Advanced: Ans: Intro: 3.1691 0, 0.9559 0; Inter: 3.6132 0, 0.9559 0; Adv: 3.8717 0, 0.9559 0; LO4.5-1; Difficulty: Medium; Type: FIB

FOR INSTRUCTOR USE ONLY

1-94 3. Describe the relationship between the response rate and the average course rating based on the additive model given above.

As the response rate increases by 0.1, the average course rating is predicted to increase by ________ for ____________ (introductory/intermediate/advanced/all) level(s).

Ans: 0.09559 0.0001, all; LO4.5-1; Difficulty: Medium; Type: FIB 4. After controlling for the response rate, which course levels are significantly different in terms of average ratings? Match the questions below to the appropriate p-values.

One of the p-values will not be used.

Is introductory different from intermediate?

Is intermediate different from advanced?

Is introductory different from advanced? .

Ans: D, C, B; LO4.5-1; Difficulty: Medium; Type: Ma

A. P-value = 0.0076 B. P-value < 0.0001 C. P-value =0.0128 D. P-value not shown in table.

5. The scatterplot below shows the relationship between average rating and response rate with a separate line of best fit for each level (introductory, intermediate, and advanced).

The output above shows that the slope corresponding to response rate is 0.9559. If level were removed from the model, how would the slope coefficient change?

A. The slope relating average rating to response rate would be higher than 0.9559.

B. The slope relating average rating to response rate would be lower than 0.9559.

C. The slope would change, because there is covariation between level and response rate, but there is not enough information to predict if it would increase or decrease.

D. The slope would not change much, because there is no significant interaction between response rate and level.

Ans: A; LO4.5-1; Difficulty: Hard; Type: MC

Questions 6 through 8: A realtor took a random sample of 50 houses currently for sale from FOR INSTRUCTOR USE ONLY

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each quadrant of the town where she works (northeast, northwest, southeast, southwest).

She wonders whether the relationship between size (in sq ft) and price (in dollars) differs based on location. The table of coefficients is given below.

Term (Intercept) size locationNW locationSE locationSW size:locationNW size:locationSE size:locationSW

Coefficient 127,739.66 129.61 -573.30 -5,092.04 -15,854.18 23.60 -60.36 -24.51

Std. Error 17,060.53 12.85 23,632.62 21,951.00 19,366.04 15.69 17.17 13.65

t-stat 7.487 10.087 -0.024 -0.232 -0.819 1.504 -3.516 -1.796

p-value <0.0001 <0.0001 0.9807 0.8168 0.4140 0.1341 0.0005 0.0741

6. Based on the graph below, how can you tell that there is an interaction between size and location in this sample?

A. Some parts of town tend to have larger houses than other parts of town.

B. Some parts of town tend to have more expensive houses than other parts of town.

C. The best fit lines for the four parts of town do not all have the same intercept.

D. The best fit lines for the four parts of town do not all have the same slope.

Ans: D; LO4.5-2; Difficulty: Medium; Type: MC 7. Which location is the reference category?

A. NE B. NW C. SE D. SW Ans: A; LO4.5-1; Difficulty: Easy; Type: MC 8. Fill in the linear equation to predict price for houses in the NW quadrant of town. Do not round.

NW: Ans: 127,166.36 0, 153.21 0; LO4.5-2; Difficulty: Medium; Type: FIB Questions 9 and 10: The plot below represents a dataset with two explanatory variables, one FOR INSTRUCTOR USE ONLY

1-96 quantitative and one categorical, that are being used to predict a quantitative response.

9. For each term of the model, predict whether the coefficient would be positive, negative, or close to 0?

Term .

Intercept X1 Indicator_B Indicator_C X1 Indicator_B X1 Indicator_C

Coeff.

Ans: positive, positive, negative, positive, close to 0, negative; LO4.5-2; Difficulty: Hard; Type: FIB 10. If you tested the significance of the interaction in this model, what would you expect to find? Assume the sample size is large enough to provide strong evidence if the interaction actually exists.

A. The p-value would probably be large, because two of the slopes are very similar.

B. The p-value would probably be large, because the slopes are not all the same.

C. The p-value would probably be small, because two of the slopes are very similar.

D. The p-value would probably be small, because the slopes are not all the same.

Ans: D; LO4.5-2; Difficulty: Medium; Type: MC 11. Suppose a regression model has a categorical variable with 5 categories. You would need to create _____ indicator variable(s) to represent this categorical variable.

Provide a numerical value for your answer.

Ans: 4

0; LO4.5-1; Difficulty: Easy; Type: FIB

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12. You are using a regression model to predict the prices of used cars based on mileage for four different types of cars (Toyota Camry, Honda Accord, Chevy Malibu, and Ford Fusion). You want to test whether there is an association between type of car and price after adjusting for mileage. What kind of test(s) should you use?

A. Conduct an ANOVA F-test for the overall model. If the p-value is small, then the test provides strong evidence of an association.

B. Conduct a partial F-test for the indicators corresponding to type of car. If the pvalue is small, then the test provides strong evidence of an association.

C. Conduct t-tests for each indicator corresponding to the type of car. If at least one of the p-values is small, then the test provides strong evidence of an association.

D. Conduct t-tests for each indicator corresponding to the type of car. If all of the pvalues are small, then the test provides strong evidence of an association.

Ans: B; LO4.5-1; Difficulty: Easy; Type: MC 13. Which of the following is an appropriate strategy for defining indicator variables to represent political identity (Republican, Democrat, or Independent)? Select all that apply.

{

A.

B.

{

{

C.

{

{

D.

{

{

{

Ans: C, D; LO4.5-1; Difficulty: Medium; Type: MS 14. You are using a multiple regression model with two explanatory variables, is quantitative and is categorical with multiple levels. You test the term in the model and find that the p-value is small. Which of the following interpretations is appropriate? Select all that apply.

A. There is strong evidence of an association between the two explanatory variables in the population.

B. There is strong evidence of an interaction between the two explanatory variables in the population.

C. Removing the interaction term would lead to a large decrease in and increase in the standard error of the residuals.

D. The difference in slopes seen in this sample would be unlikely to occur just by random chance if there were really no interaction.

Ans: B, D; LO4.5-2; Difficulty: Medium; Type: MS

FOR INSTRUCTOR USE ONLY

1-98 15. You are using a multiple regression model with two explanatory variables, is quantitative and is categorical with multiple levels. You plan to write separate prediction equations of the form for each level of .

True or False: The prediction equations will be the same, regardless of whether the categorical variable, , was recorded using indicator coding or effect coding.

Ans: True; LO4.5-1; Difficulty: Medium; Type: TF 16. Which of the following factors affect the size of the test statistic in a partial F-test for comparing two models? Select all that apply.

A. The difference in for the two models B. The difference in the number of coefficients for the two models C. The sample size Ans: A, B, C; LO4.5-1; Difficulty: Medium; Type: MS

Chapter 5 Intermediate Statistical Investigations Test Bank Question types:

FIB = Fill in the blank Ma = Matching MC = Multiple choice

Calc = Calculation MS = Multiple select TF = True-false

CHAPTER 5 TERMINAL LEARNING OUTCOMES TLO5-1: Visualize and interpret relationships among three or more quantitative variables using response surfaces.

TLO5-2: Visualize adjusted associations when adjusting for a quantitative variable, and explore potential problems when using explanatory variables that are linearly related.

TLO5-3: Fit a polynomial model for nonlinear associations, and assess the appropriateness of the model.

TLO5-4: Assess transformations of the response and or explanatory variable(s) to meet model conditions of linearity, symmetry, and equal variation.

Section 5.1: Experiments with Multiple Quantitative Explanatory Variables LO5.1-1: Consider design issues with quantitative explanatory variables.

LO5.1-2: Visualize relationships among three or more quantitative variables.

LO5.1-3: Interpret a ―response surface‖ with quantitative explanatory variables.

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LO5.1-4: Describe interactions between quantitative variables.

Questions 1 through 3: A computer science student is using an evolutionary algorithm to solve the ―traveling salesman problem,‖ which aims to find the shortest route for visiting a list of cities and returning to the original city. A number of parameters must be set in evolutionary algorithms, including tournament size and mutation rate, and the student wants to find the ideal set of algorithm parameters for this task.

The student uses a balanced, full-factorial design with three values of each parameter, tournament sizes of 8, 16, and 32, and mutation rates of 1%, 5%, and 10%. The response variable is the distance of the best (shortest) route that the algorithm finds in a fixed number of iterations. The student used a linear regression model to predict best distance: – 1. True or False: The student’s model assumes that the relationship between tournament size and best distance is the same, regardless of the mutation rate.

Ans: True; LO5.1-3; Difficulty: Medium; Type: TF

2. Do the graphs below indicate a violation of the validity conditions for a multiple linear regression model?

A. Yes, the linearity condition is severely violated.

B. Yes, the independence condition is severely violated.

C. Yes, the equal variance condition is severely violated.

D. No, these plots do not reveal any severe violations of the validity conditions.

Ans: D; LO5.1-1 Difficulty: Medium; Type: MC 3. Based on this model, what algorithmic parameters should the student choose?

Remember, the goal is to find the shortest route.

He should use a tournament size that is relatively ______ (large/small) and a mutation rate that is relatively ______ (large/small).

More research may be necessary before setting parameters outside the explanatory variable region since that would be considered ___________ (extrapolation/interpolation).

Ans: large, small, extrapolation; LO5.1-3; Difficulty: Medium; Type: FIB Questions 4 through 6: A balanced, full-factorial design was used to investigate the impact of water temperature (25°C, 35°C) and salinity (10%, 25%, 40%) on the weight gain of FOR INSTRUCTOR USE ONLY

1-100 shrimp (in mg). A 3D scatterplot of the data is shown below. The plane shows the predicted weight gains based on a linear regression model with no interaction.

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4. Using the 3D scatterplot, describe the relationships among the variables.

After adjusting for salinity, there is a _________ (positive/negative) association between temperature and weight gain.

After adjusting for temperature, there is a _________ (positive/negative) association between salinity and weight gain.

Ans: positive, positive; LO5.1-3; Difficulty: Medium; Type: FIB 5. Does this model provide accurate predictions?

For the shrimp raised in water that was 25°C with 10% salinity, the residuals are all _________ (positive/negative), which suggests that the predicted value based on a linear regression model tends to be too _________ (high/low).

Ans: negative, high; LO5.1-3; Difficulty: Medium; Type: FIB 6. Based on the scatterplot matrix below, are you concerned about the linearity condition being met?

A. Yes, because there is no linear relationship between temperature and salinity. The slope of the regression line would be 0.

B. Yes, because the spread of the weight gain values is larger for low temperatures than for high temperatures.

C. Yes, because changing from 10% to 25% salinity leads to an increase in weight gain, but changing from 25% to 40% salinity leads to a decrease in weight gain.

D. No, this scatterplot matrix does not reveal any issues with the linearity condition.

Ans: C; LO5.1-2; Difficulty: Medium; Type: MC Questions 7 through 11: Two drugs, labeled Drug A and Drug B, are often prescribed together as a treatment for a certain disease. In a randomized experiment, 120 patients were prescribed one of three dosages of Drug A (10, 20, or 30 mg) and one of four dosages of Drug B (200, 400, 600, or 800 mg). The response variable is a quantitative indicator of disease activity (DAI). The partially filled in table of coefficients is given below.

FOR INSTRUCTOR USE ONLY

1-102 Term Intercept Drug A dosage Drug B dosage Drug A dosage

Coefficient SE t-stat.

100.398 5.395 -0.969 0.250 -0.097 0.010 Drug B dosage

p-value

<0.0001

7. Fill in the degrees of freedom for this linear regression model.

Source Model Error Total

DF

Ans: 3, 116, 119; LO5.1-3; Difficulty: Medium; Type: FIB 8. Describe the interaction shown in the graph below.

In the explanatory variable region of this study, higher dosages of Drug B are associated with _________ (higher/lower) DAI values; however, the slope relating DAI to Drug B dosage is _________ (flatter/steeper) when the Drug A dosage is high.

In the table above, the missing coefficient for the interaction would be ____________ (positive/negative).

Ans: lower, flatter, positive; LO5.1-4; Difficulty: Medium; Type: FIB 9. True or False: In terms of predicted DAI, the choice of dosage for Drug A is more important than the choice of dosage for Drug B. Ans: False; LO5.1-3; Difficulty: Medium; Type: TF

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10. True or False: If the explanatory variables are standardized, the p-value corresponding to the interaction may change.

Ans: False; LO5.1-3; Difficulty: Medium; Type: TF 11. The partially-filled in table of coefficients corresponds to a model where the explanatory variables have been standardized.

Term Intercept Std. Drug A dosage Std. Drug B dosage Std. Drug A dosage

Coefficient SE t-stat. p-value 50.59 0.83 -0.41 0.84 -13.67 0.84 Std. Drug B dosage

Interpret the intercept and the slope for Drug B. The predicted DAI for a person who receives _______ mg of Drug A and _______ mg of Drug B is 50.59.

As the dosage of Drug B increases by 1 _______ (mg / SD), DAI is predicted to _______ (increase/decrease) by _______, for a person receiving the average dosage of Drug A. Ans: 20 0, 500 0, SD, decrease, 13.67 0; LO5.1-3; Difficulty: Medium; Type: FIB 12. Match each term to its definition. One of the definitions will not be used.

Full-factorial design:

A. Each treatment has the same number of observations.

B. Each factor has the same number of levels.

Balanced design: C. Each level of a factor is combined with each level of another factor.

Ans: Full-factorial: C, Balanced: A; LO5.1-1; Difficulty: Easy; Type: Ma 13. Which of the following statements is true? Select all that apply.

A. Balanced designs always eliminate the association between two explanatory variables.

B. Full-factorial designs always eliminate the association between two explanatory variables.

C. Balanced, full-factorial designs always eliminate the association between two explanatory variables.

D. It is possible to have an association between the explanatory variables in any of the designs mentioned above.

Ans: C; LO5.1-1; Difficulty: Medium; Type: MS 14. True or False: In a balanced, full-factorial design, the adjusted slope coefficients in the two-predictor model are identical to the unadjusted slope coefficients in the one-predictor models.

Ans: True; LO5.1-1; Difficulty: Medium; Type: TF FOR INSTRUCTOR USE ONLY

1-104 15. How could you represent the graph of a regression model that uses two quantitative explanatory variables and an interaction term to predict a quantitative response?

A. A line B. A plane C. A surface (which may be curved) D. A sphere Ans: C; LO5.1-2; Difficulty: Medium; Type: MC 16. If you standardize the explanatory variables in a multiple linear regression model with no interaction, which numerical values do you expect to change?

A. The intercept and slopes B. The p-values corresponding to the intercept and slopes C. The percentage of variability explained by the model D. The standard error of the residuals Ans: A; LO5.1-3; Difficulty: Medium; Type: MC 17. Which of the following are advantages of using standardized explanatory variables?

Select all that apply.

A. The intercept is more likely to have a meaningful interpretation.

B. It is easier to compare the impact of explanatory variables with different scales.

C. The validity conditions are more likely to be satisfied.

D. The interaction product variable will be independent of the explanatory variables involved in the interaction.

Ans: A, B, D; LO5.1-3; Difficulty: Medium; Type: MS

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Section 5.2: Observational Studies with Multiple Quantitative Explanatory Variables LO5.2-1: Visualize adjusted associations when adjusting for a quantitative variable.   

Create and interpret added variable plots.

Interpret model coefficients.

Interpret adjusted sums of squares.

LO5.2-2: Explore potential problems when using explanatory variables that are linearly related.

Questions 1 and 2: Suppose you want to predict a student’s final grade in a course (scale of 0100) based on their pre-test score (test on a scale of 0-100 taken before the semester began) and the number of optional study sessions they attended (0-15). In the sample, all three pairs of variables have positive correlations.

1. In a simple regression model, the unadjusted slope relating final exam score to number of study sessions attended is a positive value, . If you added pre-test scores to the model, would you expect the slope corresponding to study sessions to change?

The adjusted slope for study sessions in the multiple regression model is expected to be _________ (larger than/smaller than/equal to) the unadjusted slope for study sessions in the simple regression model.

Ans: smaller than; LO5.2-1; Difficulty: Medium; Type: FIB 2. In a simple regression model that uses number of study sessions as the only predictor of final exam scores, the value is 30%. In a simple regression model that uses pre-test scores as the only predictor of final exam scores, the value is 40%. In the multiple regression model that uses both number of study sessions and pre-test scores as predictors, how large would you expect the value to be?

A. You would expect to be 30% or lower.

B. You would expect to be between 30% and 40%.

C. You would expect to be between 40% and 70%.

D. You cannot calculate for a multiple regression model.

Ans: C; LO5.2-1; Difficulty: Medium; Type: MC Questions 3 through 7: One commonly cited metric in car commercials is time to accelerate from 0 to 60 mph. What car features might explain faster acceleration (shorter times)? The dataset includes weight (in pounds), horsepower, and acceleration (in seconds) for 392 different types of cars.

Suppose you fit a multiple regression model to predict acceleration. The equation and scatterplot are given below.

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3. Interpret the slope coefficient corresponding to weight.

As _________ (acceleration/weight) increases by 1 unit, _________ (acceleration/weight) is predicted to increase by 0.0023 units, _________ (ignoring/adjusting for) horsepower.

Ans: weight, acceleration, adjusting for; LO5.2-1; Difficulty: Medium; Type: FIB 4. If horsepower were removed from the model, how would the slope coefficient corresponding to weight change?

A. The slope would be larger than 0.0023.

B. The slope would be smaller than 0.0023, but it would still positive.

C. The slope would be negative.

D. The slope would not change.

Ans: C; LO5.2-1; Difficulty: Medium; Type: MC 5. The value for the multiple regression model is 0.602. If horsepower were removed from the model, how would the value change?

A. The value would be larger than 0.602.

B. The value would be smaller than 0.602, but it would still be positive.

C. The value would be negative.

D. value would not change.

Ans: C; LO5.2-1; Difficulty: Medium; Type: MC

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6. Does the graph below indicate any problems with model fit?

A. Yes, because the graph shows a curved pattern.

B. No, because the pattern allows you to predict the residuals and reduce error.

C. Yes, because most of the predicted values are higher than 15 (not symmetric).

D. No, because roughly half of the residuals are below 0 (symmetric).

Ans: A; LO5.2-1; Difficulty: Medium; Type: MC 7. Is there an interaction between weight and horsepower in this sample?

A. Yes, because heavier cars tend to have higher horsepower and faster acceleration times.

B. Yes, because the relationship between weight and acceleration differs slightly based on the level of horsepower.

C. Yes, because if horsepower were removed from the model, the slope relating acceleration to weight would change substantially.

D. No, because none of the regression lines cross within the explanatory variable region.

Ans: C; LO5.2-1; Difficulty: Medium; Type: MC Questions 8 and 9: Suppose you want to use age and experience (both measured in years) to predict salary (in dollars) at a small company.

8. The ANOVA table is shown below. You’ll notice that SSModel is much larger than the sum of SSExperience and SSAge. Further, the overall model is statistically significant, but neither of the individual predictors is statistically significant. Which of the following is the best explanation for these features of the ANOVA table?

Source Model Experience Age Error Total

df 2 1 1 7 9

SS 715.11 43.41 0.01 210.49 925.60

MS 357.55 43.41 0.01 30.07

F 11.89 1.44 0.00

p-value 0.0056 0.2686 0.9865

A. There is substantial covariation between experience and age.

B. There is substantial interaction between experience and age.

C. There is another confounding variable not being accounted for by the model.

D. The only reasonable explanation is a calculation error.

Ans: A; LO5.2-2; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

1-108 9. The scatterplot below is the added variable plot predicting salary from age, before and after adjusting for experience. Note that the graph shows two regression lines. Which is which?

The ______ (gray/purple) line is the original regression line relating salary to age, and the ______ (gray/purple) line is the regression line relating salary to age after adjusting for experience.

Ans: gray, purple; LO5.2-1; Difficulty: Medium; Type: FIB Questions 10 through 13: Consider a model for predicting the weight of fish (in grams) based on their length and width. The length and width were originally measured in centimeters, but the variables have been standardized. The table of coefficients is given below.

Term Intercept Std. Length Std. Width Std. Length

Std. Width

Estimate 297.01 203.52 107.55 88.99

SE 8.95 27.42 27.92 7.01

t-stat.

33.19 7.42 3.85 12.69

p-value <0.0001 <0.0001 0.0003 <0.0001

10. Is the interaction between standardized length and standardized width significant?

A. Yes, because the p-value for Std. Length Std. Width is small.

B. No, because the slope for Std. Length Std. Width is small relative to the other slope and intercept coefficients.

C. There is not enough information to answer, because the residual standard error is not given.

D. There is not enough information to answer, because no information is given about the fit of the model that excludes the interaction term.

Ans: A; LO5.2-1; Difficulty: Medium; Type: MC 11. Interpret the coefficient for Std. Length.

As length increases by 1 _______ (cm/SD), _________ (weight//width) is predicted to increase by 203.52 units, for a fish of average ___________ (weight/length/width).

Ans: SD, weight, width; LO5.2-1; Difficulty: Medium; Type: FIB 12. True or False: In this scenario, interpretation of the intercept involves extrapolation.

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Ans: False; LO5.2-1; Difficulty: Medium; Type: TF 13. Suppose the explanatory variables had not been standardized. Which of the following would you expect to change? Select all that apply.

A. The slope coefficients for the two explanatory variables, length and width.

B. The p-values for the two explanatory variables, length and width.

C. The coefficient for the length width interaction.

D. The p-value for the length width interaction.

Ans: A, B, C; LO5.2-1; Difficulty: Medium; Type: MS 14. Using data from an observational study, a regression model predicts a response variable based on two explanatory variables, and . Which of the following is true?

A. Adjusting for will not change the strength of the association between and .

B. After adjusting for , the association between and will be stronger.

C. After adjusting for , the association between and will be weaker.

D. After adjusting for , the association between and may be stronger or weaker.

Ans: D; LO5.2-1; Difficulty: Medium; Type: MC 15. Describe how a large degree of covariation among explanatory variables can impact the model.

When explanatory variables (including interactions) have a strong linear association with other explanatory variables, this can lead to __________ (larger/smaller) standard errors on the slope coefficients and a __________ (larger/smaller) residual standard error for the model overall.

Ans: larger, larger; LO5.2-2; Difficulty: Medium; Type: FIB 16. True or False: If two explanatory variables are associated with each other, using standardized versions of those variables will reduce the linear association between them.

Ans: False; LO5.2-2; Difficulty: Medium; Type: TF 17. True or False: Using standardized variables in an interaction will reduce the linear association between the interaction term and the variables involved in the interaction.

Ans: True; LO5.2-2; Difficulty: Medium; Type: TF

Section 5.3: Modeling Nonlinear Associations Part I – Polynomial models LO5.3-1: Fit a polynomial model to model a nonlinear association.

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1-110 LO5.3-2: Assess when a polynomial model is appropriate.

Questions 1 through 4: Consider three different models that could be used to predict the weight of fish (in grams) based only on fish length (in cm).

Model 1: Model 2: Model 3: 1. A line showing predicted values from Model 1 has been added to the scatterplot below.

This model tends to ________ (overestimate/underestimate) weight for the longest and shortest fish in the dataset, but it tends to ________ (overestimate/underestimate) weight for fish of average length.

Ans: underestimate, overestimate; LO5.3-2; Difficulty: Medium; Type: FIB 2. A smoother has been added to the scatterplot below. What is the purpose of adding a smoother?

A. Smoothers are helpful for exploring the form of the association.

B. Smoothers provide a simple mathematical model for making predictions.

C. Smoothers serve both of the purposes mentioned above.

D. Smothers serve neither of the purposes mentioned above.

Ans: A; LO5.3-1; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY

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3. The tables below describe Models 2 and 3.

Model 2

Term Intercept Length Length^2

Coeff.

128.35 -21.02 0.9086

SE 78.78 5.42 0.0869

t-stat.

1.63 -3.88 10.46

p-value 0.1092 0.0003 <0.0001

Model 3

Term Intercept Length Length^2 Length^3

Coeff.

304.98 -43.40 1.7699 -0.0101

SE 160.82 18.59 0.6903 0.0081

t-stat.

1.90 -2.33 2.56 -1.26

p-value 0.0635 0.0235 0.0133 0.2141

Using significance of model terms as your criterion, which model would you use choose?

A. Model 1 B. Model 2 C. Model 3 D. There is not enough information to choose, since the table for Model 1 is not given.

Ans: B; LO5.3-2; Difficulty: Medium; Type: MC 4. Based on the Model 1: Model 2: Model 3:

values given below, which model would you choose?

0.9207 0.9741 0.9749

A. Model 1, because you should always choose the model with the lowest .

B. Model 3, because you should always choose the model with the highest .

C. Model 2, because the accuracy of predictions is very similar to Model 3, but the model is simpler and easier to interpret.

D. Model 3, because the accuracy of predictions is very similar to Model 2, but the model is more flexible and allows for more ―turns‖ in the data.

Ans: C; LO5.3-2; Difficulty: Medium; Type: MC Questions 5 through 7: A quadratic model was used to predict height of children (in inches) based on age (in months). The table of coefficients is shown below.

Term Intercept Age Age^2

Coeff.

7.6557 0.5372 -0.0012

SE 4.8070 0.0622 0.0002

t-stat.

1.59 8.63 -6.27

p-value 0.1129 <0.0001 <0.0001

5. The children in the dataset are all between 99 and 221 months of age. Consider whether it is reasonable to extrapolate based on this model.

A. This model assumes that height increases at a constant rate for all ages, so extrapolation will lead to unreasonable estimates.

B. This model assumes that height increases up to a point and then starts decreasFOR INSTRUCTOR USE ONLY

1-112 ing, so extrapolation will lead to unreasonable estimates.

C. This model assumes that height increases with age but that growth slows down until height ―levels off‖ in adulthood, so extrapolation is reasonable.

D. This model assumes that height takes three ―turns‖ – increasing then decreasing then increasing again – so extrapolation will lead to unreasonable estimates.

Ans: B; LO5.3-1; Difficulty: Medium; Type: MC 6. Does the table above suggest that the quadratic model is significantly better than a linear model of degree 1 in this context?

A. The coefficient for the Age^2 term is very small, which suggests that the quadratic term is unnecessary. A linear model would be preferred.

B. The t-statistic for the Age term is larger in absolute value than the t-statistic for the Age^2 term which suggests that the linear model is better than the quadratic model.

C. The p-value for the Age^2 term is very small, which suggests that the quadratic model is significantly better than the linear model.

D. The p-value for the Age and Age^2 terms are both very small, which suggests that both the linear model and the quadratic model provide accurate predictions.

Ans: C; LO5.3-2; Difficulty: Medium; Type: MC 7. The coefficient for Age^2 is very small, so rounding decisions can have a big impact.

What should you do to address this issue?

A. Remove the Age^2 term. The coefficient is so small that the term is not significant.

B. Use standardized versions of the variable, Std. Age and (Std. Age)^2.

C. Control for confounding variables that are ―masking‖ the effect of Age^2.

D. Add an interaction term to model the changing slope between Age and Height.

Ans: B; LO5.3-1; Difficulty: Medium; Type: MC 8. Suppose you are choosing between a linear model and a quadratic model based on the residual plots below.

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Based on the residual plots, the __________ (linear/quadratic) model is preferred.

However, __________ (independence/constant variance/normality) is still a concern in the chosen model.

Ans: quadratic, constant variance; LO5.3-2; Difficulty: Medium; Type: FIB 9. The scatterplot below shows the relationship between two quantitative variables, with a smoother added to the plot.

and ,

This relationship is best modeled by a polynomial of degree _____ (1, 2, or 3), also known as a ___________ (quadratic/cubic) model.

Ans: 3, cubic; LO5.3-2; Difficulty: Medium; Type: FIB 10. What do you call models that include terms the scatterplot?

A. Additive models B. Exponential models C. Multiplicative models D. Polynomial models Ans: D; LO5.3-2; Difficulty: Easy; Type: MC

to help model nonlinear behavior in

11. True or False: A polynomial model can be considered a type of linear model, because it is of the form where , , and so on.

Ans: True; LO5.3-1; Difficulty: Easy; Type: TF 12. Which of the following are useful steps for comparing two polynomial models to decide which is most appropriate? Select all that apply.

A. Fit the more complex model and find the p-value for the highest order term.

B. Fit the simpler model and find the p-value for the highest order term.

C. Compare the residual plots for the two models to assess model validity.

D. Compare the values to see if the more complex model greatly improves the accuracy of predictions.

Ans: A, C, D; LO5.3-2; Difficulty: Medium; Type: MS FOR INSTRUCTOR USE ONLY

1-114 13. Which of the following are advantages of standardizing the explanatory variable in a quadratic model? Select all that apply.

A. It ensures that there is no association between and .

B. It ensures that the association between and is nonlinear.

C. It put explanatory variables on a more reasonable scale, so rounding is less impactful.

D. It puts explanatory variables on a more reasonable scale, so the coefficient for is larger and more likely to be statistically significant.

Ans: B, C; LO5.3-1; Difficulty: Medium; Type: MS 14. True or False: Extrapolation is less of a concern for polynomial models than it is for linear models of degree 1.

Ans: False; LO5.3-1; Difficulty: Medium; Type: TF 15. True or False: When comparing two polynomial models, you should always choose the model with the higher value.

Ans: False; LO5.3-1; Difficulty: Medium; Type: TF

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Section 5.4: Modeling Nonlinear Associations Part II - Transformations LO5.4-1: Transform the response variable to meet model conditions.

LO5.4-2: Assess different model transformations.

Questions 1 through 4: You want to create a model that predicts a country’s income (GDP/capita in dollars) based on the country’s fertility rate (babies per woman). The prediction equations and residual plots for two models are shown below.

Model 1:

Model 2:

1. Using Model 1, which of the validity conditions are violated? Select all that apply.

A. The linearity condition is violated.

B. The equal variance condition is violated.

C. The normality condition is violated.

D. None of the validity conditions are violated.

Answer: A, B, C; LO5.4-2; Difficulty: Medium; Type: MS 2. Are the predicted values based on Model 1 accurate?

For countries with the highest fertility rates (say, 6 or more babies/woman), Model 1 tends to __________ (overestimate/underestimate/accurately predict) the country’s income.

Answer: underestimate; LO5.4-1; Difficulty: Medium; Type: FIB FOR INSTRUCTOR USE ONLY

1-116 3. Using Model 2, predict the income of a country whose fertility rate is 6 babies per woman.

Note that log() denotes the natural log. Round to the nearest dollar.

Sol: Ans: 1151 1; LO5.4-1; Difficulty: Medium; Type: Calc 4. Using Model 2, the 95% confidence interval for the slope is (-0.7555, -0.5981). Which of the following interpretations is correct? Note that log() denotes the natural log.

A. As fertility rate increases by 1 unit, predicted income decreases by 0.5981 to 0.7555 units.

B. As fertility rate increases by 1 unit, predicted income decreases by 0.1218 to 0.2232 units.

C. As fertility rate increases by 1 unit, predicted income is multiplied by 1.8187 to 2.1287.

D. As fertility rate increases by 1 unit, predicted income is multiplied by 0.4698 to 0.5499.

Ans: D; LO5.4-1; Difficulty: Hard; Type: MC Questions 5 through 7: Suppose we want to use an animal’s body weight (in kg) to predict its brain weight (in kg). Consider three different models: Model 1: Model 2: Model 3: 5. You want to assess the fit of these three models. Is it appropriate to compare the models based on the standard error of the residuals?

A. Yes, we can use the standard error of the residuals to compare models. The model with the highest standard error of the residuals is preferred.

B. Yes, we can use the standard error of the residuals to compare models. The model with the lowest standard error of the residuals is preferred.

C. No, it is not appropriate to compare models based on the standard error of the residuals, because the response variable has been transformed in Models 2 and 3.

D. No, it is not appropriate to compare models based on the standard error of the residuals, because the explanatory variable has been transformed in Model 3.

Ans: C; LO5.4-2; Difficulty: Medium; Type: MC 6. Using Model 3, predict the brain weight of an animal that weighs 45 kg. Note that log() denotes the natural log.

Model 3: Sol: Answer: 0.093 0.005; LO5.4-1; Difficulty: Medium; Type: Calc

7. In the scatterplot below, the dotted line marks the 95% prediction interval for individual animals in this population based on Model 3. Is it appropriate to interpret this prediction FOR INSTRUCTOR USE ONLY

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interval?

A. Yes, it is appropriate to interpret this interval as long as you back-transform the predictions using natural logarithms: ( .

B. Yes, it is appropriate to interpret this interval as long as you back-transform the predictions using exponents: ( .

C. No, it is not appropriate to interpret this interval, because the response variable has been transformed.

D. No, it is not appropriate to interpret this interval, because the explanatory variable has been transformed.

Answer: B; LO5.4-1; Difficulty: Medium; Type: MC Questions 8 and 9: You want to build a model that describes how the population of the United States (in millions) has changed between 1790 and 2000. The residual plots for two different models are shown below.

8. Do these models provide accurate predictions?

If Model 1 were used to predict the U.S. population in 2020, we would expect it to _________ (overestimate/underestimate) the true population.

If Model 2 were used to predict the U.S. population in 2020, we would expect it to _________ (overestimate/underestimate) the true population.

Answer: underestimate, overestimate; LO5.4-2; Difficulty: Medium; Type: FIB

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1-118 9. The best transformation is often found through systematic trial and error. Having tried these two models, which transformation would you try next?

A. B.

√

C. D. Answer: B; LO5.4-1; Difficulty: Medium; Type: MC 10. Consider a petri dish that contains bacteria. The growth of the bacteria over time can be modeled using an exponential relationship of the following form:

True or False: Taking the natural logarithm of the response variable will lead to a linear relationship between the transformed variable ln(bacteria) and time.

Answer: True; LO5.4-1; Difficulty: Medium; Type: TF 11. How can data transformations be used to address the violations of validity conditions?

A. You can transform any (or all) of the explanatory variables and/or the response variable.

B. You can transform any (or all) of the explanatory variables, but you should leave the response variable on the original scale.

C. You can transform the response variable, but you should leave the explanatory variable(s) on the original scale.

D. You can transform the residuals, but you should leave the explanatory and response variables on the original scale.

Ans: A; LO5.4-1; Difficulty: Medium; Type: MC 12. You want to compare the fit of two regression models. In one of the models, the response variable has been transformed. Is it appropriate to compare the two models using ?

A. Yes, we can always use to compare the performance of two models. The model with the higher value is preferred.

B. Yes, we can always use to compare the performance of two models. The model with the lower value is preferred.

C. Yes, we can use to compare the performance of two models as long as the explanatory variables have not been transformed.

D. No, we can only use to compare the performance of two models with the same response variable.

Ans: D; LO5.4-2; Difficulty: Medium; Type: MC 13. What is the purpose of using data transformations?

A. Transformations can create a linear association between variables.

B. Transformations can reduce skewness in conditional distributions of residuals.

C. Transformations can adjust for measurement errors in the data collection process.

D. Transformations can create more similar conditional variances of residuals.

Ans: A, B, D; LO5.4-1; Difficulty: Medium; Type: MS 14. How do you decide which variable(s) to transform?

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If linearity is the only issue, it is often better to transform the __________ (explanatory/ response) variable(s); if multiple validity conditions are violated, it is often better to transform the __________ (explanatory/ response) variable(s).

Ans: explanatory, response; LO5.4-1; Difficulty: Medium; Type: FIB 15. Based on the scatterplot below, what kind of linearizing transformation would you try?

You should try a transformation that ____________ (increases/decreases) the power of the response variable or a transformation that ____________ (increases/decreases) the power of the explanatory variable.

Ans: increases, decreases; LO5.4-1; Difficulty: Medium; Type: FIB

Chapter 6 Intermediate Statistical Investigations Test Bank Question types:

FIB = Fill in the blank Ma = Matching MC = Multiple choice

Calc = Calculation MS = Multiple select TF = True-false

CHAPTER 6 TERMINAL LEARNING OUTCOMES TLO6-1: Review descriptive and inferential methods for comparing groups with a categorical response variable, including comparing and contrasting different statistics for evaluating group differences.

TLO6-2: Motivate and utilize a logistic regression model using a categorical or quantitative explanatory variable.

TLO6-3: Utilize a logistic regression model using multiple categorical and/or quantitative explanatory variables.

Section 6.1: Comparing Proportions LO6.1-1: Review descriptive and inferential methods for comparing groups with a categorical reFOR INSTRUCTOR USE ONLY

1-120 sponse variable.

LO6.1-2: Compare and contrast different statistics for evaluating group differences on a binary response variable.

Questions 1 through 4: An experiment was conducted with beagles ages 7-11 to determine whether diet and exercise can help keep old dogs mentally sharp. 20 dogs were randomly assigned to two groups: 12 were assigned to the treatment group, which was fed a special diet and given opportunities for extra exercise and social play; 8 were assigned to the control group, which received standard care and diet. After six weeks, all dogs attempted a new task. All 12 dogs in the treatment group were able to solve the task, but only 2 of the 8 dogs in the control group could do so. You plan to use a simulation-based test to decide if the difference between the groups is statistically significant.

1. Which of the following statistics could be used to summarize the sample? Select all that apply.

A. Difference in means B. Relative risk C. Chi-squared statistic D. F-statistic Ans: B, C; LO6.1-1; Difficulty: Medium; Type: MS 2. A simulation was conducted by repeatedly shuffling cards, dealing them into groups of size 12 and 8, and calculating a summary statistic. Describe the cards that would be used in this physical simulation.

A. 20 cards, each marked with the ID number of the dog attempting to solve the task B. 20 cards, each marked with the time it took the dog to solve the task C. 12 blue cards to represent the treatment group, 8 green cards to represent the control group D. 14 blue cards to represent the dogs who solved the task, 6 green cards to represent the dogs who failed to solve the task Ans: D; LO6.1-1; Difficulty: Medium; Type: MC 3. The histogram below shows 10,000 simulated differences: ̂

̂

.

Based on the histogram of shuffled differences, which of the following p-values is reasonFOR INSTRUCTOR USE ONLY

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able?

A. p-value < 0.001 B. p-value = 0.019 C. p-value = 0.211 D. p-value = 0.458 Ans: A; LO6.1-1; Difficulty: Medium; Type: MC 4. Is it reasonable to use a Normal distribution to calculate a p-value in this scenario?

A. Yes, because a sample size of 20 is large enough, as long as the data are reasonably symmetric.

B. No, because the number of successes and failures in each group is not large enough.

C. Yes, because the dogs were randomly assigned to treatment groups.

D. No, because the dogs were not randomly selected from the population.

Ans: B; LO6.1-1; Difficulty: Medium; Type: MC Questions 5 through 9: An observational study was conducted to explore the relationship between development of myopia (nearsightedness) and the use of night-lights with infants.

For each of the 404 children in the study, the researchers recorded whether they slept in darkness or with a night-light, and whether the child was nearsighted.

Darkness Night-light Total Nearsighted 18 78 96 Not Nearsighted 154 154 308 Total 172 232 404 5. Based on the segmented bar graph below, is there an association between use of nightlights and nearsightedness in this sample?

A. Yes, because in both groups, less than half of the children were nearsighted.

B. No, because in both groups, less than half of the children were nearsighted.

C. Yes, because the proportion of children who are nearsighted is lower in the darkness group than in the night-light group.

D. No, because the proportion of children who are nearsighted is lower in the darkness group than in the night-light group.

Ans: C; LO6.1-1; Difficulty: Medium; Type: MC 6. Calculate the relative risk of nearsightedness for the night-light group compared to the darkness group. Keep three decimal places in your answer.

Sol: FOR INSTRUCTOR USE ONLY

1-122 Ans: 3.213 0.01; LO6.1-2; Difficulty: Medium; Type: Calc 7. Using symbols, state the null and alternative hypotheses for testing the association between use of night-lights and nearsightedness.

A. B. C. D. Ans: B; LO6.1-1; Difficulty: Medium; Type: MC 8. Which theory-based inference method(s) could be used to test the association between use of night-lights and nearsightedness? Select all that apply.

A. Two proportion z-test B. Two sample t-test C. Chi-square test D. ANOVA F-test Ans: A, C; LO6.1-1; Difficulty: Medium; Type: MS

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9. The p-value for testing the association between use of night-lights and nearsightedness is less than 0.001. Which of the following conclusions is appropriate? You may assume that the sample is representative of some population. Select all that apply.

A. This study provides strong evidence to suggest that sleeping with a night-light causes a higher risk of nearsightedness for infants.

B. This study provides strong evidence to suggest that there is an association between night-lights and nearsightedness in the population.

C. This study suggests that the difference in the rate of nearsightedness observed in this study would be unlikely to occur by chance alone.

D. This study suggests that the difference in the rate of nearsightedness observed in this study would be likely to occur by chance alone.

Ans: B, C; LO6.1-1; Difficulty: Medium; Type: MS Questions 10 and 11: 48 cocaine abusers were randomly assigned to receive one of two treatments for six weeks: 24 subjects received the antidepressant drug desipramine hydrochloride and 24 subjects received lithium carbonate (the usual treatment). In the desipramine group, 10 had a relapse during the six-week period, compared to 18 in the lithium group.

10. Calculate the odds ratio to compare the odds of relapse for someone treated with lithium compared to the odds of relapse for someone treated with desipramine.

Sol: Ans: 4.2 0; LO6.1-2; Difficulty: Medium; Type: Calc 11. Suppose that the sample sizes were doubled but the proportion who relapsed stayed the same in each group.

The odds ratio would _______ (increase/decrease/stay the same), and the chi-square statistic would _______ (increase/decrease/stay the same), because the odds ratio measures the strength of the _________ (association/evidence), and the chi-square statistic measures the strength of the _________ (association/evidence).

Ans: stay the same, increase, association, evidence; LO6.1-2; Difficulty: Medium; Type: FIB Questions 12 and 13: Suppose we want to test whether there is an association between gender and whether or not a person supports a particular bill currently being debated in Congress. Two possible contingency tables are shown below.

Gender Male Female Total

Table 1 Support Bill Yes No 31 19 29 21 60 40

Total 50 50 100

Gender Male Female Total

Table 2 Support Bill Yes No 42 8 18 32 60 40

Total 50 50 100

12. If we assume that gender and support for the bill are independent, the expected count of males who support the bill is _____. The expected counts are the same for the two tables.

Ans: 30 0; LO6.1-1; Difficulty: Medium; Type: FIB FOR INSTRUCTOR USE ONLY

1-124 13. Which of the tables would result in a larger chi-squared statistic? Hint: You should be able to answer without calculating the chi-square statistic.

A. Table 1 B. Table 2 C. Both tables would yield the same chi-squared statistic.

D. We do not have enough information to determine which chi-squared statistic would be larger.

Ans: B; LO6.1-1; Difficulty: Medium; Type: MC 14. Researchers interviewed 159 patients with a type of cancer called mesothelioma. They also interviewed 159 controls who were similar to the mesothelioma patients in terms of age, sex, lifestyle, etc. but did not have the disease. Both groups were asked whether they had been exposed to asbestos in their home or workplace. To describe the association between asbestos and mesothelioma in this case-control study, the researchers should use the ______________ (relative risk/odds ratio).

Ans: odds ratio; LO6.1-2; Difficulty: Medium; Type: FIB 15. Describe how relative risk measures strength of association.

Relative risk values that are far from ________ (0, 1) indicate a ________ (strong/weak) association. However, the numerical value depends on which proportion is in the numerator; a relative risk of 2 indicates the same strength as a relative risk of ______ (-2, 0, 0.5).

If you change which category is considered a ―success,‖ the relative risk value ________ (changes/stays the same).

Ans: 1 0, strong, 0.5 0, changes; LO6.1-2; Difficulty: Medium; Type: FIB 16. If the probability of success is the same for two treatments, we expect the odds ratio to have a value of _____.

Ans: 1 0; LO6.1-2; Difficulty: Medium; Type: FIB 17. Which of the following is a benefit of using an odds ratio rather than a relative risk? Select all that apply.

A. The odds ratio does not change based on which proportion is in the numerator.

B. The odds ratio does not change based on which category is considered a ―success.‖ C. The odds ratio does not change based on which variable is treated as the ―response‖.

D. The odds ratio is easier to understand, because it can be directly interpreted in terms of ―likelihood‖ or ―chance of success.‖ Ans: B, C; LO6.1-2; Difficulty: Medium; Type: MS

18. True or False: The simulated two-sided p-value will be the same regardless of whether FOR INSTRUCTOR USE ONLY

Introduction to Financial Statements you use a difference of proportion, relative risk, odds ratio, z-statistic, or chi-squared statistic to summarize the data.

Ans: True; LO6.1-2; Difficulty: Medium; Type: TF

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Section 6.2: Introduction to Logistic Regression LO6.2-1: Explain the motivation and need for logistic regression.

LO6.2-2: Utilize a logistic regression model using categorical or quantitative explanatory variables.

Questions 1 through 5: The manager of a baseball team used a logistic regression model to predict the likelihood of winning a game based on the number of runs scored by the opponent. Note: log() in the formula below represents the natural log.

Predicted log-odds of winning

–

1. Calculate the odds of winning when the opponent scores 5 runs.

Sol: Ans: 1.545 0.01; LO6.2-2; Difficulty: Medium; Type: Calc 2. Calculate the probability of winning when the opponent scores 5 runs.

Sol: Ans: 0.607 0.005; LO6.2-2; Difficulty: Medium; Type: Calc 3. In this dataset, the largest number of runs scored by an opponent was 11.

True or False: This model should not be used to extrapolate when the number of runs scored by an opponent is very large, because the predicted probability of winning may be less than 0.

Ans: False; LO6.2-1; Difficulty: Medium; Type: TF 4. Interpret the slope: As the number of runs scored by the opponent increases by 1, A. the predicted odds of winning decrease by 0.598.

B. the predicted odds of winning are multiplied by -0.598.

C. the predicted odds of winning decrease by .

D. the predicted odds of winning are multiplied by .

Ans: D; LO6.2-2; Difficulty: Medium; Type: MC 5. The p-value for the slope in this logistic regression model is 0.0004. Interpret the p-value.

The data provides ________ (strong/weak) evidence that there is ___________ (no association/an association) between runs scored by the opponent and winning.

Ans: strong, an association; LO6.2-2; Difficulty: Medium; Type: FIB

FOR INSTRUCTOR USE ONLY Term

Coeff.

SE

Chi-square p-value

95% CI

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Questions 6 through 10: An experiment is done to test the effect of two toxic substances on insects. 250 insects are exposed to each substance and the number of insects that die is counted. Logistic regression is used to describe the connection between type of substance and the likelihood of death.

SubstanceB is an indicator variable (1=Substance B, 0=Substance A). A partially filled in table of coefficients is given below.

Intercept -1.9924 0.1946 SubstanceB 0.5030 0.2540

(1.6109, 2.3740) (0.0050, 1.0009)

6. Calculate the predicted odds of dying for an insect exposed to Substance A. Sol: Ans: 0.1364 0.001; LO6.2-2; Difficulty: Medium; Type: Calc 7. Interpret the slope.

A. The probability of dying when exposed to Substance B is 0.5030 higher than the probability of dying when exposed to Substance A. B. The odds of dying when exposed to Substance B are 0.5030 times as high as the odds of dying when exposed to Substance A. C. The probability of dying when exposed to Substance B is 1.6537 higher than the probability of dying when exposed to Substance A. D. The odds of dying when exposed to Substance B are 1.6537 times as high as the odds of dying when exposed to Substance A. Ans: D; LO6.2-2; Difficulty: Medium; Type: MC 8. At the significance level, is the association between death and type of substance statistically significant?

A. Yes, the association is significant, because the confidence interval for the slope does not include 0.

B. No, the association is not significant, because the confidence interval for the slope does not include 0.

C. Yes, the association is significant, because the confidence interval for the slope includes 1.

D. No, the association is not significant, because the confidence interval for the slope includes 1.

Ans: A; LO6.2-2; Difficulty: Medium; Type: MC 9. Calculate a 95% confidence interval for the population odds ratio.

The odds of death when exposed to Substance B are between _____ and ______ times as high as the odds of death when exposed to Substance A. Ans: 1.005 0.001, 2.721 0.001; LO6.2-2; Difficulty: Medium; Type: FIB 10. Suppose the indicator variable for substance had been defined differently: FOR INSTRUCTOR USE ONLY

1-128 { Calculate the intercept and slope for the logistic regression model.

Sol: (

)

Ans: 1.4894 0.001, -0.5030 0.001; LO6.2-2; Difficulty: Hard; Type: Calc 11. Use answer choices (A) – (C) to fill in the blanks. Choices may be used more than once or not at all.

In a chi-squared test, the response variable ____.

In a chi-squared test, the explanatory variable ____.

A. Must be categorical B. Must be quantitative C. May be categorical or quantitative

In logistic regression, the response variable ____.

In logistic regression, the explanatory variable ____.

Ans: A, A, A, C; LO6.2-1; Difficulty: Medium; Type: Ma 12. Which of the following models is used in logistic regression?

A. B. C. D. Ans: B; LO6.2-2; Difficulty: Easy; Type: MC 13. Based on logistic regression, which of the following formulas could be used to predict the probability of success for a particular x value?

A. ̂ ( ) B.

̂

C.

̂

D.

̂

Ans: D; LO6.2-2; Difficulty: Medium; Type: MC 14. True or False: When using a logistic regression model, the predicted probability of success is guaranteed to fall between 0 and 1.

Ans: True; LO6.2-1; Difficulty: Medium; Type: TF

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15. True or False: Given the confidence interval for the slope coefficient in logistic regression, taking the natural log of the endpoints gives you a confidence interval for the population odds ratio.

Ans: False; LO6.2-2; Difficulty: Medium; Type: TF

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Section 6.3: Multiple Logistic Regression Models LO6.3-1: Utilize a logistic regression model using multiple categorical and/or quantitative explanatory variables.

Questions 1 through 7: In schools that employ ―tracking,‖ students are assigned to classes based on their perceived ability/achievement with separate classes for high- and lowperforming students. Some schools are experimenting with ―detracking,‖ putting more heterogeneous groups of students in class together.

Burris et al. (2008) explored the effect of detracking (1 = detracked classes, 0 = tracked classed) on the likelihood of a student receiving an international baccalaureate (IB) diploma. They controlled for aptitude (a standardized quantitative variable) and whether or not the student was in special education (1 = yes, 0 = no). The researchers considered several models, finally choosing the model below.

1. Initially, the researchers included a Detracking Aptitude interaction term in the model, but they removed the term, because it was not statistically significant. Explain what the assumption of ―no interaction‖ means in this context.

A. The likelihood of a student studying in detracked classes is the same, regardless of student aptitude level.

B. The likelihood of a student studying in detracked classes changes based on student aptitude level.

C. The odds ratio comparing the odds of earning an IB diploma for students in tracked and detracked classes is the same, regardless of student aptitude level.

D. The odds ratio comparing the odds of earning an IB diploma for students in tracked and detracked classes changes based on student aptitude level.

Ans: C; LO6.3-1; Difficulty: Medium; Type: MC 2. Estimate the probability that a student with an aptitude score of 1.5 who is not in special education and studies in detracked classes will receive an IB diploma.

Sol:

̂ ̂ ̂

Ans: 0.82 0.01; LO6.3-1; Difficulty: Medium; Type: Calc 3. Interpret the slope corresponding to the Detracking variable.

The __________ (chances/odds) of earning an IB diploma for students in detracked classes are _______ times as high as the __________ (chances/odds) for someone in tracked classes, after adjusting for aptitude and special education.

Ans: odds, 1.72 0.01, chances; LO6.3-1; Difficulty: Medium; Type: FIB

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4. Interpret the slope corresponding to the Aptitude variable.

As aptitude increases by 1 ______ (unit/SD), the __________ (chances/odds) of earning an IB diploma are multiplied by _______, after adjusting for special education and detracking.

Ans: SD, odds 6.18 0.01; LO6.3-1; Difficulty: Medium; Type: FIB 5. The standard error for the Detracking coefficient was 0.158. Calculate the chi-square statistic for this variable.

Sol: (

)

Ans: 11.68 0.01; LO6.3-1; Difficulty: Medium; Type: Calc 6. What if the researchers decided to use a simple logistic regression model with only Detracking as a predictor?

A. The coefficient and p-value for the Detracking variable could both change.

B. The coefficient for the Detracking variable could change, but the p-value would stay the same.

C. The p-value for the Detracking variable could change, but the coefficient would stay the same.

D. The coefficient and p-value for the Detracking variable would both stay the same.

Ans: A; LO6.3-1; Difficulty: Medium; Type: MC 7. These data were collected at a single school before and after they detracked all their classes. Does this study support causal conclusions about the relationship between detracking and earning IB diplomas at this school?

A. Yes, because the analysis adjusted for possible confounders, aptitude and special education.

B. Yes, because students were assigned to either tracked or detracked classes. They were not allowed to choose for themselves.

C. No, because students were not randomly selected. All students in the study attended the same school.

D. No, because other changes may have occurred at this school at the same time as detracking.

Ans: D; LO6.3-1; Difficulty: Medium; Type: MC Questions 8 through 11: A multiple logistic regression model can be used to predict which passengers survive the sinking of the Titanic based on sex (1=female, 0=male) and age (in years).

8. Interpret the odds ratio for Sex. A. The predicted odds of survival for female passengers were 2.4660 times as high as the predicted odds of survival for male passengers of the same age.

B. Overall, the predicted odds of survival for female passengers were 2.4660 times as high as the predicted odds of survival for male passengers.

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1-132 C. The predicted odds of survival for female passengers were times as high as the predicted odds of survival for male passengers of the same age.

D. Overall, the predicted odds of survival for female passengers were times as high as the predicted odds of survival for male passengers.

Ans: C; LO6.3-1; Difficulty: Medium; Type: MC 9. True or False: Based on the logistic regression model above, a change of one year has the same impact on the predicted probability of survival at any point on the Age scale.

Ans: False; LO6.3-1; Difficulty: Medium; Type: TF 10. The parallel boxplots below shows the relationship between Age and Sex for passengers on the Titanic.

In the model shown above, the Age term is not statistically significant. If Age were removed from the logistic regression model, how would you expect the coefficient for Sex to change?

A. The association between Sex and Age is weak, so the coefficient for Sex would change dramatically.

B. The association between Sex and Age is weak, so the coefficient for Sex would not change dramatically.

C. The association between Sex and Age is strong, so the coefficient for Sex would change dramatically.

D. The association between Sex and Age is strong, so the coefficient for Sex would not change dramatically.

Ans: B; LO6.3-1; Difficulty: Medium; Type: MC 11. The multiple regression model above does not allow for interaction. When an Age Sex interaction term was added to the model, the term had a very small p-value. How do you interpret the p-value?

A. There is strong evidence that the relationship between Age and Survival is the same, regardless of Sex. B. There is strong evidence that the relationship between Age and Survival changes depending on Sex. C. There is weak evidence that the relationship between Age and Survival is the same, regardless of Sex. D. There is weak evidence that the relationship between Age and Survival changes depending on Sex. Ans: B; LO6.3-1; Difficulty: Medium; Type: MC 12. Suppose researchers want to use a multiple logistic regression model to estimate the FOR INSTRUCTOR USE ONLY

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probability that a person has a gym membership. What kind of explanatory variables may be used in this model?

A. All explanatory variables must be categorical.

B. All explanatory variables must be quantitative.

C. Explanatory variables may be categorical and/or quantitative.

D. Regardless of which explanatory variables are chosen, a multiple logistic regression model is not appropriate in this scenario.

Ans: C; LO6.3-1; Difficulty: Medium; Type: MC 13. A multiple logistic regression model was used to predict whether individuals would be admitted into medical school. The prediction table for the model is shown below.

Actually Admitted Actually Rejected Total

Predicted to be Admitted 80 70 150

Predicted to be Rejected 22 826 848

Total 102 896 998

Calculate the correct classification rate.

Sol: Ans: 0.908 0.005; LO6.3-1; Difficulty: Medium; Type: Calc 14. In logistic regression, what numerical summary is used to measure the unexplained variability?

A. Coefficient of covariation B. Deviance statistic C. Standard error of the residuals D. Sum of squared errors Ans: B; LO6.3-1; Difficulty: Easy; Type: MC 15. Suppose that two explanatory variables – one categorical and one quantitative – are being used to predict a categorical outcome. Which of the following conditions is necessary for Simpson’s paradox to occur?

A. A substantial association between the two explanatory variables B. A very weak association between the two explanatory variables C. A substantial interaction between the two explanatory variables D. A very weak interaction between the two explanatory variables Ans: A; LO6.3-1; Difficulty: Easy; Type: MC

Preliminaries Intermediate Statistical Investigations Test Bank Question types:

FIB = Fill in the blank Ma = Matching MC = Multiple choice

Calc = Calculation MS = Multiple select TF = True-false

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PRELIMINARIES TERMINAL LEARNING OUTCOMES TLOP-A: Explore how the relationship between two variables can be impacted by additional variables.

TLOP-B: Express the relationship between two variables using a statistical model, and calculate and report the standard error of the residuals, a measure of prediction error.

PRELIMINARIES ENABLING LEARNING OUTCOMES LOP-1: Identify and apply basic terminology of statistical studies: observational units, response variable, explanatory variable, association, confounding variable.

LOP-2: Identify potential sources and measures of variation in a response variable.

LOP-3: Produce and describe some basic visualizations and numerical summaries to compare groups and explore relationships (e.g., bar graphs, dotplots/histograms/boxplots, scatterplots, means, medians, standard deviation).

LOP-4: Explore how those comparisons and relationships can be impacted by additional variables.

LOP-5: Calculate a residual and relate it to typical prediction error.

Section P.A Questions 1 and 2: In order for students to participate in remote learning, they need access to a computer or tablet for school work. An advocacy group conducted a survey of U.S. families and analyzed the relationship between household income (less than $25,000, between $25,000 and $50,000, or greater than $50,000) and access to a computer or tablet (no access, sometimes, always).

1. Access to a computer/tablet is the __________ (explanatory/response) variable.

We can classify this variable as __________ (categorical/quantitative).

Income is the __________ (explanatory/response) variable.

We can classify this variable as __________ (categorical/quantitative).

Ans: response, categorical, explanatory, categorical; LOP-1; Difficulty: Easy; Type: FIB 2. Which type of graph is most appropriate for displaying the relationship between income and access to a computer/tablet?

A. A mosaic plot B. A histogram C. Stacked boxplots (one boxplot for each group) D. A scatterplot Ans: A; LOP-3; Difficulty: Medium; Type: MC 3. Which of the hypothetical mosaic plots below shows an association between class (first, second, third, or crew) and survival on the Titanic? Select all that apply.

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A. Graph A B. Graph B C. Graph C Ans: A; LOP-3; Difficulty: Medium; Type: MS

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1-136 Questions 4 through 7: Two hospitals, Memorial Hospital and Fairbanks Medical Center, both perform the same procedure to alleviate joint pain. Researchers surveyed a random sample of 100 patients from each hospital who had undergone this procedure and asked whether or not they had made a full recovery in a six month period after surgery. The table below shows the results of the survey. The mosaic plot further breaks down the results based on socioeconomic status (SES).

Recovery Hospital Fairbanks Memorial Total

Full 60 66 126

No 40 34 74

Total 100 100 200

4. Based on the table, is there an association between hospital and recovery in this sample?

Note: This question refers to the sample, so there is no need to consider whether this data reflects a genuine tendency in the population.

A. There is no association. More than 50% of patients recover, regardless of hospital.

B. There is no association. The recovery rate is not the same at the two hospitals.

C. There is an association. More than 50% of patients recover, regardless of hospital.

D. There is an association. The recovery rate is not the same at the two hospitals.

Ans: D; LOP-1; Difficulty: Medium; Type: MC 5. Suppose you are primarily interested in the potential association between hospital and recovery. Based on the mosaic plot, does SES satisfy the definition of a confounding variable in this scenario? _______ (Yes/No), because Memorial Hospital is more likely to serve _______ (high/low) SES patients, compared to Fairbanks, and_______ (high/low) SES patients are more likely to make a full recovery.

Ans: Yes, high, high; LOP-1; Difficulty: Medium; Type: FIB

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6. Does this scenario satisfy the definition of Simpson’s paradox? _______ (Yes/No), because overall, the recovery rates are higher at _______ (Memorial/Fairbanks), but after adjusting for SES, the conditional recovery rates are higher at _______ (Memorial/Fairbanks).

Ans: Yes, Memorial, Fairbanks; LOP-4; Difficulty: Medium; Type: FIB 7. Put the components of the study into the correct boxes in the Sources of Variation Diagram. Note: Some boxes will include more than one answer.

Observed variation in:

Sources of explained variation

Sources of unexplained variation

Inclusion criteria

A. Hospital (Memorial or Fairbanks) B. Has undergone medical procedure for joint pain C. Lifestyle factors (physicality of occupation, opportunities for rest, etc.)

D. Recovery (full or no) E. Socioeconomic status (low SES or high SES) F. Physical therapy (as recommended, less than recommended, none) Ans: Observed variation in: D, Inclusion criteria: B, Sources of explained variation: A, E, Sources of unexplained variation: C, F; LOP-2; Difficulty: Medium; Type: FIB Questions 8 through 13: In 2018, a sample of academic faculty from universities across the country were surveyed about their salaries (in US dollars). The results were classified according to each faculty member’s academic rank (instructor, assistant professor, associate professor, and full professor) and gender (male, female).

8. Identify the observational units and variables. Note: Some blanks will include more than one answer. One of the answer choices will not be used.

Observational units: Explanatory variables: Response variable:

A. Academic faculty members B. Universities C. Salary (in US dollars) D. Academic rank E. Gender

Ans: Observational units: A, Explanatory variables: D, E, Response variable: C; LOP-1; Difficulty: Easy; Type: Ma

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1-138 9. Name the graph types being used to display the distribution of salaries.

A. Bar graph B. Mosaic plot C. Histogram D. Boxplot E. Scatterplot Ans: C, D; LOP-3; Difficulty: Easy; Type: MS 10. Describe the relationship between the mean salary and the median salary.

The mean salary is ______ (greater/less) than the median salary, because the distribution of salaries is skewed ______ (right, left) Ans: greater, right; LOP-3; Difficulty: Medium; Type: FIB 11. True or False: There is an association between gender and salary in this sample.

Ans: True; LOP-3; Difficulty: Medium; Type: TF FOR INSTRUCTOR USE ONLY

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12. Use the mosaic plot to match the percentages below to the appropriate statement describing the data.

Statement 1: _____% of the faculty in this sample are male.

Statement 2: _____ % of the faculty who have reached the rank of professor are male.

Statement 3: _____% of female faculty members are at the rank of instructor.

Statement 4: _____% of male faculty members are at the rank of instructor.

A. 7% B. 22% C. 68% D. 81% Ans: 1: C, 2: D, 3: B, 4: A; LOP-3; Difficulty: Easy; Type: Ma 13. Using the tables below, compare the distribution of salaries for male and female faculty members, before and after taking rank into account.

Which of the following statements are true? Select all that apply.

A. Before taking rank into account, there is an association between gender and salary in this sample.

B. After taking rank into account, there is an association between gender and salary in this sample.

C. The nature of the relationship between gender and salary changes after adjusting for rank; thus rank satisfies the definition of a confounding variable in this scenario.

D. The direction of the relationship between gender and salary changes after adjusting for rank; thus this scenario satisfies the definition of Simpson’s paradox.

Ans: A, B, C; LOP-4; Difficulty: Hard; Type: MS FOR INSTRUCTOR USE ONLY

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Section P.B Questions 14 through 16: A sample of caregivers was selected to be representative of the U.S. population. If there was more than one child in the caregiver’s household, the child asked about was determined randomly. Each caregiver was asked to estimate how much sleep their child typically gets at night. To predict sleep times, we can use the following statistical model: Model 1: Predicted sleep = 8.26 hours, standard deviation = 1.33 hours 14. Interpret the standard deviation.

A. The range of the middle 50% of the data is 1.33 hours.

B. A typical sleep time lies about 1.33 hours from the mean sleep time.

C. 95% of the observations in this sample are in the range 8.26 1.33.

D. 95% of the observations in this sample are higher than 1.33 hours.

Ans: B; TLOP-B; Difficulty: Medium; Type: MC 15. One caregiver estimates that their child gets 7 hours of sleep per night. Calculate the residual for this observation.

Solution: 7 – 8.26 = -1.26 Ans: -1.26 0; LOP-5; Difficulty: Easy; Type: Calc 16. Suppose we add a new variable, Age, which explains a substantial amount of the variability in sleep times.

Model 2: Predicted sleep = 7.63 hours for older children (age 12-17), 8.89 hours for younger children (age 6-11) SE of the residuals = ?

Since Age explains a substantial amount of the variability in sleep times, we would expect the standard error of the residuals in Model 2 to be _______ (greater than/less than/equal to) the standard deviation in Model 1.

Ans: less than; LOP-5; Difficulty: Medium; Type: FIB

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Questions 17 and 18: For a sample of 100 colleges, four different models were used to predict the average salary in the year after graduating. The graphs below show the residuals from these models.

17. Graph 1 shows the residuals from a model that uses a school’s admittance rate to predict average salary. Graph 2 shows the residuals from a model that uses a school’s total cost per year (out-of-state) to predict average salary. Which explanatory variable do you prefer?

A. Admittance rate (Model 1), because more of the variability in average salaries is explained by the model.

B. Total cost (Model 2), because there is less unexplained variability in average salaries after applying the model.

C. Both models are equally good, because both have a distribution of residuals that is roughly bell-shaped.

D. Neither of these models is useful for predicting average salary, because both have a distribution of residuals that is centered at 0.

Ans: B; LOP-5; Difficulty: Medium; Type: MC 18. Suppose we used a model with two explanatory variables. What would the graph of the residuals look like if both admittance rate and cost were used to predict average salary?

A. Graph 3 is reasonable, because the SD of the residuals is less than 5768.

B. Graph 4 is a reasonable, because the SD of the residuals is greater than 6329.

C. Neither Graph 3 nor Graph 4 is reasonable, because the SD of the residuals should be between 5768 and 6329.

Ans: A; LOP-5; Difficulty: Medium; Type: MC

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1-142 Questions 19 and 20: A botanist collected data to measure the variation of Iris flowers of two related species. The data set consists of 50 flowers from two species of Iris – Iris setosa and Iris versicolor. Various features were measured including the length and width of the sepals, in centimeters. (Sepals are a part of the flower.)

The graphs below show the relationship between sepal length and sepal width. Graph A shows the line of best fit for the full sample. Graph B shows a separate line of fit for each species: red represents setosa and blue represents versicolor

Consider two models for predicting sepal width. Note that Model 2 accounts for Species. Some values are intentionally missing in Model 2.

Model 1: Predicted sepal width = 3.94 – 0.15(Sepal length), SE of residuals = 0.47 Model 2: Predicted sepal width = {

, SE of residuals =

19. The letter B is a placeholder for a missing value in Model 2. Would you expect this value to be positive, negative, or close to 0?

A. Positive B. Negative C. Close to 0 Ans: A; LOP-4; Difficulty: Hard; Type: MC 20. We would expect the SE of the residuals for Model 2 to be _______ (>, <, =) the SE of the residuals for Model 1.

Ans: <; LOP-5; Difficulty: Medium; Type: FIB

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