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Contents Chapter 1 Chapter 2 Chapter 3

1 7 21

Chapter 4 Chapter 5 Review of Part I

Stats Starts Here Displaying and Describing Data Relationships Between Categorical Variables—Contingency Tables Understanding and Comparing Distributions The Standard Deviation as Ruler and the Normal Model Exploring and Understanding Data

Chapter 6 Chapter 7 Chapter 8 Review of Part II

Scatterplots, Association, and Correlation Linear Regression Regression Wisdom Exploring Relationships Between Variables

89 101 125 159

Chapter 10 Chapter 11 Review of Part III

Sample Surveys Experiments and Observational Studies Gathering Data

181 189 203

Chapter 12 Chapter 13 Chapter 14 Chapter 15 Review of Part IV

From Randomness to Probability Probability Rules! Random Variables Probability Models Randomness and Probability

213 223 243 259 285

Chapter 16

303

Chapter 17 Chapter 18 Chapter 19 Review of Part V

Sampling Distribution Models and Confidence Intervals for Proportions Confidence Intervals for Means Testing Hypotheses More About Tests and Intervals Inference for One Parameter

Chapter 20 Chapter 21 Chapter 22 Chapter 23 Review of Part VI

Comparing Groups Paired Samples and Blocks Comparing Counts Inferences for Regression Inference for Relationships

393 427 447 467 489

Chapter 24 Multiple Regression Wisdom Chapter 25 Analysis of Variance Chapter 26 Multifactor Analysis of Variance Review of Part VI Inference When Variables Are Related Cumulative Review Exercises SOURCE: Browsegrades.net

37 51 73

317 339 363 375

521 525 535 543 563

Chapter 1 Stats Starts Here

Chapter 1 – Stats Starts Here Section 1.1 1.

Grocery shopping. Discount cards at grocery stores allow the stores to collect information about the products that the customer purchases, what other products are purchased at the same time, whether or not the customer uses coupons, and the date and time that the products are purchased. This information can be linked to demographic information about the customer that was volunteered when applying for the card, such as the customer’s name, address, sex, age, income level, and other variables. The grocery store chain will use that information to better market their products. This includes everything from printing out coupons at the checkout that are targeted to specific customers to deciding what television, print, or Internet advertisements to use.

Online shopping. Retailers hopes to gain all sorts of information about customer behavior, such as how long they spend looking at a page, whether or not they read reviews by other customers, what items they ultimately buy, and what items are bought together. They can then use this information to determine which other products to suggest to customers who buy similar items, to determine which advertisements to run in the margins, and to determine which items are the most popular so these items come up first in a search.

Parking lots. The owners of the parking garage can advertise about the availability of parking. They can also communicate with businesses about hours when more spots are available and when they should encourage more business.

Satellites and global climate change. This rise and fall of temperature and water levels can help in planning for future problems and guide public policy to protect our safety.

Section 1.2 5.

Super Bowl. When collecting data about the Super Bowl, the games themselves are the Who.

Nobel laureates. Each year is a case, holding all of the information about that specific year. Therefore, the year is the Who.

Health records. The sample is about 5000 people, and the population is all residents of the United States of America. The Who is the selected subjects and the What includes medical, dental, and physiological measurements and laboratory test results.

Facebook. The Who is the 350 million photos. The What might be information about the photos, for example: file format, file size, time and date when uploaded, people and places tagged, and GPS information.

Section 1.3 9.

Grade levels a) If we are, for example, comparing the percentage of first-graders who can tie their own shoes to the percentage of second-graders who can tie their own shoes, grade-level is treated as categorical. It is just a way to group the students. We would use the same methods if we were comparing boys to girls or browneyed kids to blue-eyed kids. b) If we were studying the relationship between grade-level and height, we would be treating grade level as quantitative. SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

10. ZIP codes. a) ZIP codes are categorical in the sense that they correspond to a location. The ZIP code 14850 is a standardized way of referring to Ithaca, NY. b) ZIP codes generally increase as the location gets further from the east coast of the United States. For example, one of the ZIP codes for the city of Boston, MA is 02101. Kansas City, MO has a ZIP code of 64101, and Seattle, WA has a ZIP code of 98101. 11. Voters.

13. Medicine.

The response is a categorical variable. 12. Job hunting.

The company is studying a quantitative variable. 14. Stress.

The answer is a categorical variable.

The researcher is studying a quantitative variable.

Section 1.4 15. Voting and elections. Pollsters might consider whether a person voted previously or whether he or she could name the candidates. Voting previously and knowing the candidates may indicate a greater interest in the election. 16. Weather. Meteorologists can use the models to predict the average temperature ten days in advance and compare their predictions to the actual temperatures. 17. The News. Answers will vary.

18. The Internet. Answers will vary.

19. Gaydar. Who – 40 undergraduate women; What – Whether or not the women could identify the sexual orientation of men based on a picture; Population of interest – All women. 20. Hula-hoops. Who – An unknown number of participants; What – Heart rate, oxygen consumption, and rating of perceived exertion; Population of interest – All people. 21. Bicycle Safety. Who – 2500 cars; What – Distance from the bicycle to the passing car (in inches); Population of interest – All cars passing bicyclists. 22. Investments. Who – 30 similar companies; What – 401(k) employee participation rates (in percent); Population of interest – All similar companies. 23. Honesty. Who – Coffee drinkers at a Newcastle University coffee station; What – amount of money contributed to the collection tray; Population of interest – All people in honor system payment situations. 24. Blindness. Who – 24 patients suffering from blindness; What – Whether the patient had Stargardt’s disease or dry agerelated macular degeneration, and whether or not the stem cell therapy was effective in treating the condition; Population of interest – All people with these eye conditions. 25. Not-so-diet soda. Who – 474 participants; What – whether or not the participant drank two or more diet sodas per day, waist size at the beginning of the study, and waist size at the end of the study; Population of interest – All people. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Chapter 1 Stats Starts Here

26. Molten iron. Who – 10 crankshafts at Cleveland Casting; What – The pouring temperature (in degrees Fahrenheit) of molten iron; Population of interest – All crankshafts at Cleveland Casting. 27. Weighing bears. Who – 54 bears; Cases – Each bear is a case; What – Weight, neck size, length (no specified units), and sex; When – Not specified; Where – Not specified; Why - Since bears are difficult to weigh, the researchers hope to use the relationships between weight, neck size, length, and sex of bears to estimate the weight of bears, given the other, more observable features of the bear; How – Researchers collected data on 54 bears they were able to catch; Variables – There are 4 variables; weight, neck size, and length are quantitative variables, and sex is a categorical variable. No units are specified for the quantitative variables. Concerns – The researchers are (obviously!) only able to collect data from bears they were able to catch. This method is a good one, as long as the researchers believe the bears caught are representative of all bears, in regard to the relationships between weight, neck size, length, and sex. 28. Schools. Who – Students; Cases – Each student is an individual case; What – Age (probably in years, though perhaps in years and months), race or ethnicity, number of absences, grade level, reading score, math score, and disabilities/special needs; When – This information must be kept current; Where – Not specified; Why – Keeping this information is a state requirement; How – The information is collected and stored as part of school records; Variables – There are seven variables. Race or ethnicity, grade level, and disabilities/special needs are categorical variables. Number of absences (days), age (years?), reading test score, and math test score are quantitative variables. Concerns – What tests are used to measure reading and math ability, and what are the units of measure for the tests? 29. Arby’s menu. Who – Arby’s sandwiches; Cases – Each sandwich is a case; What – type of meat, number of calories (in calories), and serving size (in ounces); When – Not specified; Where – Arby’s restaurants; Why – These data might be used to assess the nutritional value of the different sandwiches; How – Information was gathered from each of the sandwiches on the menu at Arby’s, resulting in a census. Variables – There are three variables. Number of calories and serving size (ounces) are quantitative variables, and type of meat is a categorical variable. 30. Age and party. Who – 1180 Americans; Cases – Each of the 1180 Americans surveyed is a case in this poll; What – Region, age (in years), political affiliation, and whether or not the person voted in the 2006 midterm Congressional election; When – First quarter of 2007; Where – United States; Why – The information was gathered for presentation in a Gallup public opinion poll. How – Phone Survey; Variables – There are four variables. Region, political affiliation, and whether or not the person voted in 1998 are categorical variables, and age is a quantitative variable. 31. Babies. Who – 882 births; Cases – Each of the 882 births is a case. What – Mother’s age (in years), length of pregnancy (in weeks), type of birth (caesarean, induced, or natural), level of prenatal care (none, minimal, or adequate), birth weight of baby (unit of measurement not specified, but probably pounds and ounces), gender of baby (male or female), and baby’s health problems (none, minor, major); When – 1998-2000; Where – Large city hospital; Why – Researchers were investigating the impact of prenatal care on newborn health. How – It appears that they kept track of all births in the form of hospital records, although it is not specifically stated. Variables – There are three quantitative variables: mother’s age (years), length of pregnancy (years), and birth weight of baby. There are four categorical variables: type of birth, level of prenatal care, gender of baby, and baby’s health problems.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

32. Flowers. Who – 385 species of flowers; Cases – Each of the 385 species at each of the 47 years is a case. What – Date of first flowering (in days); When – Not specified. Where – Southern England; Why – The researchers believe that this indicates a warming of the overall climate. How – Not specified. Variables – Date of first flowering is a quantitative variable. Concerns - Hopefully, date of first flowering was measured in days from January 1, or some other convention, to avoid problems with leap years. 33. Herbal medicine. Who – experiment subjects; Cases – Each subject is a case. What – herbal cold remedy or sugar solution, and cold severity (0 to 5 scale); When – Not specified. Where – Major pharmaceutical firm; Why – Scientists were testing the efficacy of an herbal compound on the severity of the common cold. How – The scientists set up a controlled experiment. Variables – There are two variables. Type of treatment (herbal or sugar solution) is categorical, and severity rating is quantitative. Concerns – The severity of a cold seems subjective and difficult to quantify. Also, the scientists may feel pressure to report negative findings about the herbal product. 34. Vineyards. Who – American Vineyards; Cases – Each vineyard is a case. What – Size of vineyard (in acres), number of years in existence, state, varieties of grapes grown, average case price (in dollars), gross sales (probably in dollars), and percent profit; When – Not specified. Where – United States; Why – Business analysts hoped to provide information that would be helpful to producers of American wines. How – Not specified. Variables – There are five quantitative variables and two categorical variables. Size of vineyard, number of years in existence, average case price, gross sales, and percent profit are quantitative variables. State and variety of grapes grown are categorical variables. 35. Streams. Who – Streams; Cases – Each stream is a case; What – Name of stream, substrate of the stream (limestone, shale, or mixed), acidity of the water (measured in pH), temperature (in degrees Celsius), and BCI (unknown units); When – Not specified; Where – Upstate New York; Why – Research is conducted for an Ecology class. How – Not specified. Variables – There are five variables. Name and substrate of the stream are categorical variables, and acidity, temperature, and BCI are quantitative variables. 36. Fuel economy. Who – Every model of automobile in the United States. Cases – Each vehicle model is a case. What – Vehicle manufacturer, vehicle type, weight (probably in pounds), horsepower (in horsepower), and gas mileage (in miles per gallon) for city and highway driving; When – This information is collected currently. Where – United States; Why – The Environmental Protection Agency uses the information to track fuel economy of vehicles. How – The data is collected from the manufacturer of each model. Variables – There are six variables. City mileage, highway mileage, weight, and horsepower are quantitative variables. Manufacturer and type of car are categorical variables. 37. Refrigerators. Who – 353 refrigerator models; Cases – Each of the 353 refrigerator models is a case. What – Brand, cost (probably in dollars), size (in cu. ft.), type, estimated annual energy cost (probably in dollars), overall rating, and repair history (in percent requiring repair over the past five years); When – 2013; Where – United States; Why – The information was compiled to provide information to the readers of Consumer Reports. How – Not specified. Variables – There are 7 variables. Brand, type, and overall rating are categorical variables. Cost, size, estimated energy cost, and repair history are quantitative variables. 38. Walking in circles. Who – 32 volunteers; Cases – Each volunteer is a case. What – Sex, height, handedness, the number of yards walked before going out of bounds, and the side of the field on which the person walked out of bounds; When – Not specified. Where – Not specified. Why – The researcher was interested in whether people walk in circles when lost. How – Data were collected by observing the people on the field, as well as by measuring and asking the participants. Variables – There are 5 variables. Sex, handedness, and side of the field are categorical variables. Height and number of yards walked are quantitative variables. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Chapter 1 Stats Starts Here

39. Kentucky Derby 2018. Who – Kentucky Derby races; What – Year, winner, jockey, trainer, owner, and time (in minutes, seconds, and hundredths of a second); When – 1875 to 2018; Where – Churchill Downs, Louisville, Kentucky; Why – Not specified. To examine the trends in the Kentucky Derby? How – Official statistics are kept for the race each year. Variables – There are 8 variables. Winner, jockey, trainer and owner are categorical variables. Date and duration are quantitative variables. 40. Indy 2018. Who – Indy 500 races; What – Year, winner, time (in minutes, seconds, and hundredths of a second), and average speed (in miles per hour); When – 1911 – 2018; Where – Indianapolis, Indiana; Why – Not specified. To examine the trends in Indy 500 races? How – Official statistics are kept for the race every year; Variables – There are 4 variables. Driver is a categorical variable. Year, time, and speed are quantitative variables. 41. Kentucky Derby 2018 on the computer. a) Fonso was the winning horse in 1880. b) The length of the race changed in 1895, from 1.5 miles to 1.25 miles. c)

The winning time in 1974 was 124 seconds.

d) Secretariat ran the Derby in under 2 minutes in 1973. 42. Indy 500 2018 on the computer. a) The average speed of the winner in 1920 was 88.618 miles per hour. b) Bill Vukovich won the Indy 500 twice in the 1950s. c)

There were only 6 Indy 500 races in the 1940s.

SOURCE: Browsegrades.net

Chapter 2 Displaying and Describing Data

Chapter 2 – Displaying and Describing Data Section 2.1 1.

Automobile fatalities. Subcompact and Mini Compact Intermediate Full Unknown

Nonoccupant fatalities.

Movie genres.

0.2658 0.2084 0.3006 0.2069 0.0183

a) A pie chart seems appropriate for the movie genre data. Each movie has only one genre, and the list of all movies constitute a “whole”. b) “Other” is the least common genre. It has the smallest region in the chart. 4.

Movie ratings. a) A pie chart seems appropriate for the movie ratings data. Each movie has only one rating, and the list of all movies constitute a “whole”. b) “R” is the most common rating. It has the largest region in the chart.

Movie ratings. i)

ii) A

iii) D

iv) B

iii) C

iv) B

Marriage in decline. i)

ii) A

Section 2.2 7.

Traffic Fatalities 2013. a) The gaps in the histogram for Year indicate that we do not have data for those years. This data set contains two variables for each case, and a histogram of the years doesn’t give us much useful information. b) All of the bars in the Year histogram are the same height because each year only appears once in the data set. c)

The distribution of passenger car fatalities has between 17,500 and 25,000 traffic fatalities per year in most years. There were also several years—possibly a second mode—with between 10,000 and 12,500 traffic fatalities.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

Traffic fatalities 2013 again. a) Two years are included in the highlighted bar. b) The years are 1975 and 1980. c)

There have been fewer traffic deaths more recently than there were in 1975 and 1980.

How big is your bicep? The distribution of the bicep measurements of 250 men is unimodal and symmetric. Based on the height of the tallest points, about 85 of these 250 men have biceps close to 13 inches around. Most are between 12 and 15 inches around. But there are two as small as 10 inches and several that are 16 inches.

10. How big is your bicep in cm? Yes, the dotplots look different. The plot based on inches has fewer values on the x-axis, so it shows less detail. This plot gives a better picture of the distribution. 11. E-mails. The distribution of the number of emails received from each student by a professor in a large introductory statistics class during an entire term is skewed to the right, with the number of emails ranging from 1 to 21 emails. The distribution is centered at about 2 emails, with many students only sending 1 email. There is one outlier in the distribution, a student who sent 21 emails. The next highest number of emails sent was only 8. 12. Adoptions. a) The distributions of the number of adoptions and population are skewed to the right. Most states have smaller populations and fewer adoptions, but some big states have substantially more of each. b) States with higher populations are likely to have more adoptions. c)

Report the number of adoptions per 100,000 people.

Section 2.3 13. Biceps revisited. The distribution of the bicep measurements of 250 men is mound-shaped and roughly symmetric. 14. E-mails II. The distribution of e-mails sent by students is skewed to the right. 15. Life expectancy. a) The distribution of life expectancies at birth in 190 countries is skewed to the left. b) The distribution of life expectancies at birth in 190 countries has one mode, at about 74 to 76 years. The fluctuations from bar to bar don’t seem to rise to the level of defining additional modes, although opinions can differ. 16. Shoe sizes. a) The distribution is roughly symmetric and possibly bimodal. b) There seem to be two modes, one between sizes 38 and 40 and another between sizes 44 and 46. This could be due to having data for both men and women. The lower mode might be for women and the upper for men. Section 2.4 17. Life expectancy II. a) The distribution of life expectancies at birth in 190 countries is skewed to the left, so the median is expected to be larger than the mean. The mean life expectancy is pulled down toward the tail of the distribution. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Chapter 2 Displaying and Describing Data

17. (continued) b) Since the distribution of life expectancies at birth in 190 countries is skewed to the left, the median is the better choice for reporting the center of the distribution. The median is resistant to the skewed shape of the distribution, so it is a better choice for most summaries. 18. Adoptions II. a) The distribution of number of adoptions is skewed to the right, so the mean is expected to be larger than the median. The mean life expectancy is pulled upwards toward the tail of the distribution. b) Since the distribution of number of adoptions is skewed to the right, the median is the better choice for reporting the center of the distribution. The median is resistant to the skewed shape of the distribution, so it is a better choice for most summaries. 19. How big is your bicep II? Because the distribution of bicep circumferences is unimodal and symmetric, the mean and the median should be very similar. The usual choice is to report the mean or to report both. 20. Shoe sizes II. Because the distribution of shoe sizes has two modes, the mean and median are not helpful in reporting the story that the data tell. It is better to report the locations of the two modes. Section 2.5 21. Life expectancy III. a) We should report the IQR. b) Since the distribution of life expectancies at birth in 190 countries is skewed to the left, the better measure of spread is the IQR. The skewness of the distribution inflates the standard deviation. 22. Adoptions III. a) We should report the IQR. b) Since the distribution of number of adoptions is skewed to the left, the better measure of spread is the IQR. The skewness of the distribution inflates the standard deviation. 23. How big is your bicep III? Because the distribution of bicep circumferences is unimodal and roughly symmetric, we should report the standard deviation. The standard deviation is generally more useful whenever it is appropriate. However, it would not be strictly wrong to use the IQR. We just prefer the standard deviation. 24. Shoe sizes III. The data combine shoe sizes for men and for women. It isn’t appropriate to summarize them as if they were a coherent collection of values. Chapter Exercises 25. Graphs in the news. Answers will vary. 26. Graphs in the news II.

29. Histogram. Answers will vary. 30. Not a histogram.

Answers will vary.

27. Tables in the news.

31. Centers in the news.

Answers will vary.

28. Tables in the news II.

32. Spreads in the news.

Answers will vary.

Part I Exploring and Understanding Data

33. Thinking about shape. a) The distribution of the number of speeding tickets each student in the senior class of a college has ever had is likely to be unimodal and skewed to the right. Most students will have very few speeding tickets (maybe 0 or 1), but a small percentage of students will likely have comparatively many (3 or more?) tickets. b) The distribution of player’s scores at the U.S. Open Golf Tournament would most likely be unimodal and slightly skewed to the right. The best golf players in the game will likely have around the same average score, but some golfers might be off their game and score 15 strokes above the mean. (Remember that high scores are undesirable in the game of golf!) c)

The weights of female babies in a particular hospital over the course of a year will likely have a distribution that is unimodal and symmetric. Most newborns have about the same weight, with some babies weighing more and less than this average. There may be slight skew to the left, since there seems to be a greater likelihood of premature birth (and low birth weight) than post-term birth (and high birth weight).

d) The distribution of the length of the average hair on the heads of students in a large class would likely be bimodal and skewed to the right. The average hair length of the males would be at one mode, and the average hair length of the females would be at the other mode, since women typically have longer hair than men. The distribution would be skewed to the right, since it is not possible to have hair length less than zero, but it is possible to have a variety of lengths of longer hair. 34. More shapes. a) The distribution of the ages of people at a Little league game would likely be bimodal because you have both players and parents. The distribution may also be skewed to the right, since parents’ ages can be higher than the mean more easily than lower. b) The distribution of the number of siblings of people in your class would likely be unimodal and skewed to the right. There are probably many students with 0 or 1 sibling and some with 2 or more. c)

The distribution of pulse rates of college-age males would likely be unimodal and symmetric. It is unusual to have either very high or low pulse rates.

d) The distribution of the number of times each face of a die shows in 100 tosses would likely be uniform. Each face of the die has the same chance of coming up, so the number of times should be about the same for each face. 35. Movie genres again. a) Thriller/Suspense has a higher bar than Adventure, so it is the more common genre. b) It is easy to tell from either chart; sometimes differences are easier to see on the bar chart because slices of the pie chart look too similar in size. 36. Movie ratings again. a) NC-17 has the lowest bar, so it is the least common rating. b) It is easy to tell from either chart; sometimes differences are easier to see on the bar chart because slices of the pie chart look too similar in size. 37. Magnet Schools. There were 1755 qualified applicants for the Houston Independent School District’s magnet schools program. 53% were accepted, 17% were wait-listed, and the other 30% were turned away for lack of space. 38. Magnet schools again. Of the 1755 students who applied for admission to the magnet schools program, 29.5% were black or Hispanic, 16.6% were Asian, and 53.9% were white.

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Chapter 2 Displaying and Describing Data

39. Causes of death 2014. a) Yes, it is reasonable to assume that heart or lung diseases caused approximately 29% of U.S. deaths in 2014, since there is no possibility for overlap. Each person could only have one cause of death. b) Since the percentages listed add up to 61.9%, other causes must account for 38.1% of US deaths. c)

A bar chart is a good choice (with the inclusion of the “Other” category). Since causes of US deaths represent parts of a whole, a pie chart would also be a good display.

40. Plane crashes. a) Yes, it is reasonable to assume that weather or mechanical failures caused approximately 37% of recent plane crashes, since there is no possibility for overlap. Each crash was assigned to only one cause category. b) The data are given to the nearest whole percent. If several of the categories were rounded up, an extra percentage point could be gained. c)

A histogram is shown below. A pie chart would also be an appropriate display.

41. Movie genres once more. a) There are too many categories to construct an appropriate display. In a bar chart, there are too many bars. In a pie chart, there are too many slices. In each case, we run into difficulty trying to display genres that only represented a few movies. b) The creators of the bar chart included a category called “Other” for many of the genres that only occurred a few times.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

42. Summer Olympics 2016. a) There are too many categories to make a meaningful bar chart or pie chart by country. b) It would be better to make a histogram of total medals, as shown below.

43. Global warming. Perhaps the most obvious error is that the percentages in the pie chart add up to 141%, when they should, of course, add up to 100%. This means that survey respondents were allowed to choose more than one response, so a pie chart is not an appropriate display. Furthermore, the three-dimensional perspective view distorts the regions in the graph, violating the area principle. The regions corresponding to “Could reduce global warming but unsure if we will” and “Could reduce global warming but people aren’t willing to so we won’t” look roughly the same size, but at 46% and 30% of respondents, respectively, they should have very different sizes. Always use simple, two-dimensional graphs. Additionally, the graph does not include a title. 44. Modalities. a) The bars have false depth, which can be misleading. This is a bar chart, so the bars should have space between them. From a simple design standpoint, running the labels on the bars one way and the vertical axis labels the other way is awkward. b) The percentages sum to 100%. This is unlikely if the respondents were asked to name three methods each. For example, it would be possible for 80% of respondents to use ice at some time and another 75% to use electric stimulation. This is a case where summing to 100% seems wrong rather than correct. 45. Cereals. a) The distribution of the carbohydrate content of breakfast cereals is bimodal, with a cluster of cereals with carbohydrate content around 13 grams of carbs and another cluster of cereals around 22 grams of carbs. The lower cluster shows a bit of skew to the left. Most cereals in the lower cluster have between 10 and 20 grams of carbs. The upper cluster is symmetric, with cereals in the cluster having between 20 and 24 grams of carbs. b) The cereals with the highest carbohydrate content are Corn Chex, Corn Flakes, Cream of Wheat (Quick), Crispix, Just Right Fruit & Nut, Kix, Nutri-Grain Almond-Raisin, Product 19, Rice Chex, Rice Krispies, Shredded Wheat ‘n’ Bran, Shredded Wheat Spoon Size, Total Corn Flakes, and Triples. 46. Run times. The distribution of run times vis skewed to the right and centered at around 30 to 31 minutes, with most observations between 29 and 32 minutes. The skewness is because it is easier to run much slower than usual but harder to run much faster than usual. Also, over a 10-year period, the runner may have slowed down due to injury or aging. 47. Heart attack stays. a) The distribution of length of stays is skewed to the right, so the mean is larger than the median. b) The distribution of the length of hospital stays of female heart attack patients is bimodal and skewed to the right, with stays ranging from 1 day to 36 days. The distribution is centered around 8 days, with the majority of the hospital stays lasting between 1 and 15 days. There are a relatively few hospital stays longer than 27 days. Many patients have SOURCE: a stay of only one day, possibly because the patient died. Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Chapter 2 Displaying and Describing Data

47. (continued) c)

The median and IQR would be used to summarize the distribution of hospital stays, since the distribution is strongly skewed.

48. Bird species 2013. a)

b) The distribution of the number of bird species is mostly symmetric, with a few values trailing off to the high end. It could be described as skewed right. 49. Super Bowl points 2016. a) The median number of points scored in the first 50 Super Bowl games is 46 points. b) The first quartile (Q1) of the number of points scored in the first 50 Super Bowl games is 37 points. The third quartile (Q3) is 55 points. c)

In the first 50 Super Bowl games, the lowest number of points scored was 21, and the highest number of points scored was 75. The median number of points scored was 46, and the middle 50% of Super Bowls has between 37 and 55 points scored, making the IQR 18 points.

50. Super Bowl edge 2016. a) The median margin is 12.5 points. b) The first quartile is 4 and the third quartile is 19. c)

The distribution of margins is strongly skewed to the right, with a median of 12.5 points. The minimum is 1 and the maximum is 45. The IQR is 15 points.

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Part I Exploring and Understanding Data

51. Test scores, large class. a) The distribution of Calculus test scores is bimodal with one mode at about 62 and one at about 78. The higher mode might be math majors, and the lower mode might be non-math majors. b) Because the distribution of Calculus test scores is bimodal, neither the mean nor the median tells much about a typical score. We should attempt to learn if another variable (such as whether or not the student is a math major) can account for the bimodal character of the distribution. 52. Test scores, small class. The distribution of Calculus test scores appears to be bimodal, with one mode at about 62 and one at about 78. The higher mode might be math majors, and the lower mode might be non-math majors. Because there are so few values in the middle bins, it is not as clear that this distribution is actually bimodal. 53. Mistake. a) As long as the boss’s true salary of $200,000 is still above the median, the median will be correct. The mean will be too large, since the total of all the salaries will decrease by $2,000,000 – $200,000 = $1,800,000, once the mistake is corrected. b) The range will likely be too large. The boss’s salary is probably the maximum, and a lower maximum would lead to a smaller range. The IQR will likely be unaffected, since the new maximum has no effect on the quartiles. The standard deviation will be too large, because the $2,000,000 salary will have a large squared deviation from the mean. 54. Sick days. a) There may be high outliers, or the distribution of sick days may be skewed to the right. The company is using the mean, which is higher because of these features of the distribution. The union negotiators are quoting the median. b) These choices make sense: The company cares about the number of days lost to sickness, so the mean (or total) is what matters. The union looks at individual workers and sees that the typical (median) worker doesn’t miss many days for sickness. 55. Floods 2015. a) The mean annual number of deaths from floods is 81.95. b) In order to find the median and the quartiles, the list must be ordered. 29 38 38 43 48 49 56 68 76 80 82 82 82 86 87 103 113 118 131 136 176 The median annual number of deaths from floods is 82, Q1 is 49 deaths, and Q3 is 103 deaths. (Some statisticians consider the median to be separate from both the lower and upper halves of the ordered list when the list contains an odd number of elements. This changes the position of the quartiles slightly. If median is excluded, Q1 = 48.5 and Q3 = 108. In practice, it rarely matters, since these measures of position are best for large data sets.) c)

The range of the distribution of deaths is Max – Min = 176 – 29 = 147 deaths. The IQR = Q3 – Q1 = 103 – 49 = 54 deaths. (Or, the IQR = 108 – 48.5 = 59.5 deaths, if the median is excluded from both halves of the ordered list.)

56. Tornadoes 2015. a) The mean annual number of deaths from tornadoes is 81.95. b) In order to find the median and the quartiles, the list must be ordered. 21 25 30 35 36 38 40 41 45 47 54 55 55 67 67 70 81 94 126 130 553 The median annual number of deaths from tornadoes is 54, Q1 is 38 deaths, and Q3 is 70 deaths. (Some statisticians consider the median to be separate from both the lower and upper halves of the ordered list when the list contains an odd number of elements. This changes the position of the quartiles slightly. If median is excluded, Q1 is 37 and Q3 is 75.5. In practice, it rarely matters, since these measures of position are best for large data sets.) SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Chapter 2 Displaying and Describing Data

56. (continued) c)

The range of the distribution of deaths is Max – Min = 553 – 21 = 532 deaths. The IQR = Q3 – Q1 = 70 – 38 = 32 deaths. (Or, the IQR = 75.5 – 37 = 38.5 deaths, if the median is excluded from both halves of the ordered list.)

57. Floods 2105 II. The distribution of deaths from floods is slightly skewed to the right and bimodal. There is one mode at about 40 deaths and one at about 80 deaths. There is one extreme value at 180 deaths. 58. Tornadoes 2015 II. The distribution of deaths from tornadoes is slightly skewed to the right, with one extreme outlier at 553. The median is 54 deaths, and the IQR is 32 deaths. 59. Pizza prices. The mean and standard deviation would be used to summarize the distribution of pizza prices, since the distribution is unimodal and symmetric. 60. Neck size. The mean and standard deviation would be used to summarize the distribution of neck sizes, since the distribution is unimodal and symmetric. 61. Pizza prices again. a) The mean pizza price is closest to $2.60. That’s the balancing point of the histogram. b) The standard deviation in pizza prices is closest to $0.15, since that is the typical distance to the mean. There are no pizza prices as far as $0.50 or $1.00. 62. Neck sizes again. a) The mean neck size is closest to 15 inches. That’s the balancing point of the histogram. b) The standard deviation in neck sizes is closest to 1 inch, because a typical value lies about 1 inch from the mean. There are few points as far as 3 inches from the mean and none as far as 5 inches, so those are too large to be the standard deviation. 63. Movie lengths 2010. a) A typical movie would be around 105 minutes long. This is near the center of the unimodal and slightly skewed histogram, with the outlier set aside. b) You would be surprised to find that your movie ran for 150 minutes. Only 3 movies (2%) ran that long. c)

The mean run time would probably be higher, since the distribution of run times is skewed to the right, and also has a high outlier. The mean is pulled towards this tail, while the median is more resistant. However, it is difficult to predict what the effect of the low outlier might be from just looking at the histogram.

64. Golf drives 2015. a) A typical drive length would be around 290 yards. This is near the center of the unimodal and reasonably symmetric histogram. b) About 25% of professional male golfers drive, on average, 280 yards or less. c)

The mean of the distribution is 288.69 yards. Any estimate around 290 is a good guess.

d) The median should be close to the mean because this distribution is symmetric. (In fact, it is 288.7, so they agree to within inches.)

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

65. Movie lengths II 2010. a) i)

The distribution of movie running times is fairly consistent, with the middle 50% of running times between 98 and 116 minutes. The interquartile range is 18 minutes.

ii) The standard deviation of the distribution of movie running times is 16.6 minutes, which indicates that movies typically varied from the mean running time by 16.6 minutes. b) Since the distribution of movie running times is skewed to the right and contains an outlier, the standard deviation is a poor choice of numerical summary for the spread since it may have an inflated value. The interquartile range is better, since it is resistant to outliers. 66. Golf drives II 2015. a) i)

The distribution of drive lengths is fairly consistent, with the middle 50% of golfers had driving averages between 282 and 294.5 yards.

ii) The standard deviation of the distribution of drive lengths is 9.8 yards, which indicates that drive lengths typically varied from the mean drive length by 9.8 yards. b) The distribution of drive lengths is symmetric, so either is appropriate. 67. Movie earnings 2015. The industry publication is using the median, while the watchdog group is using the mean. It is likely that the mean is pulled higher by a few very expensive movies. 68. Cold weather. a) The mean low temperature would probably be lower, since the distribution of low temperature is skewed to the left. The mean is pulled towards this tail, while the median is more resistant. b) The standard deviation will be higher since the data may have an outlier. The IQR won’t change, since it is resistant to outliers. 69. Gasoline 2014. a)

b) The distribution of gas prices is bimodal, with two clusters, one centered around $3.25 per gallon, and another centered around $3.45 per gallon. The lowest and highest prices were $3.11 and $3.46 per gallon. c)

There is a gap in the distribution of gasoline prices. There were no stations that charged between $3.28 and $3.39.

SOURCE: Browsegrades.net

Chapter 2 Displaying and Describing Data

70. The great one. a)

b) The distribution of games played by Gretzky us unimodal and skewed to the left. c)

Typically, Gretzky played about 80 games per season. The numbers of games played are tightly clustered in the upper 70s and low 80s.

d) Two seasons are low outliers, when Gretzky played fewer than 50 games. He might have been injured. 71. States. a) There are 50 entries in the stemplot, so the median must be between the 25th and 26th population values. Counting in the ordered stemplot gives median = 4.5 million people. The middle of the lower 50% of the list (25 state populations) is the 13th population, or 2 million people. The middle of the upper half of the list (25 state populations) is the 13th population from the top, or 7 million people. The IQR = Q3 – Q1 = 7 – 2 = 5 million people. b) The distribution of population for the 50 U.S. States is unimodal and skewed heavily to the right. The median population is 4.5 million people, with 50% of states having populations between 2 and 7 million people. There are two outliers, a state with 37 million people, and a state with 25 million people. The next highest population is only 19 million. 72. Wayne Gretzky. a) Since the distribution of number of games played is highly skewed to the left, the median is the better choice for reporting the center of the distribution. The median is resistant to the skewed shape of the distribution, so it is a better choice for most summaries. b) The median is 79 games played. c)

The mean should be lower. There are two seasons when Gretzky played an unusually low number of games. Those will pull the mean down.

73. A-Rod 2016. The distribution of the number of homeruns hit by Alex Rodriguez during the 1994 – 2016 seasons is reasonably symmetric, with the exception of a second mode around 10 homeruns. A typical number of homeruns per season was in the high 30s to low 40s. With the exception of 5 seasons in which A-Rod hit 0, 0, 5, 7, and 9 homeruns, his total number of homeruns per season was between 16 and the maximum of 57.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

74. Major hurricanes 2013. a)

b) The distribution of the yearly number of hurricanes appears to be unimodal and slightly skewed to the right, with no outliers. The mode is near 2. There is possibly a second mode near 5. 75. A-Rod again 2016. a) This is not a histogram. The horizontal axis should contain the number of home runs per year, split into bins of a convenient width. The vertical axis should show the frequency; that is, the number of years in which A-Rod hit a number of home runs within the interval of each bin. The display shown is a bar chart/time plot hybrid that simply displays the data table visually. It is of no use in describing the shape, center, spread, or unusual features of the distribution of home runs hit per year by A-Rod. b) The histogram is shown below.

76. Return of the birds 2013. a) This is not a histogram. The horizontal axis should split the number of counts from each site into bins. The vertical axis should show the number of sites in each bin. b)

77. Acid rain. a) The distribution of the pH readings of water samples in Allegheny County, Pennsylvania is skewed to the right and possibly bimodal. A roughly uniform cluster is centered around a pH of 4.4. This cluster ranges from pH of 4.1 to 4.9. Another smaller, tightly packed cluster is centered around a pH of 5.6. Two readings in the middle seem to belong to neither cluster.

SOURCE: Browsegrades.net

Chapter 2 Displaying and Describing Data

77. (continued)

b) The cluster of high outliers contains many dates that were holidays in 1973. Traffic patterns would probably be different then, which might account for the difference. 78. Marijuana 2015. The distribution of the percentages of 16-year-olds who had used marijuana is unimodal (or possibly bimodal) with center around 18%, but with France, Monaco, and the Czech Republic reporting values near 40%.

79. Final grades. The width of the bars is much too wide to be of much use. The distribution of grades is skewed to the left, but not much more information can be gathered. 80. Final grades revisited. a) The histogram bars are too narrow to show the shape of the distribution. b) The distribution of final grades is skewed to the left, with a mode near 170, and several outliers below 100. The distribution is fairly tightly clustered except for outliers. 81. Zip codes. Even though zip codes are numbers, they are not quantitative in nature. Zip codes are categories. A histogram is not an appropriate display for categorical data. The histogram the Holes R Us staff member displayed doesn’t take into account that some 5-digit numbers do not correspond to zip codes or that zip codes falling into the same classes may not even represent similar cities or towns. The employee could design a better display by constructing a bar chart that groups together zip codes representing areas with similar demographics and geographic locations. 82. ZIP codes revisited. They can’t tell very much because ZIP codes are categorical. However, there is some information in the first digit of ZIP codes. It indicates a general east (0–1) to west (8–9) direction. So, the distribution shows that a large portion of the sales occur in the West with another in the 32000 area. A bar chart of the first digits might be a better display of this. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Part I Exploring and Understanding Data

83. Math scores 2013. a) Median: 285 Mean: 284.36

IQR: 9 Standard deviation: 6.84

b) Since the distribution of Math scores is skewed to the left, it is probably better to report the median and IQR. c)

The distribution of average math achievement scores for eighth graders in the United States is skewed to the left, and roughly unimodal. The distribution is centered at 285. Scores range from 269 to 301, with the middle 50% of the scores falling between 280 and 289.

84. Boomtowns 2015. a)

b) The mean growth index is 58.26% and the median growth index is 61.7%. With the bimodal distribution, it is difficult to predict the effect on the mean. c)

It might be more informative to report one mode at about 33% and the other at about 80%.

d) The IQR is 45.1 and the standard deviation is 23.85. e)

Neither measure is ideal. It might be more informative to discuss a measure of spread for each mode.

The mean and median would be 49.23 percentage points lower. The standard deviation and IQR would not change.

g) The median and IQR won’t change very much. The middle value and the two quartiles will shift at most one data value. The mean and standard deviation will decrease. h) Answers will vary but should include a discussion of the distribution and what it tells us about growth indices. 85. Population growth 2010. The histogram shows that the distribution of Percent Change is unimodal and skewed to the right. The states vary from a minimum of –0.6% (Michigan) to 35.1% (Nevada) growth in the decade. The median was 7.8% and half of the states had growth between 4.3% and 14.1%. Not surprisingly, the top three states in terms of growth were all in the West: Nevada (35.1%), Arizona (24.6%), and Utah (23.8%).

86. Student survey. Answers will vary. The best answers will be about students and their responses, not about numbers and displays. Write what you learn about students, not about what your display looks like. Do not give the value of a summary statistic without a discussion of what it might mean. SOURCE: Browsegrades.net

Chapter 3 Relationships Between Categorical Variables – Contingency Tables

Chapter 3 – Relationships Between Categorical Variables – Contingency Tables Section 3.1 1.

College value? US Adults Presidents Total

Poor 321 32 353

Only Fair 900 222 1122

Good 750 622 1372

Excellent 107 179 286

DK/NA 64 0 64

Total 2142 1055 3197

a) The percent of college presidents who think that higher education provides a poor value is 32/1055  3%. b)

(750  107) / 2142  857 / 2142  40% of U.S. adults think that the value provided by the higher education system is either good or excellent. (622  179) / 1055  801/ 1055  75.9% of college presidents think that the value provided by the higher education system is either good or excellent.

15% of U.S. adults view college as a poor value, but only 3% of college presidents do. Similarly, U.S. adults are twice as likely to view college as an only fair value compared to the presidents (42% to 21%). Presidents are much more likely (76%) to rate colleges as a good or excellent value compared to U.S. adults (only 40%). So, in short, college presidents have a much higher opinion of the value of college than U.S. adults do.

d) In this random sample of U.S. adults, 107/2142  5.00% responded that college provides an excellent value. This is only an estimate, however, of the percentage of all U.S. adults who feel the same way. The percentage of all U.S. adults is probably close to, but not exactly 5%. 2.

Cyber comparison shopping. Yes No Total

<$30K 207 625 832

$30K – $49.9K 115 406 521

$50K – $74.9K 134 260 394

>$75K 204 417 621

Total 660 1708 2368

a) 207/832  24.9% of those earning less than $30K cyber comparison shop. 204/621  32.9% of those earning more than $75K cyber comparison shop. b) 207/660  31.4% of those who cyber comparison shop earn less than $30K. c)

660/2368  28% of those in this random sample of people who owned cell phones use their phones to cyber comparison shop. First of all, not all shoppers have cell phones, so we wouldn’t expect the percentage of all shoppers to be similar to the percentage found in this sample. Even if we were only concerned with all shoppers with phones, we would only expect the percentage of all shoppers to be reasonably close to, but not exactly, 28%.

Section 3.2 3.

College value again. a) The conditional distribution of college presidents’ opinions about the value of a college education is 3% poor; 21% only fair; 59% good; 17% excellent b) Omitting the 64 DK/NAs, the conditional distribution of the opinions U.S. adults about the value of a college education is 15% negative; 43% middle; 41% positive.

Cyber comparison shopping again. a) The conditional distribution of income for those who do not cyber compare prices is 37% <$30K; 24% $30K$49.9K; 15% $50K$74.9K; 24% >$75K. b) The conditional distribution of income for those who do cyber compare prices is 31% <$30K; 17% $30K$49.9K; 20% $50K$74.9K; 31% >$75K. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Part I Exploring and Understanding Data

Section 3.3 5.

Diet and politics. a) The distribution of political alignment among carnivores is about 25% conservative, 40% moderate, and 35% liberal. Omnivores were generally less conservative and more liberal, with about 10% conservative, 35% moderate, and 55% liberal. Vegetarians were even more liberal than the other groups, with 0% conservative, 30% moderate, and 70% liberal. b) The differences are quite large. There appears to be a strong association between diet preference and political alignment.

Diet and politics revisited. a) The distribution of diet preference among liberals is approximately 10% carnivore, 75% omnivore, and 15% vegetarian. Moderates tend to be slightly more carnivorous, with approximately 18% carnivores, 73% omnivores, and 9% vegetarians. Conservatives tend to be even more carnivorous, with 35% carnivores, 65% omnivores, and 0% vegetarians. b) Yes, the differences are real. It does appear there is a strong association between diet preference and political alignment.

Fish and prostate cancer revisited a) Looking at the horizontal axis only, approximately 1% of the men had prostate cancer. b) There are more men who didn’t have cancer and never or seldom ate fish. The rectangles are approximately the same height, but the bar for “no cancer” is much wider. c)

The percentage of men who never/seldom ate fish is lower in the group with no cancer than in the group with cancer. Disregard the width, and look only at the height to compare the conditional distribution of fish consumption within each cancer group. The bar for “never/seldom” is slightly shorter within the “no cancer” group.

College value? revisited. a) Approximately 33% (or one-third) of the respondents were college presidents. Look only at the horizontal axis to estimate this percentage. b) No, the excellence regions appear to have approximately the same areas, but because the U.S. adults region is so much wider, the regions are difficult to compare. c)

Yes, it is clear that a greater percentage of college presidents said that college provides an excellent value, when compared to the percentage of U.S. adults who said the same thing. The bar for college presidents is much taller.

d) The differences seem quite large. It does appear there is a strong association between the responses and whether the respondent was a college president. College presidents were much more likely to say that college provides an excellent or good value, and less likely to say that college provides a poor value than U.S. adults in general. Section 3.4 9.

Diet and politics III. Men Liberal Moderate Conservative Total

Carnivore 9 12 9 30

Omnivore 74 54 14 142

Vegetarian 5 1 0 6

SOURCE: Browsegrades.net

Total 88 67 23 178

Chapter 3 Relationships Between Categorical Variables – Contingency Tables 9.

(continued) Women Liberal Moderate Conservative Total

Carnivore 4 4 1 9

Omnivore 53 27 4 84

Vegetarian 12 6 0 18

Total 69 37 5 111

a) Men are more likely to be conservative carnivores. 9/178  5.1% of the men are conservative carnivores, while only 1/111  0.9% of the women are conservative carnivores. b) Liberal vegetarians are more likely to be women. Of the 17 liberal vegetarians, 12 of them are women. 12/17  70% of liberal vegetarians are women. 10. Being successful. a) The table contains column percentages, so we can add percentages within each column. 18% + 48% = 66% of young women consider it very important or one of the most important things for them to be personally successful. b) The percentage is lower for men. 11% + 47% = 58% of young men consider it very important or one of the most important things for them to be personally successful. c)

We cannot determine what percent of all women (from both age groups) feel this way, since we don’t have the sample sizes from each age group, only the percentages.

d) Young women are slightly more likely than young men (66% compared to 58%) to consider financial success important. Young men are slightly more likely to view this as somewhat important or not important. Chapter Exercises 11. Movie genres and ratings. a) 452/1529  29.6% of the films were rated R. b) 124/1529  8.1% of the films were R-rated comedies. c)

124/452  27.4% of the R-rated films were comedies.

d) 124/312  39.7% of the comedies were rated R. 12. Not the labor force. a) 2207/12872  17.1% of the unemployed were available to work now. b) 1048/12872  8.1% of the unemployed were available to work now and aged 25 to 54 years. c)

208/4158  5.0% of the unemployed 16- to 24-year-olds were in school or training.

d) 4158/12872  32.3% of the unemployed were aged 16 to 24 years. 13. Tables in the news. Answers will vary. 14. Graphs in the news. Answers will vary. 15. Poverty and region 2012. The differences in poverty are not huge, but they may be real. The Northeast and Midwest have the lowest percentages of people living below the poverty level: 12.7% and 13.7%, respectively. In the West, 15.4% live below the poverty level, and the South has the highest rate at 16.8%. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Part I Exploring and Understanding Data

16. Moviegoers and ethnicity. a) Caucasians are 66% of the population but only 56% of ticket buyers. Hispanics are only 16% of the population, but they are 26% of ticket buyers. African Americans are more consistent: 12% of the population and 11% of the moviegoers. b) There are ethnic differences among moviegoers, but they are not all that pronounced. 17. Death from the sky. a) 100 – 60 – 30 – 1 – 0.2 – 0.17 = 8.63% of estimated deaths are attributed to causes not listed here. b) Regardless of the type of display chosen, it is difficult to display causes of death with percentages as low as 0.2% and 0.17%. A display of the two largest values would not show the two smallest ones. 18. Cartoons. a) Without a fun item: 61.4%, 59.1%, 61.4%, 17.0%, 22.7% With fun item: 48.8%, 52.5%, 55.0%, 20.0%, 27.5% Using relative frequencies makes these easier to compare, since the number of students in each group is different. b) There is no evidence that cartoons help with understanding drawing a sample, the meaning of “random”, or mean versus median, since the percentage of students answering the item correctly was higher in the group without the fun item. The percentage of students correctly answering the question about categorical versus quantitative variables was higher for the group with the fun item, but not significantly higher. There appears to be some evidence that students learn better using a fun item when studying the inadequacy of mean without standard deviation. The percentage of students answering the question correct in the fun item group was 4.8 percentage points higher than in the group with no fun item, 27.5% to 22.5%. However, with only 27.5% accuracy, it can’t be said that they mastered the topic. 19. Smoking. a) The smoking rate for 18-24-year-old men was 42.1% in 1974. b) From 1974 to 2014, the smoking rate for 18-24-year-old men dropped from 42.1% to 18.5% c)

Men who were 18-24 years old in 1974 are in the 35-44 age group in 1994, the 45-54 age group in 2004, and the 2014. The smoking rate for this cohort has been decreasing through the years, from 42.1% to 31.4% to 26.5% to 19.9%. Although we don’t have data on deaths in this table, it may very well be that the smokers have a higher death rate than the non-smokers, so this decrease doesn’t necessarily mean that men in this cohort are quitting smoking.

20. Smoking women. a) In each year, the women had a lower smoking rate than the men. Women had a 18-24-year-old smoking rate of 34.0% in 1974, compared to the 42.1% rate for men of the same age. The 18-24-year-old female smoking rates decline slowly over the years, and by 2014, the smoking rate for women is 16.5%, only 2 percentage points lower than the men. b) Smokers are prone to lung cancer and other diseases. The smokers are less likely to live into old age and those left are less likely to be smokers. 21. Mothers and fathers 1965-2011. a) Fathers spend the more of their time on paid work, while mothers spend more time on child care and house work. b) The time fathers spend on paid work has decreased, and the time they spend on child care and housework has increased. For mothers, the number of hours spent on paid work has significantly increased, and they have also increased their time spent on child care while reducing housework time. c)

Parents are spending more time on child care and paid work (13 hours to 21 hours and 50 hours to 58 hours). The time spent on housework has decreased from 36 hours to 28 hours. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Chapter 3 Relationships Between Categorical Variables – Contingency Tables

21. (continued) d) Overall, parents in 2011 reported spending more time total on these tasks, a total of 107 hours in 2011 compared to 99 hours in 1965. Mothers increased their total working time by 3 hours, from 50 hours to 53 hours, while fathers increased their total working time by 5 hours, from 49 hours to 54 hours. 22. Mothers’ and fathers’ aspirations. a) The actual percentages for working full-time differ the most, with 84% of fathers and 48% of mothers working full-time. This is a difference of 36 percentage points. b) 48% of mothers work full-time, while only 27% desire to work full-time. c)

Only 7% of fathers stay at home, compared to 30% of mothers.

23. Teen smokers. According to the Monitoring the Future study, teen smoking brand preferences differ somewhat by region. Although Marlboro is the most popular brand in each region, with about 58% of teen smokers preferring this brand in each region, teen smokers from the South prefer Newports at a higher percentage than teen smokers from the West, 22.5% to approximately 10%, respectively. Camels are more popular in the West, with 9.5% of teen smokers preferring this brand, compared to only 3.3% in the South. Teen smokers in the West are also more likely to have no particular brand than teen smokers in the South. 12.9% of teen smokers in the West have no particular brand, compared to only 6.7% in the South. Both regions have about 9% of teen smokers that prefer one of over 20 other brands. 24. Being successful revisited. a) The four groups are approximately the same size, since the columns are approximately the same width. b) Age is the more important factor. The male/female distributions are more similar to each another within each age group than the male/male and female/female distributions are across age groups. The older group is more likely to respond Not important; the younger more likely to respond Very important but not the most. c)

Women aged 18–34 are most likely to respond that being successful is the most important.

25. Diet and politics IV. a) There are more men in the survey. The male columns are generally wider than the female columns. b) We can’t compare the genders within each category of political ideology since the sample sizes differ. We can, however, note that the male bars are narrowest in the Liberal category, and widest in the Conservative category, indicating that a small share of liberals and a large share of conservatives among the men. The women show the opposite association. The widest female bar is Liberal, while the narrowest is Conservative. In other words, there is an association between gender and political ideology. Males tend to be more conservative and females tend to be more liberal. c)

There is an association between politics and diet. Conservatives are more likely to be carnivores, while liberals are more likely to be vegetarians.

d) The association between politics and diet seems to differ between men and women. Differences in vegetarianism across political ideology is more pronounces in females than in males. Differences in carnivorous eating habits across political ideology is more pronounced in males than females. 26. Handguns. 76.4% of handguns involved in Milwaukee buyback programs are small caliber, while only 20.3% of homicides are committed with small caliber handguns. Along the same lines, only 19.3% of buyback handguns are of medium caliber, while 54.7% of homicides involve medium caliber handguns. A similar disparity is seen in large caliber handguns. Only 2.1% of buyback handguns are large caliber, but this caliber is used in 10.8% of homicides. Finally, 2.2% of buyback handguns are of other calibers, while 14.2% of homicides are committed with handguns of other calibers. Generally, the handguns that are involved in buyback programs are not the same caliber as handguns used in homicides in Milwaukee. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Part I Exploring and Understanding Data

27. Job satisfaction. a) This is a table of column percents. The columns add up to 100%, while the rows do not. b) i)

This can’t be found from the table. We don’t know what the percent of respondents who are very satisfied.

ii) This can’t be found from the table. We don’t know what the percent of respondents who are dissatisfied. iii) 39% of respondents who are dissatisfied with their current job are actually better off than their parents were at the same age. iv) This can’t be found from the table. We don’t know what the percent of respondents who are very satisfied. 28. A sense of identity. a) These are row percents. The rows add up to nearly 100%, while the columns do not. The differences from 100% are too large to be attributed to rounding error, so it is likely that some respondents replied something else other than the provided responses. b) i)

This can’t be found from the table. We don’t know what the percent of respondents who are selfemployed.

ii) 65% of respondents who are employed by non-profits get a sense of identity from their job. iii) This can’t be found from the table. We don’t know what the percent of respondents who think of their job as just something to do for a living. iv) This can’t be found from the table. We don’t know what the percent of respondents who get a sense of identity from their job. 29. Seniors. a) A table with marginal totals is shown below. There are 268 White graduates and 325 total graduates. 268/325 ≈ 82.5% of the graduates are white. Plans 4-year college 2-year college Military Employment Other

White 198 36 4 14 16

Minority 44 6 1 3 3

TOTAL 242 42 5 17 19

TOTAL

268

325

b) There are 42 graduates planning to attend 2-year colleges. 42/325 ≈ 12.9% c)

36 white graduates are planning to attend 2-year colleges. 36/325 ≈ 11.1%

d) 36 white graduates are planning to attend 2-year colleges and there are 268 white graduates. 36/268 ≈ 13.4% e)

There are 42 graduates planning to attend 2-year colleges, and 36 of them are white. 36/42 ≈ 85.7%

30. Politics. a) There are 192 students taking Intro Stats. Of those, 115, or about 59.9%, are male. b) There are 192 students taking Intro Stats. Of those, 27, or about 14.1%, consider themselves to be “Conservative”. c)

Chapter 3 Relationships Between Categorical Variables – Contingency Tables

30. (continued) d) There are 192 students taking Intro Stats. Of those, 21, or about 10.9%, are males who consider themselves to be “Conservative”. 31. Movies 06-15. a) This is a table of column percents. The columns add up to 100%, while the rows do not. b) Movies rated G and PG have become slightly less common, while movies rated PG-13 and R have become slightly more common. c)

For Dramas, the percentages of PG and R have decreased while the percentage of PG-13 dramas has increased significantly. For Comedies, there has been a large increase in the percentage of R-rated films.

32. Minimum wage workers. a) 7701/37972  20.3% of the woman are aged 16-24. b) The number of workers at or below minimum wage is highest for each sex at 16-24 years, with 384,000 male workers and 738,000 female workers. The number of workers at or below minimum wage decreases as the age increases for each sex, with only 22,000 male and 50,000 female workers. At each age category, the number of male workers at or below minimum wage is roughly half the number of female workers.

33. More about seniors. a) For white students, 73.9% plan to attend a 4-year college, 13.4% plan to attend a 2-year college, 1.5% plan on the military, 5.2% plan to be employed, and 6.0% have other plans. b) For minority students, 77.2% plan to attend a 4-year college, 10.5% plan to attend a 2-year college, 1.8% plan on the military, 5.3% plan to be employed, and 5.3% have other plans. c)

A side-by-side bar chart is a good display of these data.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

33. (continued) d) The conditional distributions of plans for whites and minorities are similar: White – 74% 4-year college, 13% 2-year college, 2% military, 5% employment, 6% other. Minority – 77% 4-year college, 11% 2-year college, 2% military, 5% employment, 5% other. Caution should be used with the percentages for Minority graduates, because the total is so small. Each graduate is almost 2%. Still, the conditional distributions of plans are essentially the same for the two groups. There is little evidence of an association between race and plans for after graduation. 34. Politics revisited. a) The females in this course were 45.5% Liberal, 46.8% Moderate, and 7.8% Conservative. b) The males in this course were 43.5% Liberal, 38.3% Moderate, and 18.3% Conservative. c)

A side-by-side bar chart comparing the distributions is shown below.

d) Politics and sex do not appear to be independent in this course. Although about the same fraction of males and females classified themselves as politically Liberal, females were slightly more likely to be Moderate. A much higher percentage of males (18%) considered themselves to be Conservative than did the females (8%). 35. Magnet schools revisited. a) There were 1755 qualified applicants to the Houston Independent School District’s magnet schools program. Of those, 292, or about 16.6% were Asian. b) There were 931 students accepted to the magnet schools program. Of those, 110, or about 11.8% were Asian. c)

There were 292 Asian applicants. Of those, 110, or about 37.7%, were accepted.

d) There were 1755 total applicants. Of those, 931, or about 53%, were accepted. 36. More politics. a)

b) The overall percentage of females in the class is about 40%. That’s true among Liberals and Moderates, but the percentage of females among Conservatives is only about half that. This suggests that sex and politics are not independent. SOURCE: Browsegrades.net

Chapter 3 Relationships Between Categorical Variables – Contingency Tables

37. Back to school. There were 1755 qualified applicants for admission to the magnet schools program. 53% were accepted, 17% were wait-listed, and the other 30% were turned away. While the overall acceptance rate was 53%, 93.8% of Blacks and Hispanics were accepted, compared to only 37.7% of Asians, and 35.5% of whites. Overall, 29.5% of applicants were Black or Hispanics, but only 6% of those turned away were Black or Hispanic. Asians accounted for 16.6% of applicants, but 25.3% of those turned away. It appears that the admissions decisions were not independent of the applicant’s ethnicity. 38. Parking lots. a) In order to get percentages, first we need totals. Here is the same table, with row and column totals. Foreign cars are defined as non-American. There are 45 + 102 = 147 non-American cars or 147/359 ≈ 40.95%. Origin American European Asian Total

Driver Student Staff 107 105 33 12 55 47 195

164

Total 212 45 102 359

b) There are 212 American cars of which 107 or 107/212 ≈ 50.47% were owned by students. c)

There are 195 students of whom 107 or 107/195 ≈ 54.87% owned American cars.

d) The marginal distribution of Origin is displayed in the third column of the table at the right: 59% American, 13% European, and 28% Asian.

Origin American European Asian

Totals 212 45 102

Percentages 59.05% 12.53% 28.41%

Total

359

99.99%

The conditional distributions are shown below. Student Origin distribution Origin American European Asian Total

Staff Origin distribution

Driver Student Percentage 107 54.87% 33 16.92% 55 28.21% 195

100.00%

Origin American European Asian

Staff 105 12 47

Driver Percentage 64.02% 7.32% 28.66%

Total

164

100.00%

No, the marginal distributions look slightly different. A higher percentage of staff drive American cars, 64% compared to about 55% among students. Bar charts or pie charts could be used to compare.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

39. Weather forecasts.

Forecast

Actual Weather Rain No Rain

Rain 27 7

No Rain 63 268

Total

331

Total 90 275 365

a) The table above shows the marginal totals. It rained on 34 of 365 days, or 9.3% of the days. b) Rain was predicted on 90 of 365 days, or 90/365 ≈ 24.7% of the days. c)

The forecast of Rain was correct on 27 of the days it actually rained and the forecast of No Rain was correct on 268 of the days it didn’t rain. So, the forecast was correct a total of 295 times. 295/365 ≈ 80.8% of the days.

d) On rainy days, rain had been predicted 27 out of 34 times (79.4%). On days when it did not rain, forecasters were correct in their predictions 268 out of 331 times (81.0%). These two percentages are very close. There is no evidence of an association between the type of weather and the ability of the forecasters to make an accurate prediction.

40. Twin births. a) Of the 278,000 mothers who had twins in 1995–1997, 63,000 had inadequate health care during their pregnancies, or 63,000/278,000 = 22.7%. b) There were 76,000 induced or Caesarean births and 71,000 preterm births without these procedures, or (76,000 + 71,000)/278,000 = 52.9%. c)

Among the mothers who did not receive adequate medical care, there were 12,000 induced or Caesarean births and 13,000 preterm births without these procedures. 63,000 mothers of twins did not receive adequate medical care, or (12,000 + 13,000)/63,000 = 39.7%

d) A segmented bar chart is shown below.

SOURCE: Browsegrades.net

Chapter 3 Relationships Between Categorical Variables – Contingency Tables

40. (continued) e)

52.9% of all twin births were preterm, while only 39.7% of births in which inadequate medical care was received were preterm. This is evidence of an association between level of prenatal care and twin birth outcome. If these variables were independent, we would expect the percentages to be roughly the same. Generally, those mothers who received adequate medical care were more likely to have preterm births than mothers who received intensive medical care, who were in turn more likely to have preterm births than mothers who received inadequate health care. This does not imply that mothers should receive inadequate health care do decrease their chances of having a preterm birth, since it is likely that women that have some complication during their pregnancy (that might lead to a preterm birth), would seek intensive or adequate prenatal care.

41. Blood pressure. a) The marginal distribution of blood pressure for the employees of the company is the total column of the table, converted to percentages. 20% low, 49% normal and 31% high blood pressure. Blood pressure low normal high

under 30 27 48 23

30–49 37 91 51

over 50 31 93 73

Total 95 232 147

179

197

474

Total

b) The conditional distribution of blood pressure within each age category is shown below. Blood pressure low normal high c)

under 30 27.6% 49.0% 23.5%

30–49 20.7% 50.8% 28.5%

over 50 15.7% 47.2% 37.1%

A segmented bar chart of the conditional distributions of blood pressure by age category is shown below.

d) In this company, as age increases, the percentage of employees with low blood pressure decreases, and the percentage of employees with high blood pressure increases. e)

No, this does not prove that people’s blood pressure increases as they age. Generally, an association between two variables does not imply a cause-and-effect relationship. Specifically, these data come from only one company and cannot be applied to all people. Furthermore, there may be some other variable that is linked to both age and blood pressure. Only a controlled experiment can isolate the relationship between age and blood pressure.

42. Obesity and exercise. a) Participants were categorized as Normal, Overweight or Obese, according to their Body Mass Index. Within each classification of BMI (column), participants self-reported exercise levels. Therefore, these are column percentages. The percentages sum to 100% in each column, not across each row.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

42. (continued) b) Pie charts of the conditional distributions of level of physical activity by Body Mass Index category are shown below.

No, even though the graphical displays provide strong evidence that lack of exercise and BMI are not independent. All three BMI categories have nearly the same percentage of subjects who report “Regular, not intense” or “Irregularly active”, but as we move from Normal to Overweight to Obese we see a decrease in the percentage of subjects who report “Regular, intense” physical activity (16.8% to 14.2% to 9.1%), while the percentage of subjects who report themselves as “Inactive” increases. While it may seem logical that lack of exercise causes obesity, association between variables does not imply a cause-and-effect relationship. A lurking variable (for example, overall health) might influence both BMI and level of physical activity, or perhaps lack of exercise is caused by obesity. Only a controlled experiment could isolate the relationship between BMI and level of physically activity.

43. Anorexia. These data provide no evidence that Prozac might be helpful in treating anorexia. Of the patients who took Prozac, 71.4% were diagnosed as “Healthy” (or 28.6% relapsed), while about 72.7% of the patients who took a placebo were diagnosed as “Healthy” (or 27.3% relapsed). Even though the percentage was higher for the placebo patients, this does not mean that Prozac is hurting patients. The difference between 71% and 73% is not likely to be statistically significant. 44. Antidepressants and bone fractures. These data provide evidence that taking a certain class of antidepressants (SSRI) might be associated with a greater risk of bone fractures. Approximately 10% of the patients taking this class of antidepressants experience bone fractures. This is compared to only approximately 5% in the group that were not taking the antidepressants. 45. Driver’s licenses 2014. a) There are 8.5 million drivers under 20 and a total of 214.1 million drivers in the U.S. That’s about 4.0% of U.S. drivers under 20. b) There are 105.9 million males out of 214.1 million total U.S. drivers, or about 49.5%. c)

Each age category appears to have about 50% male and 50% female drivers. At younger ages, males form the slight majority of drivers. The percentage of male drivers shrinks until female drivers hold a slight majority, among older drivers.

d) There appears to be a slight association between age and gender of U.S. drivers. Younger drivers are slightly more likely to be male, and older drivers are slightly more likely to be female. 46. Tattoos. The study by the University of Texas Southwestern Medical Center provides evidence of a strong association between having a tattoo and contracting hepatitis C. Around 33% of the subjects who were tattooed in a commercial parlor had hepatitis C, compared with 13% of those tattooed elsewhere, and only 3.5% of those with no tattoo. If having a tattoo and having hepatitis C were independent, we would have expected these percentages to be roughly the same.

SOURCE: Browsegrades.net

Chapter 3 Relationships Between Categorical Variables – Contingency Tables

46. (continued)

47. Diet and politics shuffled. a) The upper right bar chart shows the original data. b) It’s clear that shuffled data did not produce associations as obvious as the original data. The randomly scrambled data look different from the original data, and this supports the belief that there is an association between the variables. 48. College values shuffled. a) The bottom left mosaic plot shows the original data. b) It’s clear that none of the shuffled data produced associations as strong as the original data. All the mosaic plots from the randomly scrambled data look different from the original data, and this supports the belief that there is an association between the variables. 49. Hospitals.

Procedure

a) The marginal totals have been added to the table:

Major surgery Minor surgery Total

Large Hospital 120 of 800 10 of 200

Discharge delayed Small Hospital 10 of 50 20 of 250

130 of 1000

30 of 300

Total 130 of 850 30 of 450 160 of 1300

160 of 1300, or about 12.3% of the patients had a delayed discharge. b) Yes. Major surgery patients were delayed 130 of 850 times, or about 15.3% of the time. Minor Surgery patients were delayed 30 of 450 times, or about 6.7% of the time. c)

Large Hospital had a delay rate of 130 of 1000, or 13%. Small Hospital had a delay rate of 30 of 300, or 10%. The small hospital has the lower overall rate of delayed discharge.

d) Large Hospital: Major Surgery 15% delayed and Minor Surgery 5% delayed. Small Hospital: Major Surgery 20% delayed and Minor Surgery 8% delayed. Even though small hospital had the lower overall rate of delayed discharge, the large hospital had a lower rate of delayed discharge for each type of surgery. e)

No. While the overall rate of delayed discharge is lower for the small hospital, the large hospital did better with both major surgery and minor surgery.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

49. (continued) f)

The small hospital performs a higher percentage of minor surgeries than major surgeries. 250 of 300 surgeries at the small hospital were minor (83.3%). Only 200 of the large hospital’s 1000 surgeries were minor (20%). Minor surgery had a lower delay rate than major surgery (6.7% to 15.3%), so the small hospital’s overall rate was artificially inflated. Simply put, it is a mistake to look at the overall percentages. The real truth is found by looking at the rates after the information is broken down by type of surgery, since the delay rates for each type of surgery are so different. The larger hospital is the better hospital when comparing discharge delay rates.

50. Delivery service. a) Pack Rats has delivered a total of 28 late packages (12 Regular + 16 Overnight), out of a total of 500 deliveries (400 Regular + 100 Overnight). 28/500 = 5.6% of the packages are late. Boxes R Us has delivered a total of 30 late packages (2 Regular + 28 Overnight) out of a total of 500 deliveries (100 Regular + 400 Overnight). 30/500 = 6% of the packages are late. b) The company should have hired Boxes R Us instead of Pack Rats. Boxes R Us only delivers 2% (2 out of 100) of its Regular packages late, compared to Pack Rats, who deliver 3% (12 out of 400) of its Regular packages late. Additionally, Boxes R Us only delivers 7% (28 out of 400) of its Overnight packages late, compared to Pack Rats, who delivers 16% of its Overnight packages late. Boxes R Us is better at delivering Regular and Overnight packages. c)

This is an instance of Simpson’s Paradox, because the overall late delivery rates are unfair averages. Boxes R Us delivers a greater percentage of its packages Overnight, where it is comparatively harder to deliver on time. Pack Rats delivers many Regular packages, where it is easier to make an on-time delivery.

51. Graduate admissions. a) 1284 applicants were admitted out of a total of 3014 applicants and 1284/3014 = 42.6%. Program 1 2 3 4

Males Accepted (of applicants) 511 of 825 352 of 560 137 of 407 22 of 373

Females Accepted (of applicants) 89 of 108 17 of 25 132 of 375 24 of 341

Total 600 of 933 369 of 585 269 of 782 46 of 714

Total

1022 of 2165

262 of 849

1284 of 3014

b) 1022 of 2165 (47.2%) of males were admitted. 262 of 849 (30.9%) of females were admitted. c)

Since there are four comparisons to make, the table below organizes the percentages of males and females accepted in each program. Females are accepted at a higher rate in every program. Program 1 2 3 4

Males 61.9% 62.9% 33.7% 5.9%

Females 82.4% 68.0% 35.2% 7.0%

d) The comparison of acceptance rate within each program is most valid. The overall percentage is an unfair average. It fails to take the different numbers of applicants and different acceptance rates of each program. Women tended to apply to the programs in which gaining acceptance was difficult for everyone. This is an example of Simpson’s Paradox.

SOURCE: Browsegrades.net

Chapter 3 Relationships Between Categorical Variables – Contingency Tables

52. Be a Simpson. Answers will vary. The three-way table below shows one possibility. The number of local hires out of new hires is shown in each cell. Full-time New Employees Part-time New Employees

Company A 40 of 100 = 40% 170 of 200 = 85%

Company B 90 of 200 = 45% 90 of 100 = 90%

Total

210 of 300 = 70%

180 of 300 = 60%

SOURCE: Browsegrades.net

Chapter 4 Understanding and Comparing Distributions

Chapter 4 – Understanding and Comparing Distributions Section 4.1 1.

Load factors, 2017. The distribution of domestic load factors and the distribution of international load factors are both unimodal and skewed to the left. The distribution of international load factors may contain a low outlier. Because the distributions are skewed, the median and IQR are the appropriate measures of center and spread. The medians are very close, which tell us that typical international and domestic load factors are about the same. The IQRs show a bit more variability in the domestic load factors.

Load factors, 2017 by season. The distribution of Spring/Summer load factors and the distribution of Fall/Winter load factors are both unimodal and skewed to the left. Load factors in the Spring/Summer period vary less than load factors in the Fall/Winter period, but are generally higher. The center of the distribution of Fall/Winter load factors is around 77.5, while the center of the distribution of Spring/Summer load factors is around 82.

Fuel economy. If the Mitsubishi i-MiEV is removed, the standard deviation of fuel economy ratings would be much lower, dropping from 11.87 miles per gallon to 4.6 miles per gallon. The IQR would be affected very little, if at all.

Prisons 2014. a) The tallest bars on the histogram are where the vertical lines on the boxplot are closest together. b) The boxplot indicates a skewed distribution when the vertical lines on one side of the median are closer together than the vertical lines on the other side. The histogram indicates a skewed distribution when the bars on one side of the histogram are generally taller than the bars on the other side of the histogram. c)

The histogram shows clusters and gaps. The boxplot does not show these.

d) The boxplot shows the quartiles and the median, as well as showing outliers. These cannot be determined from the histogram. 5.

Load factors 2017 by month. Load factors are generally highest and least variable in the summer months (June–August). They are lower and more variable in the winter and spring. Several months have low outliers, and April has a very low outlier that is the minimum of all the data. (The low value is for April of 2003. At that time, there was concern over the deadly SARS virus, which may have depressed international travel.)

Load factors 2017 by year. Load factors have increased since 2002. In recent years the growth has leveled off and the annual variation has decreased.

Section 4.2 7.

Extraordinary months. We are aware of a possible explanation for the depressed air travel that has nothing to do with general monthly or overall annual patterns. If we want to understand these patterns, it might be best to set these months aside.

Load factors 2017 by year again. Outliers are context dependent. Nevertheless, April 2003 shows up in both views of the data. It looks like international travel was severely affected in that month.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

Section 4.3 9.

Load factors 2017 over time. a) After a period of little change in 2000–2001, load factors have been increasing steadily. b) No. These are percentages, so they can’t exceed 100%. Moreover, we have no reason to believe this trend will continue.

10. Load factors over time, a second look. a) With this smoother, the seasonal pattern is evident. Exercise 3 showed the pattern in detail with higher load factors in the summer months. b) Yes, we can expect this pattern to persist because it reflects seasonal effects such as summer vacation time that will probably continue, although the increase can’t continue for very long because these are percentages. 11. Exoplanets. It is difficult to summarize data with a distribution this skewed. The extremely large values will dominate any summary or description. 12. Exoplanets re-expressed. a) Yes, this re-expressed scale is better for understanding these distances. The log scale provides a nearly symmetric distribution, and points out that the sun was included in the data, probably accidentally. b) The sun should not be included in data about extra-solar planets. Chapter Exercises 13. In the news. Answers will vary. 14. Groups on the Internet. Answers will vary. 15. Cost of living 2016, selected cities. a) Cappuccino is the most expensive commodity, in general. The first quartile of cappuccino prices is higher than all the water prices. The middle 50% of cappuccino prices is less variable than the middle 50% of egg prices. The third quartile and maximum value of cappuccino prices are both higher than those for egg prices. b) We can’t say anything about price comparisons in individual cities from this display. The minimum price of a cappuccino is lower than the minimum price of a carton of eggs. It could be the case that those two prices are in the same city, and cappuccino costs less than eggs. 16. Cost of living 2016, more cities. a) A cappuccino is the most expensive, in general. The median price of a cappuccino is at the third quartile for a carton of eggs and above all the water prices except the high outliers. b) No, some of the water prices are high outliers and are more expensive than most of the egg prices. c)

Yes, some cartons of eggs are more expensive than a cappuccino. Several high outliers of egg prices are above all of the cappuccino prices. We have no idea, however, whether or not this occurs in the same city, or if these prices come from different cities.

SOURCE: Browsegrades.net

Chapter 4 Understanding and Comparing Distributions

17. Rock concert deaths. a) The histogram and boxplot of the distribution of “crowd crush” victims’ ages both show the distribution is essentially symmetric, but very slightly skewed to the right with two high outliers at 36 and 48. Most victims are between the ages of 16 and 24. b) This histogram shows that there may have been two modes in the distribution of ages of “crowd crush” victims, one at 18–20 years of age and another at 22–24 years of age. Boxplots, in general, can show symmetry and skewness, but not features of shape like bimodality or uniformity. Most victims were between 16 and 24 years of age. c)

The median is the better measure of center, since the distribution of ages has outliers. Median is more resistant to outliers than the mean.

d) The IQR is a better measure of spread, since the distribution of ages has outliers. IQR is more resistant to outliers than the standard deviation. 18. Slalom times 2018. a) Both distributions show that the distribution of slalom times is skewed to the high end. The histogram and boxplot of the distribution of Men’s Giant Slalom times both show that a typical time of around 175 seconds, that the range of slalom times is about 55 seconds. b) Since the distribution of slalom times is skewed, and contains possible outliers, the median is the better summary of center. c)

In the presence of skewness and possible outliers, we’d prefer the IQR to the standard deviation as a measure of spread.

19. Sugar in cereals. a) The maximum sugar content is approximately 60% and the minimum sugar content is approximately 1%, so the range of sugar contents is about 60 – 1 = 59%. b) The distribution of sugar content of cereals is bimodal, with modes centered around 5% and 45% sugar by weight. c)

Some cereals are healthy, low-sugar brands, and others are very sugary.

d) Yes. The minimum sugar content in the children’s cereals is about 35% and the maximum sugar content of adult cereals is only 34%. e)

The range of sugar contents is about the same for the two types of cereals, approximately 28%, but the IQR is larger for the adult cereals. This is an indication of more variability in the sugar content of the middle 50% of adult cereals.

20. Tendon transfers. a) The distribution of pushing strength scores is unimodal and symmetric. b) The maximum pushing strength score is 4 and the minimum is 1, so the range is a score of 4 – 1 = 3. c)

The histogram does not show that the results of the two procedures typically resulted in different strengths.

d) The distribution of biceps transfer strength scores had a higher median than the distribution of deltoid transfer strength scores. e)

The biceps transfer was not always the best. The highest strength score in the deltoid transfer was higher than the lowest 25% of biceps transfer strength scores.

The deltoid transfer produced more consistent strength scores. The IQR is much smaller for this group, even though the ranges have approximately the same value of 2.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

21. Population growth 2010, by region. a) Comparative boxplots are shown below.

b) Growth rates in NE/MW states are tightly clustered near 5%, but with two high outliers. S/W states are more variable, with a median near 13% and two outliers as well. The distributions are fairly symmetric.

22. Camp sites. a) The distribution of the number of campsites in public parks in Vermont is strongly skewed to the right, so median and IQR are appropriate measures of center and spread. b) IQR = Q3 – Q1 = 78 – 28 = 50. Upper Fence: Q3  1.5(IQR)  78  1.5(50)  78  75  153 Lower Fence: Q1  1.5(IQR)  28  1.5(50)  28  75  47, which is well below 0 campsites. There are 3 parks with greater than 180 campsites. These are definitely outliers. There are 2 parks with between 150 and 160 campsites each. These may be outliers as well. c)

A boxplot of the distribution of number of campsites is shown below.

d) The distribution is unimodal with a strong skew to the right. There are several outliers past the 1.5  IQR upper fence of 153 camp sites. The median number of camp sites is 43.5 sites. The mean is 62.8 sites. The mean is larger than the median because it has been influenced by the strong skew and the outliers. 23. Hospital stays. a) The histograms of male and female hospital stay durations would be easier to compare if the were constructed with the same scale, perhaps from 0 to 20 days. b) The distribution of hospital stays for men is skewed to the right, with many men having very short stays of about 1 or 2 days. The distribution tapers off to a maximum stay of approximately 25 days. The distribution of hospital stays for women is skewed to the right, with a mode at approximately 5 days, and tapering off to a maximum stay of approximately 22 days. Typically, hospital stays for women are longer than those for men. c)

The peak in the distribution of women’s hospital stays can be explained by childbirth. This time in the hospital increases the length of a typical stay for women, and not for men. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Chapter 4 Understanding and Comparing Distributions

24. Deaths 2014. a) The distributions of ages at death of black Americans and white Americans are both skewed unimodal and skewed to the left, towards the lower ages of death. The White distribution is strongly skewed to the left with a small peak at the low end that is infant mortality. The Black distribution is also skewed to the left, although it has three peaks: one at infancy, then at 60 to 64, and another at 85 and older. b) The distributions of death ages differ mainly in their spreads and centers. The age at death for black Americans is more variable than the age at death for white Americans. A greater proportion of black Americans die between the ages of 25 and 74 than do white Americans, while a greater proportion of white Americans dies at ages older than 74 than do black Americans. The distribution of death ages of black Americans shows a higher infant mortality rate than the distribution of death ages of white Americans. c)

The interval widths are not consistent. Most of the bars are 5 years wide, but the first bar is 1 year wide, the second bar is 4 years wide, and the last bar has width “85 and older”.

25. Women’s basketball. a) Both girls have a median score of about 17 points per game, but Scyrine is much more consistent. Her IQR is about 2 points, while Alexandra’s is over 10. b) If the coach wants a consistent performer, she should take Scyrine. She’ll almost certainly deliver somewhere between 15 and 20 points. But, if she wants to take a chance and needs a “big game”, she should take Alexandra. Alex scores over 24 points about a quarter of the time. On the other hand, she scores under 11 points about as often. 26. Gas prices 2018. a) Gas prices rose during the first nine years of the century, then dropped precipitously in 2008. They then resumed their rise, peaking in 2012. After that they have fallen and then recovered somewhat. The year 2008 was the most variable. Variability was small in 2017 and 2018. b) 2008 had both the greatest range and the largest IQR. The years 2014 and 2015 also show large variation— possibly less clear because it is spread across two years. 27. Marriage age 2016.

The distribution of marriage age of U.S. men is skewed right, with a typical man (as measured by the median) first marrying at around 24 years old. The middle 50% of male marriage ages is between about 23 and 26 years. For U.S. women, the distribution of marriage age is also skewed right, with median of around 21 years. The middle 50% of female marriage age is between about 20 and 23 years. When comparing the two distributions, the most striking feature is that the distributions are nearly identical in spread, but have different centers. Females typically seem to marry about 2 years younger than males. In fact, between 50% and 75% of the women marry at a younger age than any man. 28. Fuel economy and cylinders.

Cars with 4 cylinders generally get better gas mileage than cars with 6 cylinders, which generally get better gas mileage than cars with 8 cylinders. Additionally, the greater the number of cylinders, the more consistent the mileage becomes. 4 cylinder cars typically get between 27 – 33 mpg, 6 cylinder cars typically get between 18 – 22 mpg, and 8 cylinder cars typically get between 16 – 19 mpg. We don’t have enough data to compare 5cylinder cars. 29. Marriage age 2016 over time. a) They are time series because they report values over time. However, the values are not all equally spaced because the early values are reported only every decade, while later values are annual. b) Age at first marriage declined in the first part of the 20th century, but has been increasing for both men and women since about 1960. Throughout more than a century of data, men have typically been older at first marriage than women. c)

The increase in age cannot continue indefinitely. The pattern that men tend to be older than women at first marriage may well continue. SOURCE: Browsegrades.net Copyright © 2020 Pearson Education, Inc.

Part I Exploring and Understanding Data

30. Ozone. a) April had the highest recorded ozone level, approximately 440. b) February had the largest IQR of ozone level, approximately 50. c)

August had the smallest range of ozone levels, approximately 50.

d) January had a slightly lower median ozone level than June, 340 and 350, respectively, but June’s ozone levels were much more consistent. e)

Strong seasonal pattern with low consistent ozone concentrations in later summer/early fall and high variable concentrations in early spring. The medians follow a cyclic pattern, rising from January to April, then falling to October and rising again from October to December.

31. Test scores.

Class A is Class 1. The median is 60, but has less spread than Class B, which is Class 2. Class C is Class 3, since it’s median is higher, which corresponds to the skew to the left. 32. Eye and hair color.

The graph is not appropriate. Boxplots are for quantitative data, and these are categorical data, although coded as numbers. The numbers used for hair color and eye color are arbitrary, so the boxplot and any accompanying statistics for eye color make no sense. 33. Graduation? a) The distribution of the percent of incoming college freshman who graduate on time is probably slightly left skewed. The mean is slightly below the median and the 25th percentile is farther from the median than the 75th percentile b) Upper Fence: Q3  1.5(IQR)  74.75  1.5(74.75  59.15)  74.75  23.4  98.15 Lower Fence: Q1  1.5(IQR)  59.15  1.5(74.75  59.15)  59.15  23.4  35.75

Since the maximum value of the distribution of the percent of incoming freshmen who graduate on time is 87.4% and the upper fence is 98.15%, there are no high outliers. Likewise, since the minimum is 43.2% and the lower fence is 35.75%, there are no low outliers. Since the minimum and maximum percentages are within the fences, all percentages must be within the fences. c)

A boxplot of the distribution of the percent of incoming freshmen who graduate on time is shown below.

d) The distribution of the percent of incoming freshmen who graduate on time is roughly symmetric, with mean of approximately 68% of freshmen graduating on time. Universities surveyed had between 43% and 87% of students graduating on time, with the middle 50% of universities reporting between 59% and 75% graduating on time.

SOURCE: Browsegrades.net

Chapter 4 Understanding and Comparing Distributions

34. Vineyards. a) The distribution of size of Finger Lakes vineyards is skewed heavily to the right. The mean size is a great deal higher than the median size. b) Upper Fence: Q3  1.5(IQR)  55  1.5(55  18.5)  55  54.75  109.75 Lower Fence: Q1  1.5(IQR)  18.5  1.5(55  18.5)  18.5  54.75  36.25

The maximum of 250 acres is well above the upper fence of 109.75 acres. Therefore, there is at least one high outlier, 250 acres. Since the lower fence is negative, there are no low outliers, since it is certainly impossible to have a vineyard with negative size. c)

The boxplot of the distribution of sizes of Finger Lakes vineyards is shown below. There may be additional outliers, but we are sure that there is at least one, the maximum.

d) The vineyards range in size from 6 to 250 acres. The median size of the 36 vineyards is 33.5 acres, so half are larger and half are smaller. The middle 50% of the vineyards have sizes between 18.5 and 55 acres. The distribution of sizes is skewed to the right, with at least one outlier. 35. Caffeine. a) Who – 45 student volunteers; What – Level of caffeine consumption and memory test score; When – Not specified. Where – Not specified. Why – The student researchers want to see the possible effects of caffeine on memory. How – It appears that the researchers imposed the treatment of level of caffeine consumption in an experiment. However, this point is not clear. Perhaps they allowed the subjects to choose their own level of caffeine. b) Variables – Caffeine level is a categorical variable with three levels: no caffeine, low caffeine, and high caffeine. Test score is a quantitative variable, measured in number of items recalled correctly. c)

Comparative boxplots are shown below.

d) The groups consuming no caffeine and low caffeine had comparable memory test scores. A typical score from these groups was around 21. However, the scores of the group consuming no caffeine were more consistent, with a smaller range and smaller interquartile range than the scores of the group consuming low caffeine. The group consuming high caffeine had lower memory scores in general, with a median score of about 19. No one in the high caffeine group scored above 24, but 25% of each of the other groups scored above 24.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

36. SAT scores. a) Comparative boxplots comparing the scores of boys and girls SAT scores are shown below.

b) Females in this graduating class scored slightly higher on the Verbal SAT, with a median of 625, 25 points higher than the median of 600 for the males. Additionally, the females had higher first and third quartiles. The IQR of the males’ scores was slightly smaller, than the IQR for the females’ scores, indicating a bit more consistency in male scores. However, the overall spread of male scores was greater than that of female scores, with males having both the minimum and maximum score. Both distributions of scores were slightly skewed to the left. 37. Derby speeds 2017. a) The median speed is the speed at which 50% of the winning horses ran slower. Find 50% on the left, move straight over to the graph and down to a speed of about 36 mph. b) Q1 is at 25% on the left, and Q3 is at 75% on the left. Matching these to the ogive, Q1 = 35 mph and Q3 = 37 mph, approximately. c)

Range = Max – Min = 38 – 31 = 7 mph

IQR = Q3 – Q1 = 36.5 – 34.5 = 2 mph

d) A possible boxplot of winning Kentucky Derby Speeds is shown below.

The distribution of winning speeds in the Kentucky Derby is skewed to the left. The lowest winning speed is just under 31 mph and the fastest speed is about 38 mph. The median speed is approximately 36 mph, and 75% of winning speeds are above 35 mph. Only a few percent of winners have had speeds below 33 mph. The middle 50% of winning speeds are between 35 and 37 mph, for an IQR of 2 mph

38. Framingham cholesterol.

A boxplot for the distribution of cholesterol levels of 1400 men shown below.

The distribution of cholesterol levels is essentially symmetric, with a median near 225. The IQR is about 75 points and the extremes are about 90 and 430. SOURCE: Browsegrades.net

Chapter 4 Understanding and Comparing Distributions

39. Reading scores. a) The highest score for boys was 6, which is higher than the highest score for girls, 5.9. b) The range of scores for boys is greater than the range of scores for girls. Range = Max – Min Range(Boys) = 4 c)

The girls had the greater IQR. IQR(Boys) = Q3 – Q1 =4.9 – 3.9 = 1

Range(Girls) = 3.1

IQR(Girls) = Q3 – Q1 = 5.2 – 3.8 = 1.4

d) The distribution of boys’ scores is more skewed. The quartiles are not the same distance from the median. In the distribution of girls’ scores, Q1 is 0.7 units below the median, while Q3 is 0.7 units above the median. e)

Overall, the girls did better on the reading test. The median, 4.5, was higher than the median for the boys, 4.3. Additionally, the upper quartile score was higher for girls than boys, 5.2 compared to 4.9. The girls’ lower quartile score was slightly lower than the boys’ lower quartile score, 3.8 compared to 3.9.

The overall mean is calculated by weighting each mean by the number of students, which is 14(4.2)  11(4.6)  4.38. 25

40. Cloud seeding. a) The median and IQR, as well as the quartiles are the appropriate summary statistics, since the distribution of amount of rain produced is either skewed to the right or has high outliers. Indication of skewness and outliers can be seen in the comparison of median and mean. The mean amount of rain produced is significantly higher than the median for both seeded and unseeded clouds. Skewness or outliers pulled up the sensitive mean. b) There is evidence that that the seeded clouds produced more rain. The median and both quartiles are higher than the corresponding statistics for unseeded clouds. In fact, the median amount of rainfall for seeded clouds is 221.60 acre-feet, about 5 times the median amount for unseeded clouds. 41. Industrial experiment.

Looking at the boxplot below to the left, there is an extreme outlier in the distribution of distances for the slow speed drilling. One hole was drilled almost an inch away from the center of the target! If that distance is correct, the engineers at the computer production plant should investigate the slow speed drilling process closely. It may be plagued by extreme, intermittent inaccuracy. The outlier in the slow speed drilling process is so extreme that no graphical display can display the distribution in a meaningful way while including that outlier. That distance should be removed before looking at a plot of the drilling distances.

Looking at the boxplot above to the right, with the outlier set aside, we can determine that the slow drilling process is more accurate. The greatest distance from the target for the slow drilling process, 0.000098 inches, is still more accurate than the smallest distance for the fast drilling process, 0.000100 inches.

SOURCE: Browsegrades.net

Part I Exploring and Understanding Data

42. Cholesterol and smoking.

The smoker’s distribution has a lower median and is less spread out, but the ex-smokers group has a higher percentage of its values below 200. There is also a high outlier in the smoker’s group at 351.

In general, the cholesterol levels of smokers seem to be slightly lower than the cholesterol levels of ex-smokers. Additionally, the cholesterol levels of smokers appear more consistent than cholesterol levels of ex-smokers. 43. MPG. a) Comparative boxplots are shown below. Comparative dotplots, histograms, or stemplots would also be acceptable.

b) The distributions of fuel efficiency for all three groups have high outliers, potentially hybrid or electric cars. Asian cars generally have the highest fuel efficiency, while European cars have the lowest. The Asian median is about 32 mpg, European cars around 23 mpg, and U.S. cars 28 mpg. 44. Baseball 2016. a) Comparative boxplots are shown below. Back-to-back stem-and-leaf displays, comparative histograms, and dotplots on the same scale would also be acceptable.

b) Both distributions are roughly symmetric. The American League has a higher center but smaller variability. The Red Sox may be an outlier in the American League (although not extreme enough for the boxplot outlier nomination rule).

SOURCE: Browsegrades.net

Chapter 4 Understanding and Comparing Distributions

44. (continued) c)

The Red Sox are not nominated as an outlier (they are just below 1.5 IQRs above Q3), but their run production might be viewed as unusually high. The Rockies are well less than 1.5 IQRs above the National League Q3 so, although their run production is high, it’s probably not unusually high.

d) (Answers will vary slightly.) A histogram of the 1000 differences shows that the difference of 0.082 is fairly typical for the means of two randomly assigned groups of 15 teams. In fact, in nearly 55% of the trials, a difference greater than 0.082 was observed. There is no evidence to suggest that such a difference didn’t occur by chance.

45. Assets. a) The distribution of assets of 79 companies chosen from the Forbes list of the nation’s top corporations is skewed so heavily to the right that the vast majority of companies have assets represented in the first bar of the histogram, 0 to 10 billion dollars. This makes meaningful discussion of center and spread impossible. b) Re-expressing these data by, for example, logs or square roots might help make the distribution more nearly symmetric. Then a meaningful discussion of center might be possible. 46. Music library. a) The distribution of the number of songs students had in their digital libraries is extremely skewed to the right. That makes it difficult to determine a center. The typical number of songs in a library is probably in the first bar of the histogram. b) Re-expressing these data by, for example, logs or square roots might help make the distribution more nearly symmetric. Then a meaningful discussion of center might be possible. 47. Assets again. a) The distribution of logarithm of assets is preferable, because it is roughly unimodal and symmetric. The distribution of the square root of assets is still skewed right, with outliers. b) If c)

Assets  50 , then the company’s assets are approximately 502  $2500 million.

If log( Assets)  3 , then the company’s assets are approximately 103  $1000 million.

SOURCE: Browsegrades.net