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Stats: Modeling the World, 5th Edition Solution Manual

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    S OLUTIONS M ANUAL A DAM Y ANKAY S TATS M ODELING THE W ORLD F IFTH E DITION David E. Bock Ithaca High School (Retired) Floyd Bullard North Carolina School of Science and Mathematics Paul F. Velleman Cornell University Richard D. De Veaux Williams College

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    Contents Part I Exploring and Understanding Data Chapter 1 Stats Starts Here 1 Chapter 2 Displaying and Describing Categorical Data 5 Chapter 3 Displaying and Summarizing Quantitative Data 20 Chapter 4 Understanding and Comparing Distributions 39 Chapter 5 The Standard Deviation as a Ruler and the Normal Model 55 Review of Part I Exploring and Understanding Data 74 Part II Exploring Relationships Between Variables Chapter 6 Scatterplots, Association, and Correlation 102 Chapter 7 Linear Regression 113 Chapter 8 Regression Wisdom 138 Chapter 9 Re-expressing Data: Get It Straight! 157 Review of Part II Exploring Relationships Between Variables 168 Part III Gathering Data Chapter 10 Understanding Randomness 194 Chapter 11 Sample Surveys 204 Chapter 12 Experiments 215 Review of Part III Gathering Data 233 Part IV Randomness and Probability Chapter 13 From Randomness to Probability 249 Chapter 14 Probability Rules! 262 Chapter 15 Random Variables 284 Chapter 16 Probability Models 305 Review of Part IV Randomness and Probability 334

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    Part V From the Data at Hand to the World at Large Chapter 17 Sampling Distribution Models 358 Chapter 18 Confidence Intervals for Proportions 392 Chapter 19 Testing Hypotheses About Proportions 408 Chapter 20 More About Tests and Intervals 427 Chapter 21 Comparing Two Proportions 444 Review of Part V From the Data at Hand to the World at Large 469 Part VI Learning About the World Chapter 22 Inferences About Means 497 Chapter 23 Comparing Means 523 Chapter 24 Paired Samples and Blocks 553 Review of Part VI Learning About the World 573 Part VII Inference When Variables are Related Chapter 25 Comparing Counts 611 Chapter 26 Inferences for Regression 642 Review of Part VII Inference When Variables are Related 669 Chapter 27 Analysis of Variance 707 Chapter 28 Multiple Regression 718

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    Chapter 1 Stats Starts Here 1 Chapter 1 – Stats Starts Here 1. Voters. The response, party affiliation, is a categorical variable. 2. Job growth. The response, change in workforce size, is a categorical variable. 3. Medicine. The company is measuring minutes, a quantitative variable. 4. Stress. The researcher is measuring heart rate, a quantitative variable. 5. The news. Answers will vary. 6. The Internet. Answers will vary. 7. Bicycle and pedestrian safety. Who – pedestrians and bicyclists killed or severely injured in New York City between 2010 and 2014. What – proportion of pedestrians and bicyclists killed or injured by left-turning and right-turning vehicles. Population of interest – Answers may vary. Perhaps: All pedestrians and bicyclists killed or severely injured in New York City between 2010 and 2014 or all pedestrians and bicyclists killed or severely injured in New York City in any year or all pedestrians and bicyclists killed or severely injured in major U.S. cities. Discuss with students why these and other solutions may be problematic. 8. Investments. Who – 30 companies with similar retirement plans. What – 401(k) employee participation rates. Population of interest – All companies with similar retirement plans. 9 . Fake news. Who – Middle school, high school, and college students in 12 states. What – Ability to evaluate the quality of information found in different online resources. Population of interest – All U.S middle school, high school, and college students. 10. Biological instinct. 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. 11. Blindness. Who – 24 patients. What – Whether or not stem cell therapy was effective in treating Stargardt’s disease and/or dry age-related macular degeneration. Population of interest – All people with these eye conditions. 12. Molten iron. Who – 10 castings at Cleveland Casting. What – The pouring temperature (in degrees Fahrenheit) of molten iron. Population of interest – All castings at Cleveland Casting.

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    2 Part I Exploring and Understanding Data 13. Weighing bears. Who – 54 bears. 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. 14. Schools. Who – Students. 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, age, 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? 15. Arby’s menu. Who – Arby’s sandwiches. 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 are quantitative variables, and type of meat is a categorical variable. 16. Party and the environment. Who – American voters. What – Gender, age (in years), race, party affiliation, education, whether or not the person was “worried a great deal” about climate change, air pollution, and pollution of waterways. When – 2017. Where – United States. Why – The information was gathered for presentation in a Gallup public opinion poll. How – Poll. Variables – There are eight variables. Gender, race, party affiliation, education, whether or not the person was “worried a great deal” about climate change, air pollution, and pollution of waterways are categorical variables; age is a quantitative variable. 17. Babies. Who – 882 births. 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, gender of baby (male or female), and baby’s health problems (none, minor, major). When – 1998-2000. Where – Large city hospital. Why – Researchers were

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    Chapter 1 Stats Starts Here 3 investigating the impact of prenatal care on newborn health. How – It is not specifically stated. Variables – There are seven variables. Type of birth, level of prenatal care, gender of baby, and baby’s health problems are categorical variables; mother’s age, length of pregnancy, and birth weight of baby are quantitative variables. 18. Flowers. Who – 385 species of flowers for 47 years. 385(47) = 18,095 cases. What – Date of first flowering (in days). When – An unspecified 47 year period. 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. The number of years, 47, is also a variable. 19. Herbal medicine. Who – Patients. What – Herbal cold remedy or sugar solution, and cold severity on a scale of 0-5. 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 conducted an experiment. Variables – There are two variables. Type of treatment (herbal or sugar solution) is a categorical variable, and severity rating is a quantitative variable. The subjectivity of “cold severity” is a concern that should be raised about this study. 20. Vineyards. Who – Vineyards. 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 grapes. How – Not specified. Variables – There are seven variables. State and variety of grapes grown are categorical variables; size of vineyard, number of years in existence, average case price, gross sales, and percent profit are quantitative variables. 21. Streams. Who – Streams. What – A number of variables including: 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; acidity, temperature, and BCI are quantitative variables. 22. Fuel economy. Who – Every model of automobile. 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. Manufacturer and type of car are categorical variables; weight, horsepower, city mileage, and highway mileage are quantitative variables.

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    4 Part I Exploring and Understanding Data 23. Refrigerators. Who – 148 models of French door style refrigerators. What – Brand, price (probably in dollars), temperature performance, temperature uniformity, energy efficiency, noise, ease of use (the five previous variables are measured as Excellent, Very Good, Good, Fair, or Poor), number of doors, capacity (cu. ft.), exterior height (in.), exterior width (in.), and exterior depth (in.). When – 2017. Where – Not stated. Why – The information was compiled to provide information to the readers of Consumer Reports. How – Not specified. Variables – There are 11 variables. Brand, temperature performance, temperature uniformity, energy efficiency, noise, and ease of use are categorical variables; price, number of doors, capacity, exterior height, exterior width, and exterior depth are quantitative variables. 24. Walking in circles. Who – 32 people. 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 naturally 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. 25. Kentucky Derby 2016. Who – Kentucky Derby races. What – Year, winner, jockey, trainer, owner, and time (in minutes, seconds, and hundredths of a second. When – 1875 – 2016. Where – Churchill Downs, Louisville, Kentucky. Why – Not specified. How – Official statistics are kept for the race each year. Variables – There are 6 variables. Winner, jockey, trainer and owner are categorical variables; year and time are quantitative variables. 26. Indy 2016. 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 – 2016. Where – Indianapolis Motor Speedway. Why – Not specified. How – Official statistics are kept for the race every year. Variables – There are 4 variables. Winner is a categorical variable, while year, time, and average speed are quantitative variables.

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    Chapter 2 Displaying and Describing Categorical Data 5 Chapter 2 – Displaying and Describing Categorical Data 1. Graphs in the news. Answers will vary. 2. Graphs in the news II. Answers will vary. 3. Tables in the news. Answers will vary. 4. Tables in the news II. Answers will vary. 5. Movie genres. a) A pie chart seems appropriate from the movie genre data. Each movie has only one genre, and the 891 movies constitute a “whole”. b) “Other” is the least common genre. It has the smallest region in the chart. 6. Movie ratings. a) A pie chart seems appropriate for the movie rating data. Each movie has only one rating, and the 891 movies constitute a “whole”. b) The most common rating is R. It has the largest region on the chart. 7. Movie genres again. a) Thriller/Suspense films were more common than Adventure films. The bar for Thriller/Suspense is taller than the bar for Adventure. b) This is easier to see on the bar chart. The percentages are so close that the difference is nearly indistinguishable in the pie chart. Also, the bar chart is organized by height while the pie chart is not, making it difficult to compare genres with areas similar in proportion. 8. Movie ratings again. a) The least common rating was NC-17. It has the shortest bar. b) While it is easy in both the pie chart and the bar chart, it may be easier in the pie chart. In the pie chart, ratings are ordered clockwise by increasing area while in the bar chart, G and NC-17 are inconsistent with an increasing order. 9. Movie ratings. i. C (This chart has 4 ratings and the proportion of G ratings is smallest.) ii. A (This chart has 4 ratings and a slightly higher proportion of G ratings.) iii. D (This chart has 3 ratings with PG more common than R.) iv. B (This chart has 3 ratings with PG and R roughly equally as common.) 10. Marriage in decline. i. D (This bar and pie chart are the only ones for which Bad Thing and No Difference are not roughly equally common)

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    6 Part I Exploring and Understanding Data ii. A (Bad Thing and No Difference are roughly equal, while Don’t Know/No Response is noticeably the least frequent response) iii. B or C (These charts are indistinguishable.) iv. B or C (These charts are indistinguishable.) 11. Magnet schools. There were 1,755 qualified applicants for the Houston Independent School District’s magnet schools program. Approximately 53% were accepted, 17% were wait-listed, and the other 30% were turned away for lack of space. 12. Magnet schools again. There were 1,755 qualified applicants for the Houston Independent School District’s magnet schools program. Approximately 29.5% were Black or Hispanic, 16.6% were Asian, and 53.9% were white. 13. 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.

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    Chapter 2 Displaying and Describing Categorical Data 7 14. Plane crashes. a) As long as each plane crash had only one cause, it would be reasonable to assume that weather or mechanical failures were the causes of about 20% of recent plane crashes. b) Since the percentages listed add up to 71%, other causes (not determined) must account for 29% of recent plane crashes. c) A relative frequency bar chart is a good choice. A pie chart would also be a good display, as long as each plane crash has only one cause. 15. Oil spills as of 2016. a) Grounding, accounting for 150 spills, is the most frequent cause of oil spillage for these 460 spills. A substantial number of spills, 136, were caused by collision. Less prevalent causes of oil spillage in descending order of frequency were hull failures, other/unknown causes, fire/explosions, and equipment failure. b) If being able to differentiate between these close counts is required, use the bar chart. Since each spill only has one cause, the pie chart is also acceptable as a display, but it’s difficult to tell whether, for example, there is a greater percentage of spills caused by fire/explosions or hull failure. If you want to showcase the causes of oil spills as a fraction of all 460 spills, use the pie chart. 16. Winter Olympics 2016. 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 those countries that didn’t win many medals.

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    8 Part I Exploring and Understanding Data b) Perhaps we are primarily interested in countries that won many medals. We might choose to combine all countries that won fewer than 6 medals into a single category. This will make our chart easier to read. We are probably interested in number of medals won, rather than percentage of total medals won, so we’ll use a bar chart. A bar chart is also better for comparisons. 17. 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 each individual region and any resulting sums will occupy less area of the display than their percentages imply. Furthermore, the three-dimensional perspective view distorts the regions in the graph, violating the area principle. The regions corresponding to “Global warming isn’t happening” and “Can’t reduce global warming even if it is happening” should be the same size, at 25% of respondents. However, the “Global warming isn’t happening” region looks bigger. Always use simple, two-dimensional graphs. 18. Modalities. a) The bars have false depth, which can be misleading. This is a bar chart, so the bars should have space between them. b) Since each trainer was asked to list 3 modalities, the expected sum should be 300% rather than 100%. 19. 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 to 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 9% of teen smokers that prefer one of over 20 other brands. 20. Handguns. 76% 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

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    Chapter 2 Displaying and Describing Categorical Data 9 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. 21. Movie genres and ratings. a) 452 of these films were rated R. 452/1,529 ≈ 29.56% b) 124 of these films were R-rated comedies. 124/1,529 ≈ 8.1% c) 124 of the 452 R-rated films were comedies. 124/452 ≈ 27.43% d) 124 of the 312 comedies were R-rated. 124/312 ≈ 39.74% 22. Not the labor force. a) 2207 of the unemployed population were available to work now. 2207/12,872 ≈ 17.1% b) 1,048 of the unemployed population were available to work now and aged 25 to 54 years. 1,048/12,872 ≈ 8.14% c) 208 of 4,158 unemployed 16-24 year olds were in school or training. 208/4,158 ≈ 5% d) 4,158 of the unemployed population were aged 16-24 years. 4,158/12,872 ≈ 32.3% 23. Seniors. A table with marginal totals is to the right. a) 268 seniors were white. 268/325 ≈ 82.5% b) 42 seniors are planning to attend a 2-year college. 42/325 ≈ 13% c) 36 seniors are white and planning to attend 2-year colleges. 36/325 ≈ 11.1% d) 36 of the 268 white seniors are planning to attend 2-year colleges. 36/268 ≈ 13.4% e) There are 42 graduates planning to attend 2-year colleges. 36 are white. 36/42 ≈ 85.7% 24. Politics. a) There are 192 students taking Intro Stats. Of those, 115 are male. 115/192 ≈ 59.9%. Plans White Minority TOTAL 4-year college 198 44 242 2-year college 36 6 42 Military 4 1 5 Employment 14 3 17 Other 16 3 19 TOTAL 268 57 325

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    10 Part I Exploring and Understanding Data b) 27 students in the course consider themselves to be “Conservative”. 27/192 ≈ 14%. c) There are 115 males taking Intro Stats. Of those, 21 consider themselves to be “Conservative”. 21/115 ≈ 18.26%. d) 21 of the students in the course are males who consider themselves to be “Conservative”. 21/192 ≈ 10.94% 25. 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 segmented bar chart is a good display of these data: 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. Post High School Plans 4-year college 4-year college 2-year college 2-year college Other Other Employment Employment 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% White Minority Military

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    Chapter 2 Displaying and Describing Categorical Data 11 26. 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 segmented bar chart comparing the distributions is at the right. d) Politics and sex do not appear to be independent in this course. Although the percentage of liberals was roughly the same for each sex, females had a greater percentage of moderates and a lower percentage of conservatives than males. 27. Magnet schools revisited. a) There were 1,755 qualified applicants to the Houston Independent School District’s magnet schools program. Of those, 292 were Asian. 292/1,755 ≈ 16.6% b) There were 931 students accepted to the magnet schools program. Of those, 110, were Asian. 110/931 ≈ 11.8%. c) There were 292 Asian applicants. Of those, 110 were accepted. 110/292 ≈ 37.7%. d) There were 1,755 total applicants. Of those, 931 were accepted. 931/1,755 ≈ 53%. 28. More politics. a) 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Lib Mod Con Distribution of Sex Across Political Categories M M M F F F Percent Politics Politics of an Intro Stats Course Liberal Liberal Moderate Moderate Conservative Conservative 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Female Male Percent

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    12 Part I Exploring and Understanding Data b) The percentage of males and females varies across political categories. The percentage of self-identified Liberals and Moderates who are female is about twice the percentage of Conservatives who are female. This suggests that sex and politics are not independent. 29. Back to school. There were 1,755 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. 30. Cars. 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%. 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. e) The conditional distribution of Origin for Students is: 54.8% (107 of 195) American, 17% (33 of 195) European, and 28% (55 of 195) Asian. The conditional distribution of Origin for Staff is: 64% (105 of 164) American, 7% (12 of 164) European, and 29% (47 of 164) Asian. Driver Origin Student Staff Total American 107 105 212 European 33 12 45 Asian 55 47 102 Total 195 164 359 Origin Totals American 212 (59%) European 45 (13%) Asian 102 (28%) Total 359

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