Combining scores again The first Stat exam had a mean of

Prenatal care Results of a 1996 American Medical Association report about the infant mortality rate for twins carried for the full term of a normal pregnancy are shown on the next page, broken down by the level of prenatal care the mother had received. a) Is the overall rate the average of the other three rates? Should it be? Explain. b) Do these results indicate that adequate prenatal care is important for pregnant women? Explain. c) Do these results suggest that a woman pregnant with twins should be wary of seeking too much medical care? Explain.

Problem 1RE Bananas Here are the prices (in cents per pound) of bananas reported from 15 markets surveyed by the U.S. Department of Agriculture. a) Display these data with an appropriate graph. b) Report appropriate summary statistics. c) Write a few sentences about this distribution.

Problem 1E Stats test, part III The mean score on the Stats exam was 75 points with a standard deviation of 5 points, and Gregor’s z-score was -2. How many points did he score?

Problem 2E Cold U? A high school senior uses the Internet to get information on February temperatures in the town where he’ll be going to college. He finds a website with some statistics, but they are given in degrees Celsius. The conversion formula is °F = 9>5 °C + 32. Determine the Fahrenheit equivalents for the summary information below. Maximum temperature = 11°C Range = 33° Mean = 1° Standard deviation = 7° Median = 2° IQR = 16°

Temperatures A town’s January high temperatures average \(36^\circ F\) with a standard deviation of \(10^\circ\), while in July the mean high temperature is \(74^\circ\) and the standard deviation is \(8^\circ\). In which month is it more unusual to have a day with a high temperature of \(55^\circ\)? Explain.

Problem 4RE Rivets A company that manufactures rivets believes the shear strength (in pounds) is modeled by N(800, 50). a) Draw and label the Normal model. ________________ b) Would it be safe to use these rivets in a situation requiring a shear strength of 750 pounds? Explain. ________________ c) About what percent of these rivets would you expect to fall below 900 pounds? ________________ d) Rivets are used in a variety of applications with varying shear strength requirements. What is the maximum shear strength for which you would feel comfortable approving this company’s rivets? Explain your reasoning.

Beanstalks Beanstalk Clubs are social clubs for very tall people. To join, a man must be over 6?2? tall, and a woman over 5?10?. The National Health Survey suggests that heights of adults may be Normally distributed, with mean heights of 69.1? for men and 64.0? for women. The respective standard deviations are 2.8? and 2.5?. a) You are probably not surprised to learn that men are generally taller than women, but what does the greater standard deviation for men’s heights indicate? b) Who are more likely to qualify for Beanstalk membership, men or women? Explain.

Singers by parts The boxplots shown display the heights (in inches) of 130 members of the choir we saw in Exercise 3.40. Now we have information about what part they sing. a) It appears that the median height for sopranos is missing, but actually the median and the upper quartile are equal. How could that happen? b) Write a few sentences describing what you see.

Problem 4E Placement exams An incoming freshman took her college’s placement exams in French and mathematics. In French, she scored 82 and in math 86. The overall results on the French exam had a mean of 72 and a standard deviation of 8, while the mean math score was 68, with a standard deviation of 12. On which exam did she do better compared with the other freshmen?

Problem 5E Shipments A company selling clothing on the Internet reports that the packages it ships have a median weight of 68 ounces and an IQR of 40 ounces. a) The company plans to include a sales flyer weighing 4 ounces in each package. What will the new median and IQR be? b) If the company recorded the shipping weights of these new packages in pounds instead of ounces, what would the median and IQR be? (1 lb. = 16 oz.)

Problem 7E Guzzlers? Environmental Protection Agency (EPA) fuel economy estimates for automobile models tested recently predicted a mean of 24.8 mpg and a standard deviation of 6.2 mpg for highway driving. Assume that a Normal model can be applied. a) Draw the model for auto fuel economy. Clearly label it, showing what the 68–95–99.7 Rule predicts. b) In what interval would you expect the central 68% of autos to be found? c) About what percent of autos should get more than 31 mpg? d) About what percent of cars should get between 31 and 37.2 mpg? e) Describe the gas mileage of the worst 2.5% of all cars.

Problem 6RE Bread Clarksburg Bakery is trying to predict how many loaves to bake. In the past 100 days, they have sold between 95 and 140 loaves per day. Here is a histogram of the number of loaves they sold for the past 100 days. a) Describe the distribution. b) Which should be larger, the mean number of sales or the median? Explain. c) Here are the summary statistics for Clarksburg Bakery’s bread sales. Use these statistics and the histogram above to create a boxplot. You may approximate the values of any outliers. d) For these data, the mean was 103 loaves sold per day, with a standard deviation of 9 loaves. Do these statistics suggest that Clarksburg Bakery should expect to sell between 94 and 112 loaves on about 68% of the days? Explain.

Hotline A company’s customer service hotline handles many calls relating to orders, refunds, and other issues. The company’s records indicate that the median length of calls to the hotline is 4.4 minutes with an IQR of 2.3 minutes. a) If the company were to describe the duration of these calls in seconds instead of minutes, what would the median and IQR be? b) In an effort to speed up the customer service process, the company decides to streamline the series of pushbutton menus customers must navigate, cutting the time by 24 seconds. What will the median and IQR of the length of hotline calls become?

Problem 8E Placement exams An incoming freshman took her college’s placement exams in French and mathematics. In French, she scored 82 and in math 86. The overall results on the French exam had a mean of 72 and a standard deviation of 8, while the mean math score was 68, with a standard deviation of 12. On which exam did she do better compared with the other freshmen?

Acid rain Based on long-term investigation, researchers have suggested that the acidity (pH) of rainfall in the Shenandoah Mountains can be described by the Normal model N(4.9, 0.6). a) Draw and carefully label the model. b) What percent of storms produce rainfall with pH over 6? c) What percent of storms produce rainfall with pH under 4? d) The lower the pH, the more acidic the rain. What is the pH level for the most acidic 20% of all storms? e) What is the pH level for the least acidic 5% of all storms? f) What is the IQR for the pH of rainfall?

Problem 7RE Engines, again Horsepower is another measure commonly used to describe auto engines. Here are the summary statistics and histogram displaying horsepowers of the same group of 38 cars discussed in Exercise. Summary of Horsepower Count 38 Mean 101.7 Median 100 StdDev 26.4 Range 90 25th %tile 78 75th %tile 125 a) Describe the shape, center, and spread of this distribution. ________________ b) What is the interquartile range? ________________ c) Are any of these engines outliers in terms of horsepower? Explain. ________________ d) Do you think the 68–95–99.7 Rule applies to the horsepower of auto engines? Explain. ________________ e) From the histogram, make a rough estimate of the percentage of these engines whose horsepower is within one standard deviation of the mean. ________________ f) A fuel additive boasts in its advertising that it can “add 10 horsepower to any car.” Assuming that is true, what would happen to each of these summary statistics if this additive were used in all the cars? Exercise Engines One measure of the size of an automobile engine is its “displacement,” the total volume (in liters or cubic inches) of its cylinders. Summary statistics for several models of new cars are shown. These displacements were measured in cubic inches. Summary of Displacement Count 38 Mean 177.29 Median 148.5 StdDev 88.88 Range 275 25th %tile 105 75th %tile 231 a) How many cars were measured? ________________ b) Why might the mean be so much larger than the median? ________________ c) Describe the center and spread of this distribution with appropriate statistics. ________________ d) Your neighbor is bragging about the 227-cubic-inch engine he bought in his new car. Is that engine unusually large? Explain. ________________ e) Are there any engines in this data set that you would consider to be outliers? Explain. ________________ f) Is it reasonable to expect that about 68% of car engines measure between 88 and 266 cubic inches? (That’s 177.289 ± 88.8767.) Explain. ________________ g) We can convert all the data from cubic inches to cubic centimeters (cc) by multiplying by 16.4. For example, a 200-cubic-inch engine has a displacement of 3280 cc. How would such a conversion affect each of the summary statistics?

Winter Olympics 2010 speed skating The top 36 women’s 500-m speed skating times are listed in the table. a) The mean finishing time was 40.44 seconds, with a standard deviation of 10.03 seconds. If a Normal model were appropriate, what percent of the times should be within 2 seconds of the mean? b) What percent of the times actually fall within this interval? c) Explain the discrepancy between parts a and b.

Problem 10E Car speeds again For the car speed data of Exercise, recall that the mean speed recorded was 23.84 mph, with a standard deviation of 3.56 mph. To see how many cars are speeding, John subtracts 20 mph from all speeds. a) What is the mean speed now? What is the new standard deviation? ________________ b) His friend in Berlin wants to study the speeds, so John converts all the original miles-per-hour readings to kilometers per hour by multiplying all speeds by 1.609 (km per mile). What is the mean now? What is the new standard deviation? Exercise Car speeds John Beale of Stanford, CA, recorded the speeds of cars driving past his house, where the speed limit was 20 mph. The mean of 100 readings was 23.84 mph, with a standard deviation of 3.56 mph. (He actually recorded every car for a two-month period. These are 100 representative readings.) a) How many standard deviations from the mean would a car going under the speed limit be? ________________ b) Which would be more unusual, a car traveling 34 mph or one going 10 mph?

Problem 9RE Fraud detection A credit card bank is investigating the incidence of fraudulent card use. The bank suspects that the type of product bought may provide clues to the fraud. To examine this situation, the bank looks at the Standard Industrial Code (SIC) of the business related to the transaction. This is a code that was used by the U.S. Census Bureau and Statistics Canada to identify the type of every registered business in North America.2 For example, 1011 designates Meat and Meat Products (except Poultry), 1012 is Poultry Products, 1021 is Fish Products, 1031 is Canned and Preserved Fruits and Vegetables, and 1032 is Frozen Fruits and Vegetables. A company intern produces the following histogram of the SIC codes for 1536 transactions: He also reports that the mean SIC is 5823.13 with a standard deviation of 488.17. a) Comment on any problems you see with the use of the mean and standard deviation as summary statistics. b) How well do you think the Normal model will work on these data? Explain.

Problem 11RE Cramming One Thursday, researchers gave students enrolled in a section of basic Spanish a set of 50 new vocabulary words to memorize. On Friday, the students took a vocabulary test. When they returned to class the following Monday, they were retested—without advance warning. Both sets of test scores for the 25 students are shown below. a) Create a graphical display to compare the two distributions of scores. b) Write a few sentences about the scores reported on Friday and Monday. c) Create a graphical display showing the distribution of the changes in student scores. d) Describe the distribution of changes.

Problem 11E Music library Corey has 4929 songs in his computer’s music library. The lengths of the songs have a mean of 242.4 seconds and standard deviation of 114.51 seconds. A Normal probability plot of the song lengths looks like this: a) Do you think the distribution is Normal? Explain. b) If it isn’t Normal, how does it differ from a Normal model?

Streams As part of the course work, a class at an upstate NY college collects data on streams each year. Students record a number of biological, chemical, and physical variables, including the stream name, the substrate of the stream (limestone (L), shale (S), or mixed (M)), the pH, the temperature (\(^\circ C\)), and the BCI, a measure of biological diversity. a) Name each variable, indicating whether it is categorical or quantitative, and give the units if available. b) These streams have been classified according to their substrate—the composition of soil and rock over which they flow—as summarized in the table. What kind of graph might be used to display these data?

e-Books A study by the Pew Internet & American Life Project found that 78% of U.S residents over 16 years old read a book in the past 12 months. They also found that 21% had read an e-book using a reader or computer during that period. A newspaper reporting on these findings concluded that 99% of U.S. adult residents had read a book in some fashion in the past year. (http://libraries .pewinternet.org/2012/04/04/the-rise-of-e-reading/) Do you agree? Explain.

Problem 12E Wisconsin ACT math The histogram shows the distribution of mean ACT mathematics scores for all Wisconsin public schools in 2011. The vertical lines show the mean and one standard deviation above and below the mean. 78.8% of the data points are between the two outer lines. a) Give two reasons that a Normal model is not appropriate for these data. b) The Normal probability plot on the left shows the distribution of these scores. The plot on the right shows the same data with the Milwaukee area schools (mostly in the low mode) removed. What do these plots tell you about the shape of the distributions?

Problem 13E Prenatal care Results of a 1996 American Medical Association report about the infant mortality rate for twins carried for the full term of a normal pregnancy are shown on the next page, broken down by the level of prenatal care the mother had received. Full-Term Pregnancies, Level of Prenatal Care Infant Mortality Rate Among Twins (deaths per thousand live births) Intensive 5.4 Adequate 3.9 Inadequate 6.1 Overall 5.1 a) Is the overall rate the average of the other three rates? Should it be? Explain. ________________ b) Do these results indicate that adequate prenatal care is important for pregnant women? Explain. ________________ c) Do these results suggest that a woman pregnant with twins should be wary of seeking too much medical care? Explain.

Problem 13RE Let’s play cards You pick a card from a deck (see description in Chapter 9) and record its denomination (7, say) and its suit (maybe spades). a) Is the variable suit categorical or quantitative? b) Name a game you might be playing for which you would consider the variable denomination to be categorical. Explain. c) Name a game you might be playing for which you would consider the variable denomination to be quantitative. Explain.

Problem 15E SAT or ACT? Each year thousands of high school students take either the SAT or the ACT, standardized tests used in the college admissions process. Combined SAT Math and Verbal scores go as high as 1600, while the maximum ACT composite score is 36. Since the two exams use very different scales, comparisons of performance are difficult. A convenient rule of thumb is SAT = 40 × ACT + 150; that is, multiply an ACT score by 40 and add 150 points to estimate the equivalent SAT score. An admissions officer reported the following statistics about the ACT scores of 2355 students who applied to her college one year. Find the summaries of equivalent SAT scores. Lowest score = 19 Mean = 27 Standard deviation = 3 Q3 = 30 Median = 28 IQR = 6

Problem 14E Hams A specialty foods company sells “gourmet hams” by mail order. The hams vary in size from 4.15 to 7.45 pounds, with a mean weight of 6 pounds and standard deviation of 0.65 pounds. The quartiles and median weights are 5.6, 6.2, and 6.55 pounds. a) Find the range and the IQR of the weights. b) Do you think the distribution of the weights is symmetric or skewed? If skewed, which way? Why? c) If these weights were expressed in ounces (1 pound = 16 ounces) what would the mean, standard deviation, quartiles, median, IQR, and range be? d) When the company ships these hams, the box and packing materials add 30 ounces. What are the mean, standard deviation, quartiles, median, IQR, and range of weights of boxes shipped (in ounces)? e) One customer made a special order of a 10-pound ham. Which of the summary statistics of part d might not change if that data value were added to the distribution?

Problem 15RE Profits Here is a stem-and-leaf display showing profits as a percent of sales for 29 of the Forbes 500 largest U.S. corporations. The stems are split; each stem represents a span of 5%, from a loss of 9% to a profit of 25%. a) Find the 5-number summary. ________________ b) Draw a boxplot for these data. ________________ c) Find the mean and standard deviation. ________________ d) Describe the distribution of profits for these corporations.

Problem 16E Singers The boxplots shown display the heights (in inches) of 130 members of a choir. a) It appears that the median height for sopranos is missing, but actually the median and the upper quartile are equal. How could that happen? ________________ b) Write a few sentences describing what you see.

Hard water II The data set from England and Wales also notes for each town whether it was south or north of Derby. Here are some summary statistics and a comparative boxplot for the two regions. a) What is the overall mean mortality rate for the two regions? b) Do you see evidence of a difference in mortality rates? Explain.

Problem 14RE Accidents Progressive Insurance asked customers who had been involved in auto accidents how far they were from home when the accident happened. The data are summarized in the table. Miles from Home % of Accidents Less than 1 23 1 to 5 29 6 to 10 17 11 to 15 8 16 to 20 6 Over 20 17 a) Create an appropriate graph of these data. ________________ b) Do these data indicate that driving near home is particularly dangerous? Explain.

Problem 17E Dialysis In a study of dialysis, researchers found that “of the three patients who were currently on dialysis, 67% had developed blindness and 33% had their toes amputated.” What kind of display might be appropriate for these data? Explain.

Problem 18E Checkup One of the authors has an adopted grandson whose birth family members are very short. After examining him at his 2-year checkup, the boy’s pediatrician said that the z-score for his height relative to American 2-year-olds was -1.88. Write a sentence explaining what that means.

Music library again Corey has 4929 songs in his computer’s music library. The songs have a mean duration of 242.4 seconds with a standard deviation of 114.51 seconds. On the Nickel, by Tom Waits, is 380 seconds long. What is its z-score?

Old Faithful It is a common belief that Yellowstone’s most famous geyser erupts once an hour at very predictable intervals. The histogram below shows the time gaps (in minutes) between 222 successive eruptions. Describe this distribution.

Problem 17RE Seasons Average daily temperatures in January and July for 60 large U.S. cities are graphed in the histograms below. a) What aspect of these histograms makes it difficult to compare the distributions? b) What differences do you see between the distributions of January and July average temperatures? c) Differences in temperatures (July–January) for each of the cities are displayed in the boxplot above. Write a few sentences describing what you see.

Problem 19RE Old Faithful? Does the duration of an eruption have an effect on the length of time that elapses before the next eruption? a) The histogram below shows the duration (in minutes) of those 222 eruptions. Describe this distribution. b) Explain why it is not appropriate to find summary statistics for this distribution. c) Let’s classify the eruptions as “long” or “short,” depending upon whether or not they last at least 3 minutes. Describe what you see in the comparative boxplots.

Problem 20E Windy In the last chapter, we looked at three outliers arising from a plot of Average Wind Speed by Month in the Hopkins Forest. Each was associated with an unusually strong storm, but which was the most remarkable for its month? Here are the summary statistics for each of those three months: The outliers had values of 6.73 mph, 3.93 mph, and 2.53 mph, respectively. a) What are their z-scores? b) Which was the most extraordinary wind event?

Problem 20RE Teen drivers 2013 The National Highway Traffic Safety Administration reported that there were 3206 fatal accidents involving drivers between the ages of 15 and 19 years old the previous year, of which 65.5% involved male drivers. Of the male drivers, 18.4% involved drinking, while of the female drivers, 10.8% involved drinking. Assuming roughly equal numbers of male and female drivers, use these statistics to explain the concept of independence.

Problem 21E Acid rain Based on long-term investigation, researchers have suggested that the acidity (pH) of rainfall in the Shenandoah Mountains can be described by the Normal model N(4.9, 0.6). a) Draw and carefully label the model. ________________ b) What percent of storms produce rainfall with pH over 6? ________________ c) What percent of storms produce rainfall with pH under 4? ________________ d) The lower the pH, the more acidic the rain. What is the pH level for the most acidic 20% of all storms? ________________ e) What is the pH level for the least acidic 5% of all storms? ________________ f) What is the IQR for the pH of rainfall?

Combining scores again The first Stat exam had a mean of 80 and a standard deviation of 4 points; the second had a mean of 70 and a standard deviation of 15 points. Reginald scored an 80 on the first test and an 85 on the second. Sara scored an 88 on the first but only a 65 on the second. Although Reginald’s total score is higher, Sara feels she should get the higher grade. Explain her point of view.

Problem 21RE Liberty’s nose Is the Statue of Liberty’s nose too long? Her nose measures 4’6”, but she is a large statue, after all. Her arm is 42 feet long. That means her arm is 42/45 = 9.3 times as long as her nose. Is that a reasonable ratio? Shown in the table are arm and nose lengths of 18 girls in a Statistics class, and the ratio of arm-to-nose length for each. a) Make an appropriate plot and describe the distribution of the ratios. b) Summarize the ratios numerically, choosing appropriate measures of center and spread. c) Is the ratio of 9.3 for the Statue of Liberty unrealistically low? Explain.

Problem 23E Final exams Anna, a language major, took final exams in both French and Spanish and scored 83 on each. Her roommate Megan, also taking both courses, scored 77 on the French exam and 95 on the Spanish exam. Overall, student scores on the French exam had a mean of 81 and a standard deviation of 5, and the Spanish scores had a mean of 74 and a standard deviation of 15. a) To qualify for language honors, a major must maintain at least an 85 average for all language courses taken. So far, which student qualifies? b) Which student’s overall performance was better?

Problem 22RE Winter Olympics 2010 speed skating The top 36 women’s 500-m speed skating times are listed in the table in the next column. a) The mean finishing time was 40.72 seconds, with a standard deviation of 9.82 seconds. If the Normal model is appropriate, what percent of the times should be within 5 second of 40.72? b) What percent of the times actually fall within this interval? c) Explain the discrepancy between parts a and b.

Problem 23RE Normal cattle Using N(1152, 84), the Normal model for weights of Angus steers in Exercise, what percent of steers weigh a) over 1250 pounds? ________________ b) under 1200 pounds? ________________ c) between 1000 and 1100 pounds? Exercise Cattle The Virginia Cooperative Extension reports that the mean weight of yearling Angus steers is 1152 pounds. Suppose that weights of all such animals can be described by a Normal model with a standard deviation of 84 pounds. a) How many standard deviations from the mean would a steer weighing 1000 pounds be? ________________ b) Which would be more unusual, a steer weighing 1000 pounds or one weighing 1250 pounds?

MP3s Two companies market new batteries targeted at owners of personal music players. DuraTunes claims a mean battery life of 11 hours, while RockReady advertises 12 hours. a) Explain why you would also like to know the standard deviations of the battery lifespans before deciding which brand to buy. b) Suppose those standard deviations are 2 hours for DuraTunes and 1.5 hours for RockReady. You are headed for 8 hours at the beach. Which battery is most likely to last all day? Explain. c) If your beach trip is all weekend, and you probably will have the music on for 16 hours, which battery is most likely to last? Explain.

Problem 25RE More cattle Based on the model N(1152, 84) describing Angus steer weights, what are the cutoff values for a) the highest 10% of the weights? ________________ b) the lowest 20% of the weights? ________________ c) the middle 40% of the weights?

Cattle Using N(1152, 84), the Normal model for weights of Angus steers in Exercise 9, a) How many standard deviations from the mean would a steer weighing 1000 pounds be? b) Which would be more unusual, a steer weighing 1000 pounds or one weighing 1250 pounds?

Problem 24RE Sluggers Babe Ruth was the first great “slugger” in baseball. His record of 60 home runs in one season held for 34 years until Roger Maris hit 61 in 1961. Mark McGwire (with the aid of steroids) set a new standard of 70 in 1998. Listed below are the home run totals for each season McGwire played. Also listed are Babe Ruth’s home run totals. McGwire: 3*, 49, 32, 33, 39, 22, 42, 9*,9*, 39, 52, 58, 70, 65, 32*, 29* Ruth: 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22 a) Find the 5-number summary for McGwire’s career. ________________ b) Do any of his seasons appear to be outliers? Explain. ________________ c) McGwire played in only 18 games at the end of his first big league season, and missed major portions of some other seasons because of injuries to his back and knees. Those seasons might not be representative of his abilities. They are marked with asterisks in the list above. Omit these values and make parallel boxplots comparing McGwire’s career to Babe Ruth’s. ________________ d) Write a few sentences comparing the two sluggers. ________________ e) Create a side-by-side stem-and-leaf display comparing the careers of the two players. ________________ f) What aspects of the distributions are apparent in the stem-and-leaf displays that did not clearly show in the boxplots?

Car speeds John Beale of Stanford, California, recorded the speeds of cars driving past his house, where the speed limit read 20 mph. The mean of 100 readings was 23.84 mph, with a standard deviation of 3.56 mph. (He actually recorded every car for a two-month period. These are 100 representative readings.) a) How many standard deviations from the mean would a car going under the speed limit be? b) Which would be more unusual, a car traveling 34 mph or one going 10 mph?

Problem 26RE More IQs In the Normal model N(100, 16), what cutoff value bounds a) the highest 5% of all IQs? ________________ b) the lowest 30% of the IQs? ________________ c) the middle 80% of the IQs?

Problem 27E More cattle Recall that the beef cattle described in Exercise 25 had a mean weight of 1152 pounds, with a standard deviation of 84 pounds. a) Cattle buyers hope that yearling Angus steers will weigh at least 1000 pounds. To see how much over (or under) that goal the cattle are, we could subtract 1000 pounds from all the weights. What would the new mean and standard deviation be? b) Suppose such cattle sell at auction for 40 cents a pound. Find the mean and standard deviation of the sale prices (in dollars) for all the steers. Exercise 25: Cattle Using N(1152, 84), the Normal model for weights of Angus steers in Exercise 9, a) How many standard deviations from the mean would a steer weighing 1000 pounds be? b) Which would be more unusual, a steer weighing 1000 pounds or one weighing 1250 pounds? Exercise 9: Normal cattle The Virginia Cooperative Extension reports that the mean weight of yearling Angus steers is 1152 pounds. Suppose that weights of all such animals can be described by a Normal model with a standard deviation of 84 pounds. What percent of steers weigh a) over 1250 pounds? b) under 1200 pounds? c) between 1000 and 1100 pounds?

Problem 29E Cattle, part III Suppose the auctioneer in Exercise 27 sold a herd of cattle whose minimum weight was 980 pounds, median was 1140 pounds, standard deviation 84 pounds, and IQR 102 pounds. They sold for 40 cents a pound, and the auctioneer took a $20 commission on each animal. Then, for example, a steer weighing 1100 pounds would net the owner 0.40(1100) - 20 = $420. Find the minimum, median, standard deviation, and IQR of the net sale prices. Exercise 27: More cattle Recall that the beef cattle described in Exercise 25 had a mean weight of 1152 pounds, with a standard deviation of 84 pounds. a) Cattle buyers hope that yearling Angus steers will weigh at least 1000 pounds. To see how much over (or under) that goal the cattle are, we could subtract 1000 pounds from all the weights. What would the new mean and standard deviation be? b) Suppose such cattle sell at auction for 40 cents a pound. Find the mean and standard deviation of the sale prices (in dollars) for all the steers. Exercise 25: Cattle Using N(1152, 84), the Normal model for weights of Angus steers in Exercise 9, a) How many standard deviations from the mean would a steer weighing 1000 pounds be? b) Which would be more unusual, a steer weighing 1000 pounds or one weighing 1250 pounds? Exercise 9: Normal cattle The Virginia Cooperative Extension reports that the mean weight of yearling Angus steers is 1152 pounds. Suppose that weights of all such animals can be described by a Normal model with a standard deviation of 84 pounds. What percent of steers weigh a) over 1250 pounds? b) under 1200 pounds? c) between 1000 and 1100 pounds?

Problem 28E Cramming One Thursday, researchers gave students enrolled in a section of basic Spanish a set of 50 new vocabulary words to memorize. On Friday the students took a vocabulary test. When they returned to class the following Monday, they were retested—without advance warning. Both sets of test scores for the 25 students are shown below. Fri Mon Fri Mon 42 36 50 47 44 44 34 34 45 46 38 31 48 38 43 40 44 40 39 41 43 38 46 32 41 37 37 36 35 31 40 31 43 32 41 32 48 37 48 39 43 41 37 31 45 32 36 41 47 44 a) Create a graphical display to compare the two distributions of scores. ________________ b) Write a few sentences about the scores reported on Friday and Monday. ________________ c) Create a graphical display showing the distribution of the changes in student scores. ________________ d) Describe the distribution of changes.

Problem 28RE IQ, finis Consider the IQ model N(100, 16) one last time. a) What IQ represents the 15th percentile? ________________ b) What IQ represents the 98th percentile? ________________ c) What’s the IQR of the IQs?

Mail Here are the number of pieces of mail received at a school office for 36 days. 123 70 90 151 115 97 80 78 72 100 128 130 52 103 138 66 135 76 112 92 93 143 100 88 118 118 106 110 75 60 95 131 59 115 105 85 a) Plot these data. b) Find appropriate summary statistics. c) Write a brief description of the school’s mail deliveries. d) What percent of the days actually lie within one standard deviation of the mean? Comment.

Problem 29RE Cholesterol Assume the cholesterol levels of adult American women can be described by a Normal model with a mean of 188 mg/dL and a standard deviation of 24. a) Draw and label the Normal model. ________________ b) What percent of adult women do you expect to have cholesterol levels over 200 mg/dL? ________________ c) What percent of adult women do you expect to have cholesterol levels between 150 and 170 mg/dL? ________________ d) Estimate the IQR of the cholesterol levels. ________________ e) Above what value are the highest 15% of women’s cholesterol levels?

Problem 30E Mensa People with z-scores above 2.5 on an IQ test are sometimes classified as geniuses. If IQ scores have a mean of 100 and a standard deviation of 16 points, what IQ score do you need to be considered a genius?

Birth order revisited Consider again the data on birth order and college majors in Exercise 28. a) What is the marginal distribution of majors? b) What is the conditional distribution of majors for the oldest children? c) What is the conditional distribution of majors for the children born second? d) Do you think that college major appears to be independent of birth order? Explain.

Problem 31RE Engines One measure of the size of an automobile engine is its “displacement,” the total volume (in liters or cubic inches) of its cylinders. Summary statistics for several models of new cars are shown. These displacements were measured in cubic inches. a) How many cars were measured? b) Why might the mean be so much larger than the median? c) Describe the center and spread of this distribution with appropriate statistics. d) Your neighbor is bragging about the 227-cubic-inch engine he bought in his new car. Is that engine unusually large? Explain. e) Are there any engines in this data set that you would consider to be outliers? Explain. f) Is it reasonable to expect that about 68% of car engines measure between 88 and 266 cubic inches? (That’s 177.289 ± 88.8767.) Explain. g) We can convert all the data from cubic inches to cubic centimeters (cc) by multiplying by 16.4. For example, a 200-cubic-inch engine has a displacement of 3280 cc. How would such a conversion affect each of the summary statistics?

Problem 31E Temperatures A town’s January high temperatures average 36°F with a standard deviation of 10°, while in July the mean high temperature is 74° and the standard deviation is 8°. In which month is it more unusual to have a day with a high temperature of 55°? Explain.

Rock concerts A popular band on tour played a series of concerts in large venues. They always drew a large crowd, averaging 21,359 fans. While the band did not announce (and probably never calculated) the standard deviation, which of these values do you think is most likely to be correct: 20, 200, 2000, or 20,000 fans? Explain your choice.

Problem 32RE Body temperatures Most people think that the “normal” adult body temperature is 98.6°F. That figure, based on a 19th-century study, has recently been challenged. In a 1992 article in the Journal of the American Medical Association, researchers reported that a more accurate figure may be 98.2°F. Furthermore, the standard deviation appeared to be around 0.7°F. Assume that a Normal model is appropriate. a) In what interval would you expect most people’s body temperatures to be? Explain. ________________ b) What fraction of people would be expected to have body temperatures above 98.6°F? ________________ c) Below what body temperature are the coolest 20% of all people?

Small steer In Exercise 25, we suggested the model N(1152, 84) for weights in pounds of yearling Angus steers. What weight would you consider to be unusually low for such an animal? Explain.

Problem 33RE Age and party 2011 The Pew Research Center conducts surveys regularly asking respondents which political party they identify with or lean toward. Among their results is the following table relating preferred political party and age. (http://people-press.org/) Party Republican/Lean Rep. Democrat/Lean Dem. Neither Total Age 18–29 318 424 73 815 30–49 991 1058 203 2252 50–64 1260 1407 264 2931 65+ 1136 1087 193 2416 Total 3705 3976 733 8414 a) What percent of people surveyed were Republicans or leaned Republican? ________________ b) Do you think this might be a reasonable estimate of the percentage of all voters who are Republicans or lean Republicans? Explain. ________________ c) What percent of people surveyed were under 30 or over 65? ________________ d) What percent of people were classified as “Neither” and under the age of 30? ________________ e) What percent of the people classified as “Neither” were under 30? ________________ f) What percent of people under 30 were classified as “Neither”?

Problem 34RE Pay According to the Bureau of Labor Statistics, the mean hourly wage for Chief Executives in 2009 was $80.43 and the median hourly wage was $77.27. By contrast, for General and Operations Managers, the mean hourly wage was $53.15 and the median was $44.55. Are these wage distributions likely to be symmetric, skewed left, or skewed right? Explain.

Trees A forester measured 27 of the trees in a large woods that is up for sale. He found a mean diameter of 10.4 inches and a standard deviation of 4.7 inches. Suppose that these trees provide an accurate description of the whole forest and that a Normal model applies. a) Draw the Normal model for tree diameters. b) What size would you expect the central 95% of all trees to be? c) About what percent of the trees should be less than an inch in diameter? d) About what percent of the trees should be between 5.8 and 10.4 inches in diameter? e) About what percent of the trees should be over 15 inches in diameter?

High IQ Exercise 8, proposes modeling IQ scores with N(100, 16). What IQ would you consider to be unusually high? Explain.

Problem 35RE Age and party 2011 II Consider again the Pew Research Center results on age and political party in Exercise. a) What is the marginal distribution of party affiliation? ________________ b) Create segmented bar graphs displaying the conditional distribution of party affiliation for each age group. ________________ c) Summarize these poll results in a few sentences that might appear in a newspaper article about party affiliation in the United States. ________________ d) Do you think party affiliation is independent of the voter’s age? Explain. Exercise Age and party 2011 The Pew Research Center conducts surveys regularly asking respondents which political party they identify with or lean toward. Among their results is the following table relating preferred political party and age. (http://people-press.org/) Party Republican/Lean Rep. Democrat/Lean Dem. Neither Total Age 18–29 318 424 73 815 30–49 991 1058 203 2252 50–64 1260 1407 264 2931 65+ 1136 1087 193 2416 Total 3705 3976 733 8414 a) What percent of people surveyed were Republicans or leaned Republican? ________________ b) Do you think this might be a reasonable estimate of the percentage of all voters who are Republicans or lean Republicans? Explain. ________________ c) What percent of people surveyed were under 30 or over 65? ________________ d) What percent of people were classified as “Neither” and under the age of 30? ________________ e) What percent of the people classified as “Neither” were under 30? ________________ f) What percent of people under 30 were classified as “Neither”?

Problem 36E Combining scores again The first Stat exam had a mean of 80 and a standard deviation of 4 points; the second had a mean of 70 and a standard deviation of 15 points. Reginald scored an 80 on the first test and an 85 on the second. Sara scored an 88 on the first but only a 65 on the second. Although Reginald’s total score is higher, Sara feels she should get the higher grade. Explain her point of view.

Bike safety 2012 The Bicycle Helmet Safety Institute website includes a report on the number of bicycle fatalities per year in the United States. The table below shows the counts for the years 1994–2012. a) What are the W’s for these data? b) Display the data in a stem-and-leaf display. c) Display the data in a timeplot. d) What is apparent in the stem-and-leaf display that is hard to see in the timeplot? e) What is apparent in the timeplot that is hard to see in the stem-and-leaf display? f) Write a few sentences about bicycle fatalities in the United States.

Problem 37RE Some assembly required A company that markets build-it-yourself furniture sells a computer desk that is advertised with the claim “less than an hour to assemble.” However, through postpurchase surveys the company has learned that only 25% of its customers succeeded in building the desk in under an hour. The mean time was 1.29 hours. The company assumes that consumer assembly time follows a Normal model. a) Find the standard deviation of the assembly time model. b) One way the company could solve this problem would be to change the advertising claim. What assembly time should the company quote in order that 60% of customers succeed in finishing the desk by then? c) Wishing to maintain the “less than an hour” claim, the company hopes that revising the instructions and labeling the parts more clearly can improve the 1-hour success rate to 60%. If the standard deviation stays the same, what new lower mean time does the company need to achieve? d) Months later, another postpurchase survey shows that new instructions and part labeling did lower the mean assembly time, but only to 55 minutes. Nonetheless, the company did achieve the 60%-in-an-hour goal, too. How was that possible?

Problem 37E Old Faithful It is a common belief that Yellowstone’s most famous geyser erupts once an hour at very predictable intervals. The histogram below shows the time gaps (in minutes) between 222 successive eruptions. Describe this distribution.

Problem 38E Seasons Average daily temperatures in January and July for 60 large U.S. cities are graphed in the following histograms. a) What aspect of these histograms makes it difficult to compare the distributions? ________________ b) What differences do you see between the distributions of January and July average temperatures? ________________ c) Differences in temperatures (July–January) for each of the cities are displayed in the boxplot above. Write a few sentences describing what you see.

Problem 39E Winter Olympics 2010 downhill Fifty-nine men completed the men’s alpine downhill race in Vancouver. The gold medal winner finished in 114.3 seconds. Here are the times (in seconds) for all competitors (espn.go.com/olympics/winter/2010/results/_/sport/1 /event/2): a) The mean time was 117.34 seconds, with a standard deviation of 2.465 seconds. If the Normal model is appropriate, what percent of times will be less than 114.875 seconds? b) What is the actual percent of times less than 114.875 seconds? c) Why do you think the two percentages don’t agree? d) Make a histogram of these times. What do you see?

Profits Here is a stem-and-leaf display showing profits as a percent of sales for 29 of the Forbes 500 largest U.S. corporations. The stems are split; each stem represents a span of 5%, from a loss of 9% to a profit of 25%. a) Find the 5-number summary. b) Draw a boxplot for these data. c) Find the mean and standard deviation. d) Describe the distribution of profits for these corporations.

Problem 40E MP3s Two companies market new batteries targeted at owners of personal music players. DuraTunes claims a mean battery life of 11 hours, while RockReady advertises 12 hours. a) Explain why you would also like to know the standard deviations of the battery lifespans before deciding which brand to buy. ________________ b) Suppose those standard deviations are 2 hours for Dura Tunes and 1.5 hours for Rock Ready. You are headed for 8 hours at the beach. Which battery is most likely to last all day? Explain. ________________ c) If your beach trip is all weekend, and you probably will have the music on for 16 hours, which battery is most likely to last? Explain.

Receivers 2013 NFL data from the 2013 football season reported the number of yards gained by each of the league’s 181 wide receivers: The mean is 426.98 yards, with a standard deviation of 408.34 yards. a) According to the Normal model, what percent of receivers would you expect to gain more yards than 2 standard deviations above the mean number of yards? b) For these data, what does that mean? c) Explain the problem in using a Normal model here.

Problem 42E Old Faithful? Does the duration of an eruption have an effect on the length of time that elapses before the next eruption? a) The histogram below shows the duration (in minutes) of those 222 eruptions. Describe this distribution. ________________ b) Explain why it is not appropriate to find summary statistics for this distribution. ________________ c) Let’s classify the eruptions as “long” or “short,” depending upon whether or not they last at least 3 minutes. Describe what you see in the comparative boxplots.

Problem 43E Cattle, part III Suppose the auctioneer in Exercise sold a herd of cattle whose minimum weight was 980 pounds, median was 1140 pounds, standard deviation 84 pounds, and IQR 102 pounds. They sold for 40 cents a pound, and the auctioneer took a $20 commission on each animal. Then, for example, a steer weighing 1100 pounds would net the owner 0.40(1100) - 20 = +420. Find the minimum, median, standard deviation, and IQR of the net sale prices. Exercise More cattle Recall that the beef cattle described in Exercise had a mean weight of 1152 pounds, with a standard deviation of 84 pounds. a) Cattle buyers hope that yearling Angus steers will weigh at least 1000 pounds. To see how much over (or under) that goal the cattle are, we could subtract 1000 pounds from all the weights. What would the new mean and standard deviation be? ________________ b) Suppose such cattle sell at auction for 40 cents a pound. Find the mean and standard deviation of the sale prices for all the steers. Exercise1 Cattle The Virginia Cooperative Extension reports that the mean weight of yearling Angus steers is 1152 pounds. Suppose that weights of all such animals can be described by a Normal model with a standard deviation of 84 pounds. a) How many standard deviations from the mean would a steer weighing 1000 pounds be? ________________ b) Which would be more unusual, a steer weighing 1000 pounds or one weighing 1250 pounds?

Problem 45E Cattle, finis Consider the Angus weights model N(1152, 84) one last time. a) What weight represents the 40th percentile? b) What weight represents the 99th percentile? c) What’s the IQR of the weights of these Angus steers?

Problem 44E Sample A study in South Africa focusing on the impact of health insurance identified 1590 children at birth and then sought to conduct follow-up health studies 5 years later. Only 416 of the original group participated in the 5-year follow-up study. This made researchers concerned that the follow-up group might not accurately resemble the total group in terms of health insurance. The table in the next column summarizes the two groups by race and by presence of medical insurance when the child was born. Carefully explain how this study demonstrates Simpson’s paradox. (Birth to Ten Study, Medical Research Council, South Africa) Number (%) Insured Follow-Up Not Traced Race Black 36 of 404 (8.9%) 91 of 1048 (8.7%) White 10 of 12 (83.3%) 104 of 126 (82.5%) Overall 46 of 416 (11.1%) 195 of 1174 (16.6%)

Cholesterol Assume the cholesterol levels of adult American women can be described by a Normal model with a mean of 196 mg/dL and a standard deviation of 27. a) Draw and label the Normal model. b) What percent of adult women do you expect to have cholesterol levels over 200 mg/dL? c) What percent of adult women do you expect to have cholesterol levels between 170 and 180 mg/dL? d) Estimate the IQR of the cholesterol levels. e) Above what value are the highest 15% of women’s cholesterol levels?

Problem 48E Rock concerts A popular band on tour played a series of concerts in large venues. They always drew a large crowd, averaging 21,359 fans. While the band did not announce (and probably never calculated) the standard deviation, which of these values do you think is most likely to be correct: 20, 200, 2000, or 20,000 fans? Explain your choice.

Problem 46E Caught speeding Suppose police set up radar surveillance on the Stanford street described in Exercise. They handed out a large number of tickets to speeders going a mean of 28 mph, with a standard deviation of 2.4 mph, a maximum of 33 mph, and an IQR of 3.2 mph. Local law prescribes fines of $100, plus $10 per mile per hour over the 20 mph speed limit. For example, a driver convicted of going 25 mph would be fined 100 + 10152 = +150. Find the mean, standard deviation, maximum, and IQR of all the potential fines. Exercise Car speeds John Beale of Stanford, CA, recorded the speeds of cars driving past his house, where the speed limit was 20 mph. The mean of 100 readings was 23.84 mph, with a standard deviation of 3.56 mph. (He actually recorded every car for a two-month period. These are 100 representative readings.) a) How many standard deviations from the mean would a car going under the speed limit be? ________________ b) Which would be more unusual, a car traveling 34 mph or one going 10 mph?

Problem 49E Guzzlers? Environmental Protection Agency (EPA) fuel economy estimates for automobile models tested recently predicted a mean of 24.8 mpg and a standard deviation of 6.2 mpg for highway driving. Assume that a Normal model can be applied. a) Draw the model for auto fuel economy. Clearly label it, showing what the 68–95–99.7 Rule predicts. ________________ b) In what interval would you expect the central 68% of autos to be found? ________________ c) About what percent of autos should get more than 31 mpg? ________________ d) About what percent of cars should get between 31 and 37.2 mpg? ________________ e) Describe the gas mileage of the worst 2.5% of all cars.

Problem 50E Mail Here are the number of pieces of mail received at a school office for 36 days. 123 70 90 151 115 97 80 78 72 100 128 130 52 103 138 66 135 76 112 92 93 143 100 88 118 118 106 110 75 60 95 131 59 115 105 85 a) Plot these data. ________________ b) Find appropriate summary statistics. ________________ c) Write a brief description of the school’s mail deliveries. ________________ d) What percent of the days actually lie within one standard deviation of the mean? Comment.

Eggs Hens usually begin laying eggs when they are about 6 months old. Young hens tend to lay smaller eggs, often weighing less than the desired minimum weight of 43 grams. a) The average weight of the eggs produced by the young hens is 48.3 grams, and only 26% of their eggs exceed the desired minimum weight. If a Normal model is appropriate, what would the standard deviation of the egg weights be? b) By the time these hens have reached the age of 1 year, the eggs they produce average 67.8 grams, and 98% of them are above the minimum weight. What is the standard deviation for the appropriate Normal model for these older hens? c) Are egg sizes more consistent for the younger hens or the older ones? Explain.

Problem 52E High IQ Exercise proposes modeling IQ scores with N(100, 16). What IQ would you consider to be unusually high? Explain.