Chapter 12 Exercises

Section 12-1 . CONCEPT CHECK Write true or false for each statement. 1. Usually a sample has fewer members than the population. 2. A frequency distribution and a relative frequency distribution are the same thing. 3. A grouped frequency distribution has the same number of groups as there are data items. 4. A grouped frequency distribution should have from 5 to 12 classes. 5. A histogram has numerical scales on both vertical and horizontal axes. 6. Some data sets are more accurately depicted in a circle graph than in a pie chart. In each case, use the given data to do the following: (a) Construct frequency and relative frequency distributions, in a table similar to Table 2. (b) Construct a histogram. (c) Construct a frequency polygon. 7. Preparation for Summer A magazine identified the five "maintenance" activities often performed by people as summer approaches. 1. Prep the car for road trips. 2. Clean up the house or apartment. 3. Groom the garden. 4. Exercise the body. 5. Organize the wardrobe. The following data are the responses of 30 people who were asked, on June 1, how many of the five activities they had accomplished. 1 1 3 1 0 3 0 0 2 1 2 2 0 0 5 3 4 0 1 0 4 2 0 2 0 1 0 1 2 3 8. Favorite Numbers The following data are the “favorite numbers” (single digits, 1 through 9) of the 24 pupils of a second grade class. 4 7 2 7 6 1 7 2 9 8 5 1 4 3 8 9 5 5 9 2 6 5 2 7 In each case, use the given data to do the following: (a) Construct grouped frequency and relative frequency distributions, in a table similar to Table 3. (Follow the suggested guidelines for class limits and class width.) (b) Construct a histogram. (c) Construct a frequency polygon. 9. Exam Scores The scores of the 48 members of a sociology lecture class on a 50-point exam are as follows. 40 43 44 33 41 40 33 39 42 36 43 42 45 42 23 41 43 40 37 47 44 48 31 46 38 31 46 38 39 42 47 40 47 41 48 42 45 29 32 38 45 28 49 46 47 36 48 46 Use six classes with a uniform class width of 5 points, where the lower limit of the first class is 21 points. 10. Charge Card Account Balances The following raw data represent the monthly account balances (to the nearest dollar) for a sample of 50 brand-new charge card users. 51 175 46 138 79 118 90 163 88 107 75 154 85 60 42 54 62 128 114 73 108 119 116 145 129 130 81 105 96 71 83 145 117 60 125 130 94 88 136 112 62 165 118 84 74 62 81 110 108 71 Use seven classes with a uniform width of 20 dollars, where the lower limit of the first class is 40 dollars. 11. Daily High Temperatures The following data represent the daily high temperatures (in degrees Fahrenheit) for the month of June in a town in the Sacramento Valley of California. 79 84 88 96 102 104 99 97 92 94 85 92 100 99 101 104 97 108 106 106 90 82 74 72 83 107 98 102 97 94 Use eight classes with a uniform width of 5 degrees, where the lower limit of the first class is 70 degrees. 12. IQ Scores of College Freshmen The following data represent IQ scores of a group of 50 college freshmen. 121 109 118 92 130 112 114 117 122 115 121 107 108 113 124 112 111 106 116 118 104 107 118 118 110 124 115 103 100 114 96 124 116 123 104 135 113 126 116 111 127 134 98 129 102 103 107 113 117 112 Use nine classes with a uniform width of 5, where the lower limit of the first class is 91.

In each case, construct a stem-and-leaf display for the given data. Treat the ones digits as the leaves. For any single-digit data, use a stem of 0. 13. Games Won in the National Basketball Association Approaching midseason, the teams in the National Basketball Association had won the following numbers of games. 27 20 29 11 26 11 12 7 26 18 22 19 14 13 22 9 25 11 10 15 38 10 22 23 31 8 24 15 24 15 14. Accumulated College Units The students in a biology class were asked how many college units they had accumulated to date. Their responses follow. 22 4 13 12 21 33 15 17 12 24 32 42 26 11 53 62 42 25 13 8 54 18 21 14 19 17 38 17 20 10 15. Distances to School The following data are the daily round-trip distances to school (in miles) for 30 randomly chosen students attending a community college in California. 16 30 10 11 18 26 34 18 8 12 21 14 5 22 4 25 9 10 6 21 12 18 9 16 44 23 4 13 36 8 16. Yards Gained in the National Football League The following data represent net yards gained per game by National Football League running backs who played during a given week of the season. 25 19 36 73 37 88 67 33 54 79 19 39 45 22 58 7 30 43 24 36 65 43 33 55 40 29 112 60 86 62 52 29 18 25 41 3 49 16 32 46 Sample Masses in a Geology Laboratory Stem-and-leaf displays can be modified in various ways in order to obtain a reasonable number of stems. The following data, representing the measured masses (in grams) of thirty mineral samples in a geology lab, are shown in a double- stem display in Table 7 . 60.7 41.4 50.6 39.5 46.4 58.1 49.7 38.8 61.6 55.2 47.3 52.7 62.4 59.0 44.9 35.6 36.2 40.6 56.9 42.6 34.7 48.3 55.8 54.2 33.8 51.3 50.1 57.0 42.8 43.7 Table 7 Stem-and-Leaf Display for Mineral Sample Masses (30-34) 3 | 4.7 3.8 (35-39) 3 | 9.5 8.8 5.6 6.2 (40-44) 4 | 1.4 4.9 0.6 2.6 2.8 3.7 (45-49) 4 | 6.4 9.7 7.3 8.3 (50=54) 5 | 0.6 2.7 4.2 1.3 0.1 ______ (55-59) 5 | 8.1 5.2 9.0 6.9 5.8 7.0___ (60-64) 6 | 0.7 1.6 2.4 Write a short answer for each problem. 35. Describe how the stem-and-leaf display of Table 7 was constructed. 36. Explain why Table 7 is called a "double-stem" display. 37. In general, how many stems (total) are appropriate for a stem-and-leaf display? Explain your reasoning. Solve each problem. 38. Record Temperatures According to the National Climatic Data Center, U.S. Department of Commerce, the highest temperatures (in degrees Fahrenheit) ever recorded in the 50 states (as of June 15, 2017) were as follows. 112 100 128 120 134 114 106 110 109 112 100 118 117 116 118 121 114 114 105 109 107 112 115 115 118 117 118 125 106 110 122 108 110 121 113 120 119 111 104 113 120 113 120 117 107 110 118 112 114 115 Present these data in a double-stem display

Measuring Elapsed Times While doing an experiment, a physics student recorded the following sequence of elapsed times (in seconds) in a lab notebook. 2.16, 22.2, 2.96, 2.20, 2.73, 2.22, 2.39 21. Find the mean. 22. Find the median. When reviewing the calculations later, the student decided that the entry 22.2 should have been recorded as 2.22 and made that change in the listing. 23. Find the mean for the new list. 24. Find the median for the new list. 25. Which measure, the mean or the median, was affected more by correcting the error? 26. Find the mode(s), if any, for the original data and the changed data. Scores on Management Examinations Thao earned the following scores on her six management exams last semester. 79, 81, 44, 89, 79, 90 27. Find the mean, median, and mode for Thao's scores. 28. Which of the three averages probably is the best indicator of Thao's ability? 29. Thao has a chance to replace her score of 44 by taking a "make-up" exam. What must she score on the make-up exam to get an overall average (mean) of 85? For each of the following frequency distributions, find (a) the mean (to the nearest tenth), (b) the median, and (c) the mode or modes (if any). 30. 31. 32. Average Employee Salary A company has 15 employees with a salary of $21,500 11 employees with a salary of $23,000 17 employees with a salary of $25,800 2 employees with a salary of $31,500, 4 employees with a salary of $38,900, 1 employee with a salary of $147,500. Find the mean salary for the employees (to the nearest hundred dollars). Grade-Point Averages Find the grade-point average for each of the following students. Assume A = 4, B = 3, C = 2, D = 1, and F = 0. Round to the nearest hundredth. 33. 34. Federal Budget Totals The table gives federal receipts and outlays for the years 2013—2017. (The 2017 figures are estimates.) Use this information for Exercises 35 and 36. Over the five-year period, find (a) the mean and (b) the median for each of the following 35. receipts 36. outlays World Cell Phone Use In 2016, just the top six countries accounted for over 50% of cell phone subscriptions worldwide. Use the data in the table for Exercises 37 and 38. Value Frequency 615 540 605 579 586 600 13 7 9 14 7 5 Value Frequency 6 7 8 9 3 1 8 4 Unit s Grad e 8 3 5 A B C Unit s Grad e 2 3 7 2 C B A F Fiscal Year Receipts (in billions of dollars) Outlays (in billions of dollars) 2013 2014 2015 2016 2017 $2775.1 3021.5 3249.9 3268.0 3459.7 $3454.6 3506.1 3688.4 3852.6 4062.2

Country Cell Phone Subscriptions (in millions)___________ China 1364.9 India 1127.8 United States 416.7 Indonesia 385.6 Russia 244.1 Brazil 231.4 37. Find the approximate mean number of cell phone subscriptions for these six countries in 2016. 38. The Unite States accounted for about 5.547% of world- wide subscriptions. About how many subscriptions were active in the world that year? 39. Find the approximate mean number of cell phone subscriptions for the BRIC countries (Brazil, Russia, India, and China) in 2016. 40. Find the approximate mean number of cell phone subscriptions for the non-BRIC countries in the list in 2016. Crew, Passengers, and Entertainers on cruise ships. The table shows, for four cruises, the numbers of crew members, passengers, and entertainers (not included as passengers). For each quantity in Exercises 41—43, find (a) the mean, and (b) the median. Cruise Crew Passengers Entertainers Alaska 185 1900 35 Mexico 223 3000 75 Scandinavia 175 1200 20 Mediterranean 215 2700 50 41. number of crew members per cruise 42. number of passengers per cruise 43. total number of persons per cruise Olympic Medal Standings The top ten medal-winning nations in the 2018 Winter Olympics at PyeongChang, South Korea, are shown in the table. Use the given information for Exercises 44 - 47 Medal Standings for the 2018 Winter Olympics Place Nation Gold Silver Bronze Total 1 Norway 14 14 11 39 2 Germany 14 10 7 31 3 Canada 11 8 10 29 4 United States 9 8 6 23 5 Netherlands 8 6 6 20 6 Korea 5 8 4 17 7 OAR 2 6 9 17 8 Switzerland 5 6 4 15 9 France 5 4 6 15 10 Sweden 7 6 1 14 Calculate the following for all nations shown. 44. the mean number of gold medals 45. the median number of bronze medals 46. the mode, or modes, for the number of silver medals 47. each of the following for the total number of medals a) mean b) median c) mode or modes In Exercises 48 and 49, use the given stem-and-leaf display to identify (a) the mean, (b) the median, and (c) the mode or modes (if any) for the data represented. 48. Online Sales The display here represents prices (to the nearest dollar) charged by 23 different online sellers for a new car alternator. Give answers to the nearest dollar. 9| 9 10| 2 3 10| 5 8 9 11| 1 2 3 3 11| 5 6 7 8 8 12| 0 2 4 12| 5 6 7 9 13| 4 49. Sores on a Biology Exam The display her represents scores achieved on a 100-point biology exam by the 34 members of a class. 4| 7 5| 1 3 6 6| 2 5 5 6 7 8 8 7| 0 4 5 6 7 7 8 8 8 8 9 8| 0 1 1 3 4 5 5 9| 0 0 0 1 6

Consider the following sample. 13, 14, 17, 19, 21, 22, 25 33. Compute the mean and standard deviation for the sample (each to the nearest hundredth). 34. Now add 5 to each item of the given sample and compute the mean and standard deviation for the new sample. 35. Go back to the original sample. This time subtract 10 from each item, and compute the mean and standard deviation of the new sample. 36. Based on your answers for the previous three exercises, what happens to the mean and standard deviation when all items of a sample have the same constant k added or subtracted? 37. Go back to the original sample again. This time multiply each item by 3, and compute the mean and standard deviation of the new sample. 38. What happens to the mean and standard deviation when all items of a sample are multiplied by the same constant k? Section 12-4 . CONCEPT CHECK Write true or false for each statement. 1. The z-score is the frequency of the item z in a frequency distribution. 2. The nth percentile is the item that exceeds n percent of the items in a distribution. 3. Every quartile can be found by computing a instead. 4. Within any very large distribution, there are 100 percentiles, 10 deciles, and 4 quartiles. 5. A box plot specifically shows the minimum and maximum data values, as well as all of the quartiles. 6. A lack of symmetry in a distribution can show up in either the box or the whiskers of a box plot. Suppose a distribution of 400 items has mean 125 and standard deviation 5. 7. Find the z-score of the item x = 133. 1.60 Solve each problem. 39. Comparing Water Heater Lifetimes Two brands of electric water heaters, both carrying 6-year warranties, were sampled and tested under controlled conditions. Five of each brand failed after the numbers of months shown here. Brand A: 74, 65, 70, 64, 71 Brand B: 69, 70, 62, 72, 60 (a) Calculate both sample means. (b) Calculate both sample standard deviations. (c) Which brand apparently lasts longer? (d) Which brand has the more consistent lifetime? 8. Find the z-score of the item x = 113. -2.40 9. How many items are less than Q1? 10. How many items are greater than D1? Numbers of Restaurant Customers Refer to the dinner customers data of Example 3 . Approximate each of the following. Use the methods illustrated in this section. 11. the fifteenth percentile 12. the thirty-fifth percentile 13. the second decile 14. the eighth decile In the following exercises, make use of z-scores. 15. Relative Positions on Sociology Quizzes In a sociology class, Neil scored 5 on a quiz for which the class mean and standard deviation were 4.6 and 2.1, respectively. Janet scored 6 on another quiz for which the class mean and standard deviation were 4.9 and 2.3, respectively. Relatively speaking, which student did better? Janet

16. Relative Performances in Track Events In Saturday's track meet Ramon, a high jumper, jumped 6 feet 5 inches. Conference high jump marks for the past season had a mean of 6 feet even and a standard deviation of 3.5 inches. Ric, Ramon's teammate, achieved 18 feet 8 inches in the long jump. In that event the conference season average (mean) and standard deviation were 16 feet 6 inches and 1 foot 10 inches, respectively. Relative to this past season in this conference, which athlete had a better performance on Saturday? Ramon 17. Relative Lifetimes of Tires The lifetimes of Brand A tires are distributed with mean 45,000 miles and standard deviation 4500 miles, while Brand B tires last for only 38,000 miles on the average (mean) with standard deviation 2080 miles. Nicole's Brand A tires lasted 37,000 miles, and Yvette's Brand B tires lasted 35,000 miles. Relatively speaking, within their own brands, which driver got the better wear? Yvette 18. Relative Ratings of Fish Caught In a certain lake, the trout average 12 inches in length with a standard deviation of 2.65 inches. The bass average 4 pounds in weight with a standard deviation of 0.9 pound. If Tobi caught an 18-inch trout and Katrina caught a 6-pound bass, then relatively speaking, which catch was the better trophy? Tobi’s trout World's Largest Energy Producers and Consumers The table includes only countries in the top ten in 2013 for both production and consumption of energy. (Energy units are quadrillion Btu,) Population is for midyear 2017, in millions. Use this information for Exercises 19—30. Country Population Production Consumption China 1379 97.3 122.5 United States 327 76.4 97.1 Russia 142 54.0 30.5 Canada 36 15.8 14.4 India 1282 13.4 24.1 Compute z-scores (accurate to one decimal place) for the following quantities. 19. Russia's population -0.8 20. U.S. production 0.7 21. China's consumption 1.3 22. India's consumption -0.7 In each case, determine which country occupied the given position. 23. the twenty-fifth percentile in population Russia 24. the second decile in production Canada 25. the third quartile in consumption United States 26. the first decile in production India Solve each problem. 27. Determine which was relatively higher: Canada in production or India in consumption. India 28. Construct box plots for both production and consumption, one above the other in the same drawing. 29. What does your box plot of Exercise 28 for consumption indicate about the following characteristics of the consumption data? (a) the central tendency (b) the dispersion (c) the location of the middle half of the data items 30. Comparing your two box plots of Exercise 28, what can you say about energy among the world's top producers and consumers of 2013? The ratings for the ten leading passers in the league for 2017 regular season play are ranked in the following table. Rank NFL Passer Rating Points 1 Carson Wentz, Philadelphia 75.9 2 Case Keenum, Minnesota 69.7 3 Tom Brady, New England 67.4 4 Dak Prescott, Dallas 66.7 5 Matt Ryan, Atlanta 63.7 6 Ben Roethlisberger, Pittsburgh 63.2 7 Matthew Stafford, Detroit 61.7 8 Alex Smith, Kansas City 61.6 9 Drew Brees, New Orleans 59.0 10 Russell Wilson, Seattle 58.3 Find the measures (to one decimal place) in Exercises 50- 55. 50. the sixth decile 51. the three quartiles 52. the midrange (See Exercise 33.) 53. the ninety-fifth percentile 54. the interquartile range (See Exercise 35.) 55. the midquartile (see Exercise 34.) 56. Construct a box plot for the rating points data.

distributed with a mean 14 ounces and standard deviation 0.6 ounce. Among a sample of 1000 of those peaches, about how many could be expected to have weights as follows? 31. more than 12 ounces 1000 32. between 13.5 and 15.5 ounces 791 33. at least 14 ounces 500 34. between 14 and 15.2 ounces 477 35. between 14.9 and 15.5 ounces 61 36. less than 13.4 ounces 159 IQs of Employees A large company employs workers whose IQs are distributed normally with mean 105 and standard deviation 12.5. Management uses this information to assign employees to projects that will be challenging, but not too challenging. What percent of the employees would have IQs that satisfy the following criteria? 37. less than 90 11.5% 38. more than 120 11.5% 39. between 100 and 120 54.0% 40. between 85 and 100 29% Net Weight of Cereal Boxes A certain dry cereal is packaged in 12-oz boxes. The machine that fills the boxes is set so that, on the average, a box contains 12.4 oz. The machine filled boxes have content weight that can be closely approximated by a normal curve. What is the probability that a randomly selected box will be underweight (net weight less than 12 oz) if the standard deviation is as follows? 41. 0.5 oz 42. 0.4 oz 43. 0.3 oz 44. 0.2 oz 21.2% 15.9% 9.2% 2.3% 45. Recommended Daily Vitamin Allowances In nutrition, the recommended daily allowance of vitamins is a number set by the government to guide an individual's daily vitamin intake. Actually, vitamin needs vary dramatically from person to person, but the needs are closely approximated by a normal curve. To calculate the recommended daily allowance, the government first finds the average (mean) need for vitamins among people in the population and the standard deviation. The recommended daily allowance is then defined as the mean plus 2.5 times the standard deviation. What fraction of the population would receive adequate amounts of vitamins under this plan? 0.994, or 99.4% Recommended Daily Vitamin Allowances Find the recommended daily allowance for each vitamin if the mean need and standard deviation are as follows. (See Exercise 45.) 46. mean need = 1600 units; standard deviation = 140 units 1950 units 47. mean need 146 units; standard deviation 8 units 166 units Assume the following distributions are all normal, and use the areas under the normal curve given in Table 16 to find the appropriate areas. 48. Assembling Cell Phones The times taken by workers to assemble a certain kind of cell phone are normally distributed with mean 18.5 minutes and standard deviation 3.8 minutes. Find the probability that one such phone will require less than 12.8 minutes in assembly. 0.067 49. Finding Blood-Clotting Times The mean clotting time of blood is 7.47 sec, with a standard deviation of 3.6 sec. What is the probability that an individual’s blood-clotting time will be less than 7 sec or greater than 8 sec? 0.888 50. Sizes of Fish The average length of the fish caught in Vernal Lake is 11.6 in., with a standard deviation of 3.5 in. Find the probability that a fish caught there will be longer than 16 in. 0.104 51. Size Grading of Eggs To be graded extra-large, an egg must weigh at least 2.2 oz. If the average weight for an egg is 1.5 oz, with a standard deviation of 0.4 oz, how many of five dozen randomly chosen eggs would you expect to be extra-large? about 2 eggs

Extension Section . EXTENSION EXERCISES Correlating Fertilizer and Corn Ear Size In a study to determine the linear relationship between the length (in decimeters) of an ear of corn (y) and the amount (in tons per acre) of fertilizer used (x), the following values were determined. n = 10 ∑ xy = 75 ∑ x = 30 ∑ x 2 = 100 ∑ y = 24 ∑ y 2 = 80 1. Find an equation for the least squares line. 2. Find the correlation coefficient. 3. Jf 3 tons per acre of fertilizer are used, what length (in decimeters) would the regression equation predict for an ear of corn? Correlating Celsius and Fahrenheit Temperatures In an experiment to determine the linear relationship between temperatures on the Celsius scale (y) and on the Fahrenheit scale (x), a student got the following results. n = 5 ∑ xy = 28,050 ∑ x = 376 ∑ x 2 = 62,522 ∑ y = 120 ∑ y 2 = 13,450 4. Find an equation for the least squares line. 5. Find the reading on the Celsius scale that corresponds to a reading of 1200 Fahrenheit, using the equation of Exercise 4. 6. Find the correlation coefficient. Correlating Heights and Weights of Adult Men A sample of 10 adult men gave the following data on their heights and weights. Height (inches) (x) | 62 62 63 65 66 Weight (pounds) (y) |120 140 130 150 142 Height (inches) (x) | 67 68 68 70 72 Weight (pounds) (y) |130 135 175 149 168 7. Find the equation of the least squares line. 8. Using the results of Exercise 7, predict the weight of a man whose height is 60 inches. 9. What would be the predicted weight of a man whose height is 70 inches? 10. Compute the correlation coefficient. Correlating Reading Ability and IQs The table below gives reading ability scores and IQs for a group of 10 individuals. Reading (x) | 83 76 75 85 74 IQ (y) |120 104 98 115 87 Reading (x) | 90 75 78 95 80 IQ (y) |127 90 110 134 119 11. Plot a scatter diagram with reading on the horizontal axis. 12. Find the equation of a regression line. 13. Use your regression line equation to estimate the IQ of a person with a reading score of 65. Correlating Yearly Sales of a Company Sales, in thousands of dollars, of a certain company are shown here. Year (x) | 0 1 2 3 4 5_ Sales (y)| 48 59 66 75 80 90 14. Find the equation of the least squares line. 15. Find the correlation coefficient. 16. If the linear trend displayed by this data were to continue beyond year 5, what sales amount would you predict in year 7? Comparing the Ages of Dogs and Humans It often is said that a dog's age can be multiplied by 7 to obtain the equivalent human age. A more accurate correspondence (through the first 14 years) is shown in this table from The Old Farmer's Almanac, edition, page 180. Dog age (x) | ½ 1 2 3 4 5 6 7___ Equivalent | human age (y) |10 15 24 28 32 36 40 44 Dog age (x) | 8 9 10 11 12 13 Equivalent | human age (y) |48 52 56 60 64 68