112 CHAPTER 2 Descriptive Statistics Finding z-Scores The distribution of the ages of the winners of the Tour de France from 1903 to 2020 is approximately bell-shaped. The mean age is 27.9 years, with a standard deviation of 3.4 years. In Exercises 43–48, use the corresponding z-score to determine whether the age is unusual. Explain your reasoning. (Source: Le Tour de France) Winner Year Age 43. Christopher Froome 2016 31 44. Jan Ullrich 1997 24 45. Antonin Magne 1931 27 46. Firmin Lambot 1922 36 47. Henri Cornet 1904 20 48. Tadej Pogacˇar 2020 21 49. Life Spans of Tires A brand of automobile tire has a mean life span of 35,000 miles, with a standard deviation of 2250 miles. Assume the life spans of the tires have a bell-shaped distribution. (a) The life spans of three randomly selected tires are 34,000 miles, 37,000 miles, and 30,000 miles. Find the z-score that corresponds to each life span. Determine whether any of these life spans are unusual. (b) The life spans of three randomly selected tires are 30,500 miles, 37,250 miles, and 35,000 miles. Using the Empirical Rule, find the percentile that corresponds to each life span. 50. Life Spans of Fruit Flies The life spans of a species of fruit fly have a bell-shaped distribution, with a mean of 33 days and a standard deviation of 4 days. (a) The life spans of three randomly selected fruit flies are 34 days, 30 days, and 42 days. Find the z-score that corresponds to each life span. Determine whether any of these life spans are unusual. (b) The life spans of three randomly selected fruit flies are 29 days, 41 days, and 25 days. Using the Empirical Rule, find the percentile that corresponds to each life span. Comparing z-Scores from Different Data Sets The table shows population statistics for the ages of Best Actor and Best Supporting Actor winners at the Academy Awards from 1929 to 2020. The distributions of the ages are approximately bell-shaped. In Exercises 51–54, compare the z-scores for the actors. Best actor Best supporting actor m ≈ 43.8 yr m ≈ 50.2 yr s ≈ 8.7 yr s ≈ 13.5 yr 51. Best Actor 2018: Gary Oldman, Age: 59 Best Supporting Actor 2018: Sam Rockwell, Age: 49 52. Best Actor 2005: Jamie Foxx, Age: 37 Best Supporting Actor 2005: Morgan Freeman, Age: 67 53. Best Actor 1970: John Wayne, Age: 62 Best Supporting Actor 1970: Gig Young, Age: 56 54. Best Actor 1982: Henry Fonda, Age: 76 Best Supporting Actor 1982: John Gielgud, Age: 77
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