6. Normal Distribution and Continuous Random Variables

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

0.6736

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

0.94

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)

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)

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 1970: John Wayne, Age: 62

Best Supporting Actor 1970: Gig Young, Age: 56

The towing capacities (in pounds) of all the pickup trucks at a dealership have a bell-shaped distribution, with a mean of 11,830 pounds and a standard deviation of 2370 pounds. In Exercises 45– 48, use the corresponding z-score to determine whether the towing capacity is unusual. Explain your reasoning.

5,500 pounds

The mean gestational length of a sample of 208 horses is 343.7 days, with a standard deviation of 10.4 days. The data set has a bell-shaped distribution.

b. Determine whether a gestational length of 318.4 days is unusual.

The per capita disposable income for residents of a U.S. city in a recent year is normally distributed, with a mean of about $44,000 and a standard deviation of about $2450. Use this information in Exercises 7–10.

Out of 800 residents, about how many would you expect to have a disposable income of between $40,000 and $42,000?

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.

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.

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.

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.

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(z < - 1.11)

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P33

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P1.5

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P91

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

" style="" width="260">