BackGraphical Analysis In Exercises 41 and 42, the midpoints A, B, and C are marked on the histograms at the left. Match them with the indicated z-scores. Which z-scores, if any, would be considered unusual?
z = 0, z = 2.14, z = −1.43
The mean annual salary for a sample of electrical engineers is $86,500, with a standard deviation of $1500. The data set has a bell-shaped distribution.
b. The salaries of three randomly selected electrical engineers are $93,500, $85,600, and $82,750. Find the z-score that corresponds to each salary. Determine whether any of these salaries are unusual.
Explain how to transform a given x-value of a normally distributed variable x into a z-score.
Computing and Interpreting z-Scores In Exercises 39 and 40, (a) find the z-score that corresponds to each value and (b) determine whether any of the values are unusual.
Stanford-Binet IQ Scores The test scores for the Stanford-Binet Intelligence Scale are normally distributed with a mean score of 100 and a standard deviation of 16. The test scores of four students selected at random are 98, 65, 106, and 124.
Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.
MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (a) less than 490. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)
Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.
Health Club Schedule The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (a) less than 17 minutes.
Graphical Analysis In Exercises 13–16, a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded region of the graph. Assume the variable x is normally distributed.
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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.
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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?