6-4 The Central Limit Theorem 293 16.Aircraft Cockpit The overhead panel in an aircraft cockpit typically includes controls for such features as landing lights, fuel booster pumps, and oxygen. It is important for pilots to be able to reach those overhead controls while sitting. Seated adult males have overhead grip reaches that are normally distributed with a mean of 51.6 in. and a standard deviation of 2.2 in. a. If an aircraft is designed for pilots with an overhead grip reach of 53 in., what percentage of adult males would not be able to reach the overhead controls? Is that percentage too high? b. If the cockpit is designed so that 95% of adult males would be able to reach the overhead controls, what is the overhead grip reach distance? c. A small regional airline employs 40 male pilots. An engineer wants to design for an overhead grip reach that satisfies this criterion: There is a 0.95 probability that 40 randomly selected male pilots have a mean overhead grip reach that is greater than or equal to the designed overhead reach distance. What overhead grip reach distance satisfies that design? Why should this engineer be fired? Hypothesis Testing. In Exercises 17–20, apply the central limit theorem to test the given claim. (Hint: See Example 3.) 17.Freshman 15 The term “Freshman 15” refers to the claim that college students gain 15 lb during their freshman year at college. Data Set 13 “Freshman 15” includes measurements from 67 college students from their freshman year, and they had weight gains with a mean of 2.6 lb and a standard deviation of 8.6 lb. Assume that the mean weight gain really is 15 lb and find the probability that a random sample of 67 college students would have a mean weight gain of 2.6 lb or less. What does the result suggest about the claim of the “Freshman 15”? 18.Adult Sleep Times (hours) of sleep for randomly selected adult subjects included in the National Health and Nutrition Examination Study are listed below. Here are the statistics for this sample: n = 12, x = 6.8 hours, s = 2.0 hours. The times appear to be from a normally distributed population. A common recommendation is that adults should sleep between 7 hours and 9 hours each night. Assuming that the mean sleep time is 7 hours, find the probability of getting a sample of 12 adults with a mean of 6.8 hours or less. What does the result suggest about a claim that “the mean sleep time is less than 7 hours”? 4844869771078 19.Weight Watchers Diet When 40 people used the Weight Watchers diet for one year, their mean weight loss was 3.0 lb, and the standard deviation was 4.9 lb (based on data from “Comparison of the Atkins, Ornish, Weight Watchers, and Zone Diets for Weight Loss and Heart Disease Reduction,” by Dansinger et al., Journal of the American Medical Association, Vol. 293, No. 1). Assuming that the diet has no effect so the true mean amount of lost weight is 0 lb, find the probability of getting a sample of 40 subjects with a mean weight loss of 3.0 lb or higher. Based on the result, is the mean weight loss of 3.0 lb significantly high? What do the results suggest about the effectiveness of the diet? 20.Correcting for a Finite Population In a study of babies born with very low birth weights, 275 children were given IQ tests at age 8, and their scores approximated a normal distribution with m = 95.5 and s = 16.0 (based on data from “Neurobehavioral Outcomes of School-age Children Born Extremely Low Birth Weight or Very Preterm,” by Anderson et al., Journal of the American Medical Association, Vol. 289, No. 24). Fifty of those children are to be randomly selected without replacement for a follow-up study. a. When considering the distribution of the mean IQ scores for samples of 50 children randomly selected from a population of 275 children, should sx be corrected by using the finite population correction factor? Why or why not? What is the value of sx? b. Find the probability that the mean IQ score of the follow-up sample is between 95 and 105. 6-4 Beyond the Basics
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