6-4 The Central Limit Theorem 291 6. a. If 1 male college student is randomly selected, find the probability that he gains at least 2.0 kg during his freshman year. b. If 16 male college students are randomly selected, find the probability that their mean weight gain during their freshman year is at least 2.0 kg. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? 7. a. If 1 male college student is randomly selected, find the probability that he gains between 0 kg and 3 kg during freshman year. b. If 9 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? 8. a. If 1 male college student is randomly selected, find the probability that he gains between 0.5 kg and 2.5 kg during freshman year. b. If 4 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0.5 kg and 2.5 kg. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem. 9. Safe Loading of Elevators The elevator in the car rental building at San Francisco International Airport has a placard stating that the maximum capacity is “4000 lb—27 passengers.” Because 4000>27 = 148, this converts to a mean passenger weight of 148 lb when the elevator is full. We will assume a worst-case scenario in which the elevator is filled with 27 adult males. Based on Data Set 1 “Body Data” in Appendix B, assume that adult males have weights that are normally distributed with a mean of 189 lb and a standard deviation of 39 lb. a. Find the probability that 1 randomly selected adult male has a weight greater than 148 lb. b. Find the probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb. c. What do you conclude about the safety of this elevator? 10. Designing Manholes According to the website www.torchmate.com, “manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter.” Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder widths that are normally distributed with a mean of 18.2 in. and a standard deviation of 1.0 in. (based on data from the National Health and Nutrition Examination Survey). a. What percentage of men will fit into the manhole? b. Assume that the Connecticut’s Evergreen company employs 36 men who work in manholes. If 36 men are randomly selected, what is the probability that their mean shoulder width is less than 18.5 in.? Does this result suggest that money can be saved by making smaller manholes with a diameter of 18.5 in.? Why or why not? 11. Water Taxi Safety Passengers died when a water taxi sank in Baltimore’s Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 “Body Data” in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb. a. Given that the water taxi that sank was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the boat is filled to the stated capacity of 25 passengers?
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