An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 130 lb and 191 lb. The new population of pilots has normally distributed weights with a mean of 139 lb and a standard deviation of 33.6 lb.

a. If a pilot is randomly selected, find the probability that his weight is between 130 lb and 191 lb. The probability is approximately 0.5447. (Round to four decimal places as needed.)

b. If 32 different pilots are randomly selected, find the probability that their mean weight is between 130 lb and 191 lb. The probability is approximately (Round to four decimal places as needed.)

The probabilities in this question, you apply the principles of probability within a normal distribution for the first part, and the central limit theorem for the second part. The probabilities are then found through Z-scores and relevant lookups on Z-table.

Explanation:

This question is about probability in a normal distribution, a topic in statistics, a branch of mathematics. The first part of the question asks for the probability of a single randomly selected pilot having a weight between 130 lb and 191 lb. This calculation has been done and given in the question as approximately 0.5447.

In the second part, the question is about the sampling distribution (a probability distribution of a statistic) of means. Since the mean weight of all pilots is 139 lb, and the standard deviation is 33.6 lb, the central limit theorem tells us that the mean of 32 randomly selected pilots will also be normally distributed, but with a standard deviation of 33.6/sqrt(32). Here, you can use Z-score formulas and look up the Z-scores in the Z-table to calculate the probability of mean weight of the 32 pilots falling between 130 lb and 191 lb.

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