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------------------------------------------------ An engineer is going to redesign an ejection seat. The seat was originally designed for pilots weighing between 150 lb and 191 lb. The new population of pilots has normally distributed weights with a mean of 156 lb and a standard deviation of 32.6 lb.

a. If a pilot is randomly selected, find the probability that his weight is between 150 lb and 191 lb. The probability is approximately 0.4315. (Round to four decimal places as needed.)

b. If 37 different pilots are randomly selected, find the probability that their mean weight is between 150 lb and 191 lb. (Round to four decimal places as needed.)

The probability that a randomly selected pilot's weight is between 150 lb and 191 lb is approximately 0.4291. The probability that the mean weight of 37 randomly selected pilots is between 150 lb and 191 lb is approximately 0.8699.

Explanation:

To solve part a, we need to find the probability that a randomly selected pilot's weight is between 150 lb and 191 lb. Since the weights of the pilots are normally distributed with a mean of 156 lb and a standard deviation of 32.6 lb, we can use the standard normal distribution table to find the corresponding probabilities.

First, we need to calculate the z-scores for the given weights. The z-score is calculated by subtracting the mean from the observed value and dividing it by the standard deviation.

For 150 lb: z = (150 - 156) / 32.6 = -0.1837

For 191 lb: z = (191 - 156) / 32.6 = 1.0748

Next, we can use the standard normal distribution table to find the probabilities corresponding to these z-scores. From the table, we find that the probability for a z-score of -0.1837 is approximately 0.4286, and the probability for a z-score of 1.0748 is approximately 0.8577.

Therefore, the probability that a randomly selected pilot's weight is between 150 lb and 191 lb is approximately 0.8577 - 0.4286 = 0.4291 (rounded to four decimal places).

To solve part b, we need to find the probability that the mean weight of 37 randomly selected pilots is between 150 lb and 191 lb. Since the mean weight is calculated from a sample, we need to use the standard error of the mean instead of the standard deviation.

The standard error of the mean is calculated by dividing the standard deviation by the square root of the sample size. In this case, the standard error of the mean is 32.6 / sqrt(37) = 5.34 lb.

Using the standard normal distribution table, we can find the z-scores for the given weights using the formula z = (x - mean) / standard error.

For 150 lb: z = (150 - 156) / 5.34 = -1.1243

For 191 lb: z = (191 - 156) / 5.34 = 6.5597

From the standard normal distribution table, we find that the probability for a z-score of -1.1243 is approximately 0.1301, and the probability for a z-score of 6.5597 is approximately 1.0000.

Therefore, the probability that the mean weight of 37 randomly selected pilots is between 150 lb and 191 lb is approximately 1.0000 - 0.1301 = 0.8699 (rounded to four decimal places).

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