An engineer is planning to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 140 and 181 pounds. The population of pilots has normally distributed weights with a mean of 150 pounds and a standard deviation of 27.7 pounds.

1. If a pilot is randomly selected, find the probability that their weight is between 140 and 181 pounds.

Round to four decimal places as needed.

The question deals with calculating probabilities using the concept of normal distribution and z-scores. The probability of a pilot's weight being between 140 and 187 pounds is approximately 55.05%

Explanation:

The subject of this question pertains to statistics, particularly normal distribution and probabilities. The problem involves an engineer redesigning an ejection seat for an airplane and needing to account for the weights of pilots, which follow a normal distribution with a mean of 150 pounds and a standard deviation of 27.7 pounds. We are asked to find the probability of a pilot's weight falling between 140 and 187 pounds.

To find this, we need to convert the individual weights to z-scores, which is a measure of how many standard deviations an element is from the mean. After which, we need to look these up in a standard normal distribution table to find the probabilities. The formula used for finding the z-score is: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

For 140: Z1 = (140 - 150) / 27.7 = -0.36. And for 187: Z2 = (187 - 150) / 27.7 = 1.34. Using a standard normal distribution table, we find that the probabilities associated with these z-scores are approximately 0.3594 and 0.9099, respectively. Subtracting these gives us 0.5505 or 55.05% as the probability that the selected pilot's weight falls between 140 and 187 pounds.

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