An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 140 lb and 201 lb. The new population of pilots has normally distributed weights with a mean of 145 lb and a standard deviation of 29.7 lb.

a. If a pilot is randomly selected, find the probability that his weight is between 140 lb and 201 lb.

The probability is approximately _______ (Round to four decimal places as needed).

The probability that a randomly selected pilot's weight is between 140lb and 201lb is approximately 0.6284, calculated using the conversion to z-scores and the standard normal distribution.

Explanation:

In this question of probability, we are given that the weights of the new population of pilots are normally distributed with a mean of 145lb and a standard deviation of 29.7lb. We are asked to find the probability that a randomly selected pilot weighs between 140lb and 201lb.

To solve this, we first convert the weights into z-scores (a z-score gives us an idea of how far from the mean a data point is). For 140lb and 201lb, the z-scores can be calculated as follows: Z = (X - μ) / σ, where X is the weight, μ is the mean weight and σ is the standard deviation. This gives us z1 = (140 - 145) / 29.7 = -0.1683 and z2 = (201 - 145) / 29.7 = 1.8855.

The probability that a randomly selected pilot's weight is between 140lb and 201lb is the area under the standard normal curve between these z-scores, which can be found in the standard normal distribution table or calculated with a statistical program. The required probability is approximately 0.6284 (rounded to four decimal places).

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