Questions: When women were finally allowed to become pilots of fighter aircraft, the military needed to redesign the ejection seats because they had been designed for men only. The ejection seats were designed for weights between 130 lb and 171 lb. Weights of women are now normal with a mean of 172 lb and a standard deviation of 37 lb. Complete through (c) below. a. If 1 woman is randomly selected, find the probability that her weight is between 130 lb and 171 lb. The probability is approximately (Round to four decimal places as needed.)

Solution Steps

To find the probability that a randomly selected woman's weight is between \(130 \, \text{lb}\) and \(171 \, \text{lb}\), we first calculate the Z-scores for the lower and upper bounds using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

For the lower bound \(X = 130 \, \text{lb}\):

\[ Z_{start} = \frac{130 - 172}{37} \approx -1.1351 \]

For the upper bound \(X = 171 \, \text{lb}\):

\[ Z_{end} = \frac{171 - 172}{37} \approx -0.027 \]

Next, we find the probabilities corresponding to these Z-scores using the cumulative distribution function \( \Phi(Z) \):

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) \]

Substituting the Z-scores:

\[ P = \Phi(-0.027) - \Phi(-1.1351) \]

Using the values obtained:

\[ P \approx 0.3611 \]

Final Answer