Redesign of Ejection SeatsWhen women were finally allowed to become pilotsoffighter jets, engineers needed to redesign the ejection seats because they had been originallydesigned for men only. The ACES-II ejection seats were designed for men weighing between

140 lb and 211 lb. Weights of women are now normally distributed with a mean of 171 lb and astandard deviation of 46 lb (based on Data Set 1 “Body Data”Appendix B).

a.If 1 woman is randomly selected, find the probability that her weight is between 140 lb and 211 lb.

b.If 25 different women are randomly selected, find the probability that their mean weight is between 140 lb and 211 lb.

c.When redesigning the fighter jet ejection seats to better accommodate women, which probability is more relevant: the result from part (a) or the result from part (b)? Why?

The weights of the women are normally distributed with a mean$\left(\mu \right)$ equal to 171 lb and a standard deviation$\left(\sigma \right)$ equal to 46 lb.

a.

Let X denote the weight of the women.

The probability that a randomly selected woman has a weight between 140 lb and 211 lb is computed using the standard normal table, as shown below.

$$\begin{gathered} P\left(140<x<211\right)=P\left(\frac{140-\mu }{\sigma }<\frac{x-\mu }{\sigma }<\frac{211-\mu }{\sigma }\right) \\ =P\left(\frac{140-171}{46}<z<\frac{211-171}{46}\right) \\ =P\left(-0.67<z<0.87\right) \\ =P\left(z<0.87\right)-P\left(z<-0.67\right) \end{gathered}$$

$$\begin{gathered} =0.8078-0.2514 \\ =0.5564 \end{gathered}$$

Therefore, the probability that a randomly selected woman has a weight between 140 lb and 211 lb is equal to 0.5564.

b.

Let $\overline{x}$denote the sample mean weight of the women.

The sample mean weight of the women follows a normal distribution with a mean equal to ${\mu }_{\overline{x}}=\mu$and a standard deviation equal to ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$.

The sample size is equal to n=25.

The probability that the sample mean weight of the women is between 140 lb and 211 lb is computed using the standard normal table, as shown below.

$$\begin{gathered} P\left(140<\overline{x}<211\right)=P\left(\frac{140-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{211-\mu }{\frac{\sigma }{\sqrt{n}}}\right) \\ =P\left(\frac{140-170}{\frac{46}{\sqrt{25}}}<z<\frac{211-170}{\frac{46}{\sqrt{25}}}\right) \\ =P\left(-3.37<z<4.35\right) \\ =P\left(z<4.35\right)-P\left(z<-3.37\right) \\ =0.9999-0.0004 \\ =0.9995 \end{gathered}$$

Therefore, the probability that the sample mean weight of the women is between 140 lb and 211 lb is equal to 0.9995.

c.

The probability computed in part (a) represents the probability of an individual woman to suitably fit the ejection seat.

The probability computed in part (b) represents the probability of a group of women to suitably fit the ejection seat.

However, the ejection seat is to be occupied by one specific woman and not a set of women.

Therefore, the probability of part (a) is more relevant for redesigning the fighter jet ejection seats to better accommodate women.