Question

When women were finally allowed to become pilots of fighter​ jets, engineers needed to redesign the ejection seats because they had been originally designed for men only. The ejection seats were designed for men weighing between 140 lb and 201 lb. Weights of women are now normally distributed with a mean of 162 and a standard deviation of 46 lb . Complete parts​ (a) through​ (c) below.

Answer 1

(a) What percentage of women would not be able to use the original ejection seats designed for men?

To answer this question, we need to find the percentage of women whose weight is outside the range of 140 lb to 201 lb, which is the weight range for which the original ejection seats were designed. We can do this by standardizing the weight distribution of women using the formula:

z = (x - μ) / σ

where z is the standard score, x is the weight of a woman, μ is the mean weight of women, and σ is the standard deviation of the weight of women.

Substituting the given values, we get:

z = (140 - 162) / 46 = -0.478

z = (201 - 162) / 46 = 0.848

Using a standard normal table or a calculator, we can find that the percentage of women whose weight is below 140 lb (corresponding to a z-score of -0.478) is 31.45%. Similarly, the percentage of women whose weight is above 201 lb (corresponding to a z-score of 0.848) is 19.34%. Therefore, the percentage of women who would not be able to use the original ejection seats designed for men is:

31.45% + 19.34% = 50.79%

(b) What weight range would accommodate 99.7% of the female pilots?

To find the weight range that would accommodate 99.7% of the female pilots, we need to find the z-scores that correspond to the 0.15% and 99.85% percentiles of the standard normal distribution. Using a standard normal table or a calculator, we can find that these z-scores are approximately -2.97 and 2.97, respectively.

Substituting these z-scores into the standardization formula, we get:

-2.97 = (x - 162) / 46

x = 63.88 lb

and

2.97 = (x - 162) / 46

x = 260.12 lb

Therefore, the weight range that would accommodate 99.7% of the female pilots is approximately 63.88 lb to 260.12 lb.

(c) What percentage of women would need ejection seats redesigned for them?

To find the percentage of women who would need ejection seats redesigned for them, we need to find the percentage of women whose weight is outside the weight range that would accommodate 99.7% of the female pilots. Using the values from part (b), we can see that the weight range that would accommodate 99.7% of the female pilots is approximately 63.88 lb to 260.12 lb. Therefore, the percentage of women who would need ejection seats redesigned for them is:

100% - 99.7% = 0.3%