A survey found that women's heights are normally distributed with mean 63.4 in and standard deviation 2.4 in. A branch of the military requires women's heights to be between 58 in and 80 in.
a. Find the percentage of women meeting the height requirement. Are many women being denied the opportunity to join this branch of the military because they are too short or too tall?
b. If this branch of the military changes the height requirements so that all women are eligible except the shortest $1 \%$ and the tallest $2 \%$, what are the new height requirements?
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Are many women being denied the opportunity to join this branch of the military because they are too short or too tall?
A. No, because the percentage of women who meet the height requirement is fairly small.
B. Yes, because the percentage of women who meet the height requirement is fairly large.
C. Yes, because a large percentage of women are not allowed to join this branch of the military because of their height.

D. No, because only a small percentage of women are not allowed to join this branch of the military because of their height.

Step 1 ：Given that women's heights are normally distributed with a mean of 63.4 inches and a standard deviation of 2.4 inches.

Step 2 ：The height requirement for a branch of the military is between 58 inches and 80 inches.

Step 3 ：We need to find the percentage of women who meet this height requirement.

Step 4 ：This is a problem of finding the probability that a normally distributed random variable falls within a certain range.

Step 5 ：We can use the cumulative distribution function (CDF) of the normal distribution to solve this problem.

Step 6 ：The CDF at a point x gives the probability that the random variable is less than or equal to x.

Step 7 ：So, to find the probability that the height is between 58 and 80 inches, we need to find the difference between the CDF at 80 and the CDF at 58.

Step 8 ：Calculating the z-scores for the lower and upper bounds, we get \(z_{lower} = -2.25\) and \(z_{upper} = 6.92\).

Step 9 ：Using these z-scores, we find the probabilities \(p_{lower} = 0.0122\) and \(p_{upper} = 1.00\).

Step 10 ：The probability that a woman's height is between 58 and 80 inches is the difference between these two probabilities, which is \(p_{between} = 0.9878\).

Step 11 ：This means that approximately 98.78% of women meet the height requirement.

Step 12 ：So, only a small percentage of women are not allowed to join this branch of the military because of their height.

Step 13 ：Final Answer: \(\boxed{\text{D. No, because only a small percentage of women are not allowed to join this branch of the military because of their height.}}\)