A survey found that women's heights are normally distributed with mean 63.6 in and standard deviation 2.3 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 tflis branch of the military because they are too short or too tall?
A. No, because only a small percentage of women are not allowed to join this branch of the military because of their height.
B. Yes, because a large percentage of women are not allowed to join this branch of the military because of their height.
C. Yes, because the percentage of women who meet the height requirement is fairly large.
D. No, because the percentage of women who meet the height requirement is fairly small.
b. For the new height requirements, this branch of the military requires women's heights to be at least $\square$ in and at most $\square$ in. (Round to one decimal place as needed.)

Step 1 ：Calculate the z-scores for the lower and upper height limits. The z-score is calculated as follows: \(z = \frac{X - \mu}{\sigma}\), where X is the value from the dataset, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 2 ：For the lower limit (58 in): \(z1 = \frac{58 - 63.6}{2.3} = -2.43\)

Step 3 ：For the upper limit (80 in): \(z2 = \frac{80 - 63.6}{2.3} = 7.13\)

Step 4 ：Find the area between these z-scores on the standard normal distribution. This can be found using a standard normal distribution table or a calculator with a normal distribution function.

Step 5 ：The area to the left of \(z1\) (-2.43) is 0.0075 and the area to the left of \(z2\) (7.13) is almost 1.

Step 6 ：The percentage of women meeting the height requirement is the area between these z-scores, which is \(1 - 0.0075 = 0.9925\) or 99.25%.

Step 7 ：\(\boxed{\text{So, the answer to the question "Are many women being denied the opportunity to join this branch of the military because they are too short or too tall?" is A. No, because only a small percentage of women are not allowed to join this branch of the military because of their height.}}\)

Step 8 ：If the military changes the height requirements so that all women are eligible except the shortest 1% and the tallest 2%, we need to find the z-scores that correspond to these percentages and then convert these z-scores back to heights.

Step 9 ：The z-score for the shortest 1% is -2.33 and for the tallest 2% is 2.05.

Step 10 ：So, the new height requirements are: Lower limit = \(\mu + z1*\sigma = 63.6 + (-2.33)*2.3 = 58.7\) in, Upper limit = \(\mu + z2*\sigma = 63.6 + 2.05*2.3 = 68.4\) in

Step 11 ：\(\boxed{\text{So, for the new height requirements, this branch of the military requires women's heights to be at least 58.7 in and at most 68.4 in.}}\)