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? Click to view page 1 of the table. Click to view page 2 of the table. b. For the new height requirements, this branch of the military requires women's heights to be at least in and at most in. (Round to one decimal place as needed.)

Step 1 ：The mean height of women is given as 63.4 inches with a standard deviation of 2.4 inches. The current height requirement for the military branch is between 58 inches and 80 inches.

Step 2 ：To find the percentage of women meeting the height requirement, we calculate the z-scores for these heights and then use the standard normal distribution table to find the corresponding probabilities. The z-score is calculated as \((x - \text{mean}) / \text{standard deviation}\).

Step 3 ：Using the given values, the z-scores for the lower and upper bounds are approximately -2.25 and 6.92 respectively.

Step 4 ：Using the standard normal distribution table, we find that the percentage of women meeting the height requirement is approximately 98.78%. This means that only a small percentage of women are being denied the opportunity to join this branch of the military because they are too short or too tall.

Step 5 ：For the second part of the question, we need to find the heights that correspond to the shortest 1% and the tallest 2% of women. This can be done by finding the z-scores that correspond to these percentiles and then converting these z-scores back to heights.

Step 6 ：The z-scores that correspond to the shortest 1% and the tallest 2% of women are approximately -2.33 and 2.05 respectively.

Step 7 ：Converting these z-scores back to heights, we find that the new height requirements for this branch of the military are approximately 57.8 inches and 68.3 inches. This means that all women are eligible except the shortest 1% and the tallest 2%.

Step 8 ：Final Answer: The new height requirements for this branch of the military are \(\boxed{57.8 \text{ inches and } 68.3 \text{ inches}}\).