Question

A survey found that women's heights are normally distributed with mean 63.5in and standard deviation 2.2in. 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?

Answer 1

Let X is a random variable shows the women height. Here X is normally distributed with mean

(a)

The z-score for X = 58 in. is

The z-score for X = 80 in. is

The percentage of women meeting the height requirment is

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(b)

Lowest allowable height: Here we need z-score that has 0.01 area to its left. The z-score -2.33 has 0.01 area to its left. So required X is

X = 58.374

That is this branch of the military requires womens heights to be at least 58.374 inches.

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upper bound on height: Here we need z-score that has 0.02 area to its right. The z-score 2.05 has 0.02 area to its right. So required X is

X = 68.01

That is this branch of the military requires womens heights to be at most 68.01 inches.