Question

 A survey found that women's heights are normally distributed with mean 63.7 in and standard deviation 2.2 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?

 a. The percentage of women who meet the height requirement is _______  %

 (Round to two decimal places as needed.)

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

 D. Yes, because a large percentage of women are not allowed to join this branch of the military because of their height. 

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.)

Answer 1

a.

Given,

= 63.7 ,   = 2.2

Let X be the random variable denote the women's height. Then X ~ N( = 63.7 ,   = 2.2)

P(58 < X < 80) = P(X < 80) - P(X < 58)

= P[Z < (80 - 63.7)/2.2] - P[Z < (58 - 63.7)/2.2]

= P[Z < 7.4] - P[Z < -2.59]

= 1 - 0.0048

= 0.9952

= 99.52 %

The women's height which does not meet military requirements is 1 - 0.9952 = 0.0048.

So, the answer is,

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

b.

z value for 1% percentile is -2.326

Lower limit of height = 63.7 - 2.2 * 2.326 = 58.6 inch

z value for (100 - 2) = 98% percentile is  2.054

Upper limit of height = 63.7 + 2.2 * 2.054 = 68.2 inch

So, the women's height bet at least 58.6 in and at most 68.2 in