Question

a survey found that womens heights are normally distributed with mean 63.9 in and standard deviation 2.4 in. a branch of the military requires womens 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

a. the percentage of women who meet the height requirement is 99.31%.
(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 only a small percentage of women are not allowed to join this branch of the military because of their height.
b. no, because the percentage of women who meet the height requirement is fairly small.
c. yes, because a large percentage of women are not allowed to join this branch of the military because of their height.
d. yes, because the percentage of women who meet the height requirement is fairly large.
b. for the new height requirements, this branch of the military requires womens heights tobe at least □ in and at most □ in.
(round to one decimal place as needed.)

Explanation:

Step1: Find the z - scores

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 63.9\) (mean) and \(\sigma=2.4\) (standard deviation).
For the lower bound (shortest 1%):
We look for the z - score \(z_1\) such that \(P(Z < z_1)=0.01\). Using the standard normal table, \(z_1\approx - 2.33\)
For the upper bound (tallest 2%):
We look for the z - score \(z_2\) such that \(P(Z>z_2) = 0.02\), then \(P(Z\leq z_2)=1 - 0.02=0.98\). Using the standard normal table, \(z_2\approx2.05\)

Step2: Calculate the height values

Using the formula \(x=\mu+z\sigma\)
For the lower bound:
\(x_1=\mu+z_1\sigma=63.9+( - 2.33)\times2.4\)
\(x_1=63.9-5.592 = 58.308\approx58.3\)
For the upper bound:
\(x_2=\mu+z_2\sigma=63.9 + 2.05\times2.4\)
\(x_2=63.9+4.92=68.82\approx68.8\)

Answer:

a. A. No, because only a small percentage of women are not allowed to join this branch of the military because of their height.
b. The branch of the military requires women’s heights to be at least \(58.3\) in and at most \(68.8\) in.