Question

a survey found that womens heights are normally distributed with mean 63.8 in and standard deviation 2.3 in. a branch of the military requires womens heights to be between 58 in and 8 a. find the percentage of women meeting the height requirement. b. if this branch of the military changes 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? 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 99.41 %. (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. yes, because a large percentage of women are not allowed to join this branch of the military because of their height. b. no, because only a small 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 womens heights to be at least in and at most in. (round to one decimal place as needed.)

Explanation:

Step1: Find z - scores for new requirements

We know that for the shortest 1% (left - tail), the z - score $z_1$ has an area of 0.01 to its left. Looking up in the standard normal distribution table, $z_1\approx - 2.33$. For the tallest 2% (right - tail), the area to the left of the z - score $z_2$ is $1 - 0.02=0.98$. Looking up in the table, $z_2\approx2.05$.

Step2: Use z - score formula to find heights

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 63.8$ in and $\sigma = 2.3$ in. For the lower height $x_1$:
\[

$$\begin{align*} -2.33&=\frac{x_1 - 63.8}{2.3}\\ x_1-63.8&=-2.33\times2.3\\ x_1&=63.8-2.33\times2.3\\ x_1&=63.8 - 5.359\\ x_1&\approx58.4 \end{align*}$$

\]
For the upper height $x_2$:
\[

$$\begin{align*} 2.05&=\frac{x_2 - 63.8}{2.3}\\ x_2-63.8&=2.05\times2.3\\ x_2&=63.8+2.05\times2.3\\ x_2&=63.8 + 4.715\\ x_2&\approx68.5 \end{align*}$$

\]

Answer:

a. B. No, because only a small percentage of women are not allowed to join this branch of the military because of their height.
b. 58.4 in, 68.5 in