Question

in a certain city, the average 20- to 29-year old man is 69.8 inches tall, with a standard deviation of 3.1 inches, while the average 20- to 29-year old woman is 64.5 inches tall, with a standard deviation of 3.9 inches. who is relatively taller, a 75-inch man or a 70-inch woman?

find the corresponding z-scores. who is relatively taller, a 75-inch man or a 70-inch woman? select the correct choice below and fill in the answer boxes to complete your choice. (round to two decimal places as needed.)
a. the z-score for the man, , is larger than the z-score for the woman, , so he is relatively taller.
b. the z-score for the man, , is smaller than the z-score for the woman, , so he is relatively taller.
c. the z-score for the woman, , is larger than the z-score for the man, , so she is relatively taller.
d. the z-score for the woman, , is smaller than the z-score for the man, , so she is relatively taller.

Explanation:

Step1: Recall z-score formula

The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $x$ is the observed value, $\mu$ is the population mean, and $\sigma$ is the population standard deviation.

Step2: Calculate man's z-score

For the 75-inch man: $\mu = 69.8$, $\sigma = 3.1$, $x=75$
$z_{\text{man}} = \frac{75 - 69.8}{3.1} = \frac{5.2}{3.1} \approx 1.68$

Step3: Calculate woman's z-score

For the 70-inch woman: $\mu = 64.5$, $\sigma = 3.9$, $x=70$
$z_{\text{woman}} = \frac{70 - 64.5}{3.9} = \frac{5.5}{3.9} \approx 1.41$

Step4: Compare z-scores

$1.68 > 1.41$, so the man's z-score is larger, meaning he is relatively taller.

Answer:

A. The z-score for the man, 1.68, is larger than the z-score for the woman, 1.41, so he is relatively taller.