Question

the heights of adult men in america are normally distributed, with a mean of 69.3 inches and a standard deviation of 2.61 inches. the heights of adult women in america are also normally distributed, but with a mean of 64.5 inches and a standard deviation of 2.57 inches.
a) if a man is 6 feet 3 inches tall, what is his z - score (to two decimal places)?
z =
b) if a woman is 5 feet 11 inches tall, what is her z - score (to two decimal places)?
z =
c) who is relatively taller?
the 5 foot 11 inch american woman
the 6 foot 3 inch american man

Explanation:

Step1: Convert height to inches for man

1 foot = 12 inches, so 6 feet 3 inches = 6*12 + 3 = 75 inches.

Step2: Calculate z - score for man

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean and $\sigma$ is the standard deviation. For men, $\mu = 69.3$ and $\sigma=2.61$. So $z=\frac{75 - 69.3}{2.61}=\frac{5.7}{2.61}\approx2.18$.

Step3: Convert height to inches for woman

5 feet 11 inches = 5*12+11 = 71 inches.

Step4: Calculate z - score for woman

For women, $\mu = 64.5$ and $\sigma = 2.57$. So $z=\frac{71 - 64.5}{2.57}=\frac{6.5}{2.57}\approx2.53$.

Step5: Compare z - scores

Since 2.53>2.18, the woman has a higher z - score.

Answer:

a) 2.18
b) 2.53
c) The 5 foot 11 inch American woman