Question

The heights of adult men in America are normally distributed, with a mean of 69.8 inches and a standard deviation of 2.69 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.1 inches and a standard deviation of 2.55 inches.
Required:
a. If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)?
b. What percentage of men are SHORTER than 6 feet 3 inches?
c. If a woman is 5 feet 11 inches tall, what is her z-score (to two decimal places)?
d. What percentage of women are TALLER than 5 feet 11 inches?

Answer 1

Answer:

a) 1.93

b) 97.32% of men are SHORTER than 6 feet 3 inches

c) 2.71

d) 0.34% of women are TALLER than 5 feet 11 inches

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

a. If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)?

For man, $\mu = 69.8, \sigma = 2.69$

A feet has 12 inches, so this is Z when X = 6*12 + 3 = 75. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{75 - 69.8}{2.69}$

$Z = 1.93$

b. What percentage of men are SHORTER than 6 feet 3 inches?

Z = 1.93 has a pvalue of 0.9732

97.32% of men are SHORTER than 6 feet 3 inches

c. If a woman is 5 feet 11 inches tall, what is her z-score (to two decimal places)?

For woman, $\mu = 64.1, \sigma = 2.55$

Here we have X = 5*12 + 11 = 71.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{71 - 64.1}{2.55}$

$Z = 2.71$

d. What percentage of women are TALLER than 5 feet 11 inches?

Z = 2.71 has a pvalue of 0.9966

1 - 0.9966 = 0.0034

0.34% of women are TALLER than 5 feet 11 inches