Questions: The distribution of heights of adult men in the U.S. is approximately Normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a Normal distribution and the 68-95-99.7 rule to answer the following. (a) About what percent of men are between 64 and 66.5 inches? (b) About what percent of men are shorter than 66.5 inches? (c) About what percent of men are taller than 74 inches?

Solution Steps

To find the probability that men are between 64 and 66.5 inches, we calculate:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-1.0) - \Phi(-2.0) = 0.1359 \]

Thus, the probability that men are between 64 and 66.5 inches is:

\[ \text{Probability} = 13.59\% \]

Next, we determine the probability that men are shorter than 66.5 inches:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-1.0) - \Phi(-\infty) = 0.1587 \]

Therefore, the probability that men are shorter than 66.5 inches is:

\[ \text{Probability} = 15.87\% \]

Finally, we calculate the probability that men are taller than 74 inches:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(2.0) = 0.0228 \]

Thus, the probability that men are taller than 74 inches is:

\[ \text{Probability} = 97.72\% \]

Final Answer