Question \#4 (10 points)
Question
The height of men is a normally distrubuted variable with a mean of 67.6 inches and a standard deviation of 3.1 inches.
$\star *$ Round answers to ONE decimal place**
a.) What is the minimum height you could be to be considered in the top $5 \%$ of tallest men?
b.) What is the maximum height you could be and still be considered in the shortest $25 \%$ of men?

Step 1 ：To find the minimum height to be considered in the top 5% of tallest men, we need to find the z-score that corresponds to the 95th percentile. The z-score for the 95th percentile is approximately 1.645.

Step 2 ：The formula to convert a z-score to a raw score (X) is \(X = \mu + Z\sigma\), where \(\mu\) is the mean, Z is the z-score, and \(\sigma\) is the standard deviation.

Step 3 ：Substituting the given values into the formula, we get \(X = 67.6 + 1.645(3.1) = 67.6 + 5.0995 = 72.7\) inches.

Step 4 ：\(\boxed{72.7}\) inches is the minimum height to be considered in the top 5% of tallest men.

Step 5 ：To find the maximum height to still be considered in the shortest 25% of men, we need to find the z-score that corresponds to the 25th percentile. The z-score for the 25th percentile is approximately -0.674.

Step 6 ：Using the same formula as above, we get \(X = 67.6 - 0.674(3.1) = 67.6 - 2.0894 = 65.5\) inches.

Step 7 ：\(\boxed{65.5}\) inches is the maximum height to still be considered in the shortest 25% of men.