The height of men is a normally distrubuted variable with a mean of 68 inches and a standard deviation of 3 inches. **Round answers to ONE decimal place** a.) What is the minimum height you could be to be considered in the top $15 \%$ of tallest men? b.) What is the tallest you could be to be considered in the shortest $20 \%$ of men? Question Help: Message instructor Submit Question

Step 1 ：To find the minimum height to be considered in the top 15% of tallest men, we need to find the z-score that corresponds to the 85th percentile. Using a standard normal distribution table or a z-score calculator, we find that the z-score for the 85th percentile is approximately 1.0.

Step 2 ：We then use the formula for a z-score, which is: \(Z = \frac{X - \mu}{\sigma}\) where \(Z\) is the z-score, \(X\) is the value we're trying to find, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 3 ：Rearranging the formula to solve for \(X\) gives us: \(X = Z\sigma + \mu\)

Step 4 ：Substituting the given values into the formula gives us: \(X = 1.0 * 3 + 68 = 71\) inches

Step 5 ：\(\boxed{71}\) inches is the minimum height to be considered in the top 15% of tallest men.

Step 6 ：To find the tallest height to be considered in the shortest 20% of men, we need to find the z-score that corresponds to the 20th percentile. Using a standard normal distribution table or a z-score calculator, we find that the z-score for the 20th percentile is approximately -0.8.

Step 7 ：Substituting the given values into the formula gives us: \(X = -0.8 * 3 + 68 = 66.6\) inches

Step 8 ：\(\boxed{66.6}\) inches is the tallest height to be considered in the shortest 20% of men.