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.