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Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the heights of men are assumed to follow a normal distribution, which allows for the calculation of probabilities related to height.

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are essential for determining the percentage of a normally distributed variable that falls within a certain range, such as the height requirements set by the U.S. Coast Guard.

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in estimating the percentage of men whose heights fall within the specified range of 60 in. to 80 in., facilitating the analysis of the height requirement.