In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):


Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.
Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.


Mickey Mouse Disney World requires that people employed as a Mickey Mouse character must have a height between 56 in. and 62 in.


a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as Mickey Mouse characters?

Verified step by step guidance

1. Step 1: Identify the problem as one involving the normal distribution. The goal is to find the percentage of men whose heights fall between 56 inches and 62 inches, given that men's heights are normally distributed with a mean (μ) of 68.6 inches and a standard deviation (σ) of 2.8 inches.
2. Step 2: Standardize the height values (56 inches and 62 inches) into z-scores using the z-score formula: z = (X - μ) / σ. For each height, substitute the values of X (the height), μ (68.6), and σ (2.8) into the formula to calculate the corresponding z-scores.
3. Step 3: Use a standard normal distribution table (or a statistical software/calculator) to find the cumulative probabilities corresponding to the z-scores calculated in Step 2. These probabilities represent the area under the normal curve to the left of each z-score.
4. Step 4: Subtract the cumulative probability of the lower z-score (corresponding to 56 inches) from the cumulative probability of the upper z-score (corresponding to 62 inches). This difference gives the proportion of men whose heights fall within the specified range.
5. Step 5: Multiply the proportion obtained in Step 4 by 100 to convert it into a percentage. Interpret the result in the context of the problem, noting that a very small percentage of men meeting the height requirement suggests that most Mickey Mouse characters are likely not men.

Key Concepts

Normal Distribution

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. Understanding this concept is crucial for analyzing the heights of men and women in the given problem, as it allows us to calculate probabilities and percentages related to height.

Z-Score

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. In this context, Z-scores will help determine how many standard deviations a specific height (like the Mickey Mouse character height requirement) is from the mean height of men, allowing for the calculation of the percentage of men who meet the requirement.

Percentile and Probability

Percentiles are measures that indicate the value below which a given percentage of observations in a group falls. In this scenario, calculating the percentage of men who meet the height requirement involves finding the corresponding percentile for the Z-scores derived from the height limits. This concept is essential for interpreting the results and understanding the implications regarding the gender distribution of those employed as Mickey Mouse characters.