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.


Snow White Disney World requires that women employed as a Snow White character must have a height between 64 in. and 67 in.


a. Find the percentage of women meeting the height requirement.

Verified step by step guidance

1. Step 1: Recognize that the problem involves a normal distribution. The heights of women are normally distributed with a mean (μ) of 63.7 inches and a standard deviation (σ) of 2.9 inches. The goal is to find the percentage of women whose heights fall between 64 inches and 67 inches.
2. Step 2: Standardize the height values (64 inches and 67 inches) to z-scores using the z-score formula: z = (X - μ) / σ. For each height, substitute the values of X (64 and 67), μ (63.7), and σ (2.9) into the formula.
3. Step 3: Use a standard normal distribution table or a statistical software to find the cumulative probabilities corresponding to the z-scores calculated in Step 2. These cumulative probabilities represent the area under the standard normal curve to the left of each z-score.
4. Step 4: To find the percentage of women meeting the height requirement, subtract the cumulative probability of the lower z-score (corresponding to 64 inches) from the cumulative probability of the higher z-score (corresponding to 67 inches). This difference gives the proportion of women whose heights fall within the specified range.
5. Step 5: Convert the proportion obtained in Step 4 to a percentage by multiplying it by 100. This percentage represents the proportion of women meeting the height requirement for the Snow White character.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting 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 data that follows this distribution, such as heights in this question.

Z-scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are essential for determining the relative position of a data point within a normal distribution, allowing us to find probabilities and percentages associated with specific height ranges.

Percentile and Area Under the Curve

In statistics, the percentile indicates the value below which a given percentage of observations fall. The area under the normal distribution curve represents probabilities, and calculating the area between two Z-scores allows us to find the percentage of women whose heights fall within a specified range. This concept is key to solving the problem of determining how many women meet the height requirement for the Snow White character.