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.
If the Navy changes the height requirements so that all women are eligible except the shortest 3% and the tallest 3%, what are the new height requirements for women?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the height range for women that excludes the shortest 3% and the tallest 3%. This means we are looking for the 3rd percentile and the 97th percentile of the normal distribution for women's heights.
2. Step 2: Recall the properties of a normal distribution. The z-scores corresponding to the 3rd percentile and the 97th percentile can be found using a z-table or statistical software. For the 3rd percentile, the z-score is approximately -1.88, and for the 97th percentile, the z-score is approximately 1.88.
3. Step 3: Use the z-score formula to calculate the corresponding heights. The formula is: \( z = \frac{x - \mu}{\sigma} \), where \( z \) is the z-score, \( x \) is the height, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Rearrange the formula to solve for \( x \): \( x = z \cdot \sigma + \mu \).
4. Step 4: Plug in the values for the 3rd percentile. Using \( z = -1.88 \), \( \mu = 63.7 \), and \( \sigma = 2.9 \), calculate the height corresponding to the 3rd percentile: \( x_{3\%} = -1.88 \cdot 2.9 + 63.7 \).
5. Step 5: Plug in the values for the 97th percentile. Using \( z = 1.88 \), \( \mu = 63.7 \), and \( \sigma = 2.9 \), calculate the height corresponding to the 97th percentile: \( x_{97\%} = 1.88 \cdot 2.9 + 63.7 \). The resulting range of heights will be \( [x_{3\%}, x_{97\%}] \).

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. In this context, women's heights follow a normal distribution, allowing us to use statistical methods to determine specific percentiles.

Percentiles

A percentile is a measure used in statistics indicating the value below which a given percentage of observations fall. For example, the 3rd percentile is the height below which 3% of women fall. In this question, we need to find the heights corresponding to the 3rd and 97th percentiles to establish the new height requirements for women.

Z-scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It allows for the comparison of scores from different distributions. To find the height requirements for women, we will calculate the Z-scores that correspond to the 3rd and 97th percentiles and then convert these Z-scores back to height values using the given mean and standard deviation.