Question

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?

Accepted Answer

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. 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. 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 $$ x $$: $$ x = z \cdot \sigma + \mu . $$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: $$ x_{3\%} = -1.88 \cdot 2.9 + 63.7 . $$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 $$ [x_{3\%}, x_{97\%}] .$$

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Normal Distribution

Percentiles

Z-scores