The heights of American women aged 18 to 24 are normally distributed with a mean of 66 inches and a standard deviation of 2.5 inches. In order to serve in the U.S. Army, women must be between 57 inches and 79 inches tall. What percentage of women are ineligible to serve based on their height?
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That is a pretty powerful applet to check your answers.
To find the percentage of women who are ineligible to serve based on their height, we need to determine the proportion of women whose height falls outside the range of 57 to 79 inches.
First, we need to standardize the height range by subtracting the mean (66 inches) from each boundary and dividing by the standard deviation (2.5 inches):
Lower boundary: (57 - 66) / 2.5 = -3.6
Upper boundary: (79 - 66) / 2.5 = 5.2
Next, we need to determine the proportion of women whose heights fall below the lower boundary and above the upper boundary in a standard normal distribution. We can use a Z-table or a calculator to find these probabilities.
The probability of a Z-score less than -3.6 is virtually zero (approximately 0.0001).
The probability of a Z-score greater than 5.2 is also almost zero (approximately 0.9999).
Now, we need to calculate the combined proportion of women whose heights fall outside the given range. Since the normal distribution is symmetric, we can calculate the proportion below the lower boundary and above the upper boundary and then double it.
Proportion below the lower boundary: 0.0001
Proportion above the upper boundary: 0.0001
Total proportion outside the range: 2 * 0.0001 = 0.0002
To convert this proportion to a percentage, we multiply by 100:
Percentage of women ineligible to serve based on height: 0.0002 * 100 = 0.02%
Therefore, approximately 0.02% of women are ineligible to serve in the U.S. Army based on their height.