Question

The U.S. Navy requires that fighters pilots have heights between 62in. and 78 in. Women's heights are normally distributed with a mean of 63.8 in. and a standard deviation of 63.8 in. and standard deviation of 2.6 in. Find the percentage of women meeting the height requirement. If the Navy changes the height requirements so that all women are eligible except the shortest 2% and tallest 2%, w hat are the new height requirements for women?

Answer 1

The percentage of women meeting the height requirement is 100%. The new height requirements for women are approximately 57.7 inches to 69.9 inches.

Step-by-step explanation:

The percentage of women meeting the height requirement can be found by calculating the z-scores for the lower and upper bounds of the height requirement and then finding the area under the normal distribution curve between those z-scores.

First, we calculate the z-score for the lower bound:

z = (lower bound - mean) / standard deviation = (62 - 63.8) / 2.6 = -0.6923

Next, we calculate the z-score for the upper bound:

z = (upper bound - mean) / standard deviation = (78 - 63.8) / 2.6 = 5.4615

Using a standard normal distribution table or a calculator, we can find the area under the curve between these two z-scores. The area between -0.6923 and 5.4615 is approximately 1. The percentage of women meeting the height requirement is therefore 100%.

If the Navy changes the height requirements so that all women are eligible except the shortest 2% and tallest 2%, we need to find the z-scores that correspond to these percentiles. The z-score for the 2nd percentile is approximately -2.05 and the z-score for the 98th percentile is approximately 2.05. We can then find the actual height values corresponding to these z-scores by using the formula: height = mean + (z * standard deviation). For the new height requirements, the lower bound is approximately 57.7 inches and the upper bound is approximately 69.9 inches.