A survey found that women’s heights are normally distributed with mean 63.4 in and standard deviation 2.5 in. A branch of the military requires women’s heights to be between 58 in and 80 in.

a. Find the percentage of women meeting the height requirement. Are many women being denied the opportunity to join this branch of the military because they are too short or too tall? The percentage of women who meet the height requirement is %? (Round to two decimals places as needed).
b. If this branch of the military changes the height requirements so that all women are eligible except the shortest 1% and the tallest 2%, what are the new height requirements?

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To find the percentage of women meeting the height requirement, we need to find the area under the normal distribution curve between 58 inches and 80 inches. We can do this by standardizing the values using the formula:

z = (x - μ) / σ

where x is the height value, μ is the mean height, and σ is the standard deviation.

a. Calculation for the percentage of women meeting the height requirement:
Step 1: Standardize the lower and upper height values using the formula:
z1 = (58 - 63.4) / 2.5
z2 = (80 - 63.4) / 2.5

Step 2: Use a standard normal distribution table or a calculator to find the areas under the curve:

Area = P(z1 < Z < z2)

You can use a standard normal distribution table, or you can use online calculators or statistical software. For this example, we will use an online calculator.

Using the calculator, the area under the curve is approximately 0.9938.

Step 3: Convert the result to a percentage:
Percentage = 0.9938 * 100 = 99.38%

Therefore, approximately 99.38% of women meet the height requirement.

In this case, it seems that very few women would be denied the opportunity to join this branch of the military because they are too short or too tall.

b. If the new height requirements exclude the shortest 1% and the tallest 2%, we need to find the new height values that correspond to these percentages.

Step 1: Find the z-scores corresponding to the desired percentages.

For the shortest 1%:
z1 = z-score corresponding to a cumulative area of 0.01

For the tallest 2%:
z2 = z-score corresponding to a cumulative area of 0.98

These z-scores can also be found using a standard normal distribution table or calculator.

Step 2: Convert the z-scores back to height values using the formula:
x = (z * σ) + μ

For the shortest 1%:
x1 = (z1 * 2.5) + 63.4

For the tallest 2%:
x2 = (z2 * 2.5) + 63.4

Calculate the values using the formula and round as necessary.

These new height requirements will determine the range of heights that are now eligible to join the branch of the military.

a. Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores. Multiply by 100 to get %.

b. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.01 and .02) to get the Z scores. Insert in equation above.

98.75 was the answer my hw accepted