A survey found that women\'s heights are normally distributed with mean 63.4 in

Question

A survey found that women's heights are normally distributed with mean 63.4 in and standard deviation 2.2 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?

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?

a. The percentage of women who meet the height requirement is __%. (Round to two decimal places as needed.) Are many women being denied the opportunity to join this branch of the military because they are too short or too tall?

A. No, because only a small percentage of women are not allowed to join this branch of the military because of their height.

B. Yes, because the percentage of women who meet the height requirement is fairly large.

C. No, because the percentage of women who meet the height requirement is fairly small.

D. Yes, because a large percentage of women are not allowed to join this branch of the military because of their height.

b. For the new height requirements, this branch of the military requires women's heights to be at least __ in and at most __ in. (Round to one decimal place as needed.)

Explanation / Answer

We are given that,

A survey found that women's heights are normally distributed with mean 63.4 in and standard deviation 2.2 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?

So you want to draw a Normal Bell Curve with the mean down the middle at 63.4 and to the left a line at 58 and to the right a line at 80. You are looking at this area in between. You can find this by finding the area up to 58 and then the area up to 80 and subtract these. So we need z values to look up.

Lower z = (58-63.4) / 2.2 = -2.45 which I look up in table gives area 0.0071

Upper z = (80-63.4)/2.2 = 7.55 ch is 1

Subtract these 1 - 0.0071 = 0.9929 which is the percentage of women who meet the height requirement.

Percentage = 0.9929*100 = 99.29%

This means that certainly many women are not being denied because only a small percentage of women are not allowed to join this branch of the military because of their height.

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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?

Now we know the area 1% or 0.01 that we find the z value so we can manipulate our formula

z = (x – µ) / and solve for x to get x = µ + z

So you can see we know the mean and the standard deviation we just need the two z values for 1% and 2%.

At 1% or 0.01 look at the four digit values (as this is the area) and I find the closest to 0.01 is 0.0099 at a z value of -2.33. So I have the lower height as

x = 63.4 + (-2.33)(2.2) = 58.28

And now for the upper height, if we are 2% in the top, that means 98% or 0.9800 is shaded beneath that we look up. I find 0.9798 for a z value of 2.05. So I have the upper height as

x = 63.4 + (2.05)(2.2) = 67.91

The new requirements are 58.28 and 67.91.

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So you want to draw a Normal Bell Curve with the mean down the middle at 63.4 and to the left a line at 58.28 and to the right a line at 67.91. You are looking at this area in between. You can find this by finding the area up to 58.28 and then the area up to 67.91 and subtract these. So we need z values to look up.

Lower z = (58.28-63.4) / 2.2 = -2.33 which I look up in table gives area 0.010

Upper z = (67.91-63.4)/2.2 = 2.05  which I look up in table gives area 0.9798.

Subtract these 0.9798 - 0.010 = 0.9698 which is the percentage of women who meet the height requirement.

Percentage = 0.9698*100 = 96.98%

This means that certainly many women are not being denied because only a small percentage of women are not allowed to join this branch of the military because of their height.

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