Question

In a certain country the heights of adult men are normally distributed with a mean of 66.1 inches and a standard deviation of 2.7 inches. The​ country's military requires that men have heights between 62 inches and 73 inches. Determine what percentage of this​ country's men are eligible for the military based on height.

Answer 1

To determine the percentage of men eligible for the military based on height, we need to calculate the proportion of men whose heights fall between 62 inches and 73 inches. First, convert the height values to z-scores using the formula z = (x - mean) / standard deviation. Then, look up the corresponding z-scores in the standard normal distribution table to find the proportion of men within the desired height range.

Step-by-step explanation:

To determine the percentage of men eligible for the military based on height, we need to calculate the proportion of men whose heights fall between 62 inches and 73 inches.

First, we convert the height values to z-scores using the formula z = (x - mean) / standard deviation.

Then, we look up the corresponding z-scores in the standard normal distribution table to find the proportion of men within the desired height range. We can subtract this proportion from 1 to find the percentage of men who are eligible for the military based on height.

Answer 2

93.03%

Step-by-step explanation:

Population mean (μ) = 66.1 inches

Standard deviation (σ) = 2.7 inches

The z-score for a given 'X' value is:

`z = (X- \mu)/(\sigma)`

For X = 62 inches

`z = (62- 66.1)/(2.7)\\z=-1.5185`

A z-score of -1.5185 corresponds to the 6.44-th percentile of a normal distribution.

For X = 62 inches

`z = (73- 66.1)/(2.7)\\z=2.5555`

A z-score of 2.5555 corresponds to the 99.47-th percentile of a normal distribution.

The total percentage of men eligible for the military is the percentage within those two values, therefore:

`E = 99.47-6.44\\E=93.03 \%`

93.03% this​ country's men are eligible for the military based on height.