In a certain country the heights of adult men are normally distributed with a mean of 69.4 inches and a standard deviation of 2.4 inches. The country's military requires that men have heights between 66 inches and 76 inches. Determine what percentage of this country's men are eligible for the military based on height.
The percentage of men that are eligible for the military based on height is
%.
(Round to two decimal places as needed.
87.50
To determine the percentage of men eligible for the military based on height, we need to find the area under the normal distribution curve between the heights of 66 inches and 76 inches.
First, we need to standardize these values using the formula:
z = (x - μ) / σ
Where:
- z is the standardized score
- x is the original height
- μ is the mean height
- σ is the standard deviation of height
For 66 inches:
z1 = (66 - 69.4) / 2.4
For 76 inches:
z2 = (76 - 69.4) / 2.4
Next, we need to find the corresponding areas under the standard normal distribution curve using the z-scores.
Using a standard normal distribution table or a calculator, we can find the following probabilities:
- P(Z < z1) = P(Z < (66 - 69.4) / 2.4)
- P(Z < z2) = P(Z < (76 - 69.4) / 2.4)
Then, we subtract P(Z < z1) from P(Z < z2) to find the percentage of men eligible for the military:
Percentage = (P(Z < z2) - P(Z < z1)) * 100
Now, let's calculate the values:
z1 = (66 - 69.4) / 2.4 = -1.4167
z2 = (76 - 69.4) / 2.4 = 2.7500
Using a standard normal distribution table or a calculator, we find:
P(Z < -1.4167) ≈ 0.0788
P(Z < 2.7500) ≈ 0.9972
Now, let's calculate the percentage:
Percentage = (0.9972 - 0.0788) * 100
Percentage ≈ 91.84%
Therefore, approximately 91.84% of men in this country are eligible for the military based on height.
To determine the percentage of men eligible for the military based on height, we need to calculate the proportion of men with heights between 66 inches and 76 inches in relation to the total population.
First, we need to standardize the height values using the formula for z-score:
z = (x - μ) / σ
where:
x = individual height
μ = mean of the population (69.4 inches in this case)
σ = standard deviation of the population (2.4 inches in this case)
For the lower limit of 66 inches:
z1 = (66 - 69.4) / 2.4
For the upper limit of 76 inches:
z2 = (76 - 69.4) / 2.4
Calculating the z-scores:
z1 = -1.42
z2 = 2.75
Next, we need to find the proportion of men within this range by looking up the corresponding z-scores in the standard normal distribution table.
The standard normal distribution table gives us the area under the curve to the left of the specified z-score. Since we want the proportion between the two z-scores, we subtract the area corresponding to the lower z-score from the area corresponding to the higher z-score.
To find the percentage, we multiply the resulting proportion by 100.
The percentage of men eligible for the military based on height can be calculated as follows:
percentage = (area between z1 and z2) * 100
Using the standard normal distribution table, we find that the area corresponding to z1 is 0.0838 and the area corresponding to z2 is 0.9970.
percentage = (0.9970 - 0.0838) * 100
percentage = 91.32%
Therefore, approximately 91.32% of men in this country are eligible for the military based on height.