Women’s heights are normally distributed with a mean of 63.6 in and a sd of 2 .5in. The US Army requires women’s heights to be between 58 in and 80 in.
Find the percentage of women meeting that height requirement.
Find the Z scores for those two heights and use the same table indicated in previous post.
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To find the percentage of women meeting the height requirement, we can use the concept of the standard normal distribution.
1. First, we need to standardize the height requirement using the formula:
z = (X - μ) / σ
Where:
X = height requirement (minimum or maximum)
μ = mean height
σ = standard deviation
For the minimum height requirement of 58 inches:
z_min = (58 - 63.6) / 2.5
For the maximum height requirement of 80 inches:
z_max = (80 - 63.6) / 2.5
2. Next, we need to find the corresponding areas under the standard normal curve for the standardized values from step 1. These areas represent the percentage of women meeting the height requirement.
We can use a standard normal table or a statistical calculator to find these areas. For example, if using a standard normal table, we can find the area corresponding to z_min and z_max.
Let's assume the area corresponding to z_min is A_min and the area corresponding to z_max is A_max.
3. Finally, we can calculate the percentage of women meeting the height requirement by subtracting A_min from A_max and multiplying by 100:
Percentage = (A_max - A_min) * 100
This will give us the percentage of women whose heights fall within the specified range of 58 inches to 80 inches.
To find the percentage of women meeting the height requirement, we need to calculate the proportion of women whose heights fall within the specified range.
First, let's find the z-scores for the lower and upper bounds of the height requirement using the formula:
z = (x - μ) / σ
where:
x = the value we want to convert to a z-score (lower or upper bound)
μ = the mean of the distribution (63.6 in)
σ = the standard deviation of the distribution (2.5 in)
For the lower bound:
z_lower = (58 - 63.6) / 2.5
For the upper bound:
z_upper = (80 - 63.6) / 2.5
Now, we can use a standard normal table or a statistical calculator to find the corresponding area under the normal curve for these z-scores. The area represents the proportion of women meeting the height requirement.
Alternatively, we can use a calculator or an online tool to find the cumulative probability associated with the z-scores. The cumulative probability is the percentage of values below a particular z-score.
Let's calculate the percentages using a statistical calculator or online tool.
Using an online calculator, the percentage of women meeting the height requirement is approximately:
Percentage = P(z_lower < Z < z_upper)
Note: Z represents the standard normal distribution.
You can substitute the calculated values of z_lower and z_upper into the above expression and find the result.