Assume the heights of women are normally distributed with a mean given by 63.6 inches and a standard deviation given by 2.5 inches, the US Army requires women's height to be between 58 and 80 inches. Find the percentage of women meeting that height requirement
To find the percentage of women meeting the height requirement, we can use the standard normal distribution.
Step 1: Standardize the lower and upper limits using the formula:
z = (x - μ) / σ
For the lower limit:
z_lower = (58 - 63.6) / 2.5 = -2.24
For the upper limit:
z_upper = (80 - 63.6) / 2.5 = 6.64
Step 2: Find the area under the standard normal curve between these two z-scores.
Using a standard normal distribution table or a calculator, we can find these areas
For the lower limit, the percentage is the area to the right of -2.24:
P(z < -2.24) = 0.9871
For the upper limit, the percentage is the area to the left of 6.64:
P(z < 6.64) = 1.0
Step 3: Find the percentage of women meeting the height requirement:
Since the percentage meeting the height requirement is between 58 and 80 inches, we need to calculate the difference between the areas:
P(height requirement) = P(z < 6.64) - P(z < -2.24)
P(height requirement) = 1.0 - 0.9871
P(height requirement) = 0.0129
Therefore, the percentage of women meeting the height requirement is approximately 0.0129 or 1.29%.
To find the percentage of women meeting the height requirement, we need to calculate the area under the normal distribution curve between the heights of 58 and 80 inches.
First, we need to standardize the heights using the formula: z = (x - μ) / σ, where x is the specific height, μ is the mean, and σ is the standard deviation.
For the lower bound of 58 inches:
z(lower) = (58 - 63.6) / 2.5 = -2.24
For the upper bound of 80 inches:
z(upper) = (80 - 63.6) / 2.5 = 6.56
Now, we can use a standard normal distribution table or a statistical calculator to find the areas corresponding to these z-scores.
From the table or calculator, we find the area to the left of z = -2.24 is approximately 0.0122 (or approximately 1.22%).
The area to the left of z = 6.56 is approximately 1 (as the standard normal distribution table approximates to 1 for extreme values).
To find the area between these two z-scores, we subtract the area for the lower bound from the area for the upper bound:
Area = 1 - 0.0122 = 0.9878 (or approximately 98.78%)
Therefore, approximately 98.78% of women meet the height requirement of the US Army.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.