the heights of young women are approximately Normal with 65 inches and standard deviation 2.5 inches. The heights of men in the same age group have mean 70 inches standard deviation 2 inches. What percentage of women are taller than a man of average (mean) height?
To find the percentage of women who are taller than a man of average height, we need to compare the heights of women to the average height of men.
First, we need to find the Z-score for the average height of men. The Z-score is calculated using the formula:
Z = (x - μ) / σ
Where:
- Z is the standard score
- x is the value being standardized (average height of men)
- μ is the mean of the population (mean height of men)
- σ is the standard deviation of the population (standard deviation of men's heights)
Using the given values, we can calculate the Z-score for the average height of men:
Z = (70 - 70) / 2 = 0
Since the Z-score is 0, this means that the average height of men is exactly at the mean of the distribution of women's heights.
We can now determine the percentage of women who are taller than a man of average height by finding the proportion of the area under the normal distribution curve that is greater than the Z-score of 0.
Since the normal distribution is symmetrical, we only need to find the area in the upper tail of the distribution.
To find this percentage, we can use a standard normal distribution table (also known as a Z-table) or a statistical calculator.
Using a Z-table, we look up the area to the right of the Z-score of 0, which is 0.5000. Since the area under the entire normal distribution curve is 1, the area to the left of the Z-score of 0 is also 0.5000.
Therefore, the percentage of women who are taller than a man of average height is 50%.