Question

Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.15 and a standard deviation of 1.51. Using the empirical rule, what percentage of American women have shoe sizes that are less than 5.13? Please do not round your answer.

Answer 1

Less than 2.5% of American women have shoe sizes less than 5.13 according to the empirical rule, given the z-score of approximately -2 for a shoe size 5.13 with a mean of 8.15 and a standard deviation of 1.51.

Step-by-step explanation:

The question asks about using the empirical rule to find the percentage of American women with shoe sizes less than 5.13 given a mean of 8.15 and a standard deviation of 1.51. Considering the empirical rule states that approximately 68% of data lies within one standard deviation of the mean, 95% within two standard deviations, and over 99% within three standard deviations for a bell-shaped distribution, we can calculate the z-score for a shoe size of 5.13.

Z = (X - μ) / σ = (5.13 - 8.15) / 1.51 = -2.01

Here, the z-score is approximately -2, which means the shoe size 5.13 is two standard deviations below the mean. According to the empirical rule, less than 2.5% of the distribution is more than two standard deviations below the mean. Therefore, we expect less than 2.5% of American women to have shoe sizes less than 5.13.