Question 5 of 15, Step 1 of 1
$4 / 15$
Correct
3
transcript
Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.18 and a standard deviation of 1.5 . Using the empirical rule, what percentage of American women have shoe sizes that are no more than 3.68? Please do not round your answer.

Answer
How to enter your answer (opens in new window)
Keypad
Keyboard Shortcu
$\%$

Step 1 ：Given that the mean shoe size of American women is 8.18 and the standard deviation is 1.5, we are asked to find the percentage of American women who have shoe sizes no more than 3.68.

Step 2 ：First, we calculate the z-score, which is the number of standard deviations a data point is from the mean. The formula for the z-score is \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 3 ：Substituting the given values into the formula, we get \(Z = \frac{3.68 - 8.18}{1.5} = -3.0\). This means that a shoe size of 3.68 is 3 standard deviations below the mean.

Step 4 ：According to the empirical rule, about 0.3% of data falls beyond three standard deviations from the mean. Since we are looking for the percentage of women who have shoe sizes no more than 3.68, we need to consider the left half of the distribution.

Step 5 ：Therefore, we divide 0.3% by 2 to get 0.15%. This is the percentage of American women who have shoe sizes no more than 3.68.

Step 6 ：Final Answer: \(\boxed{0.15\%}\)