Question

Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.42 and a standard deviation of 1.52. Using the empirical rule, what percentage of American women have shoe sizes that are between 6.9 and 9.94?

Answer 1

Answer: 68%

Work Shown

mu = 8.42 = mean

sigma = 1.52 = standard deviation

Let's find the z score when x = 6.9

z = (x - mu)/sigma

z = (6.9 - 8.42)/(1.52)

z = -1

Do the same for x = 9.94

z = (x - mu)/sigma

z = (9.94 - 8.42)/(1.52)

z = 1

So P(6.9 < x < 9.94) is the same as P(-1 < z < 1) when mu = 8.42 and sigma = 1.52

We need to find the area under the curve between z = -1 and z = 1.

The Empirical Rule chart is shown below. Mark -1 and 1 on the horizontal axis. Add up the areas between those endpoints. Those would be the pink regions marked with 34% each. Therefore, about 34+34 = 68% of the women have shoe sizes between 6.9 and 9.94

Roughly 68% of the shoe sizes are within 1 standard deviation of the mean.

To get a more accurate percentage, you would need to use a Z table or a stats calculator. A stats calculator is the best option I think.