Questions: 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 ?

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 ?

Transcript text: 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 ?

Solution

Solution Steps

Step 1: Identify the Range

The range of interest is between 6.9 and 9.94.

Step 2: Calculate Z-Scores

The Z-score for the lower bound (6.9) is calculated using the formula \(Z = \frac{X - \mu}{\sigma}\), resulting in \(Z_L = -1\). Similarly, the Z-score for the upper bound (9.94) is \(Z_U = 1\).

Step 3: Use the Empirical Rule or Standard Normal Distribution

Since the range may not exactly fit within the standard deviations covered by the empirical rule, we use the standard normal distribution to find the exact percentage of the population within the specified range.