Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.368.36 and a standard deviation of 1.491.49.  Using the empirical rule, what percentage of American women have shoe sizes that are no more than 6.876.87?  Please do not round your answer.

To solve this problem, we'll use the empirical rule, which is also known as the 68-95-99.7 rule. This rule states that for a bell-shaped (normal) distribution:

- About 68% of the data falls within 1 standard deviation of the mean.
- About 95% of the data falls within 2 standard deviations of the mean.
- About 99.7% of the data falls within 3 standard deviations of the mean.

Given:
- Mean (\(\mu\)) = 8.36
- Standard deviation (\(\sigma\)) = 1.49
- Shoe size of interest = 6.87

### Step 1: Calculate the z-score

The z-score is calculated using the formula:

\[
z = \frac{X - \mu}{\sigma}
\]

Where:
- \(X\) is the value of interest (6.87),
- \(\mu\) is the mean (8.36),
- \(\sigma\) is the standard deviation (1.49).

\[
z = \frac{6.87 - 8.36}{1.49} = \frac{-1.49}{1.49} = -1
\]

### Step 2: Interpret the z-score using the empirical rule

A z-score of -1 corresponds to being 1 standard deviation below the mean. According to the empirical rule:

- 68% of the data lies within 1 standard deviation of the mean.
- This means 34% of the data lies between the mean and 1 standard deviation below the mean.
- Since the distribution is symmetric, 50% of the data lies below the mean.

Thus, the percentage of American women with shoe sizes no more than 6.87 (which is 1 standard deviation below the mean) is:

\[
50\% - 34\% = 16\%
\]

So, **16% of American women have shoe sizes that are no more than 6.87**.

Would you like more details or have any questions? 

Here are five related questions you might consider:
1. What percentage of American women have shoe sizes between 6.87 and 10.85?
2. How would the result change if the standard deviation was larger?
3. What is the percentage of women with shoe sizes more than 10.85?
4. How can we calculate the exact percentage using the z-table instead of the empirical rule?
5. What assumptions must be satisfied for the empirical rule to be accurate?

**Tip:** The empirical rule is a quick way to estimate probabilities for normally distributed data, but for more precise calculations, using z-scores and a z-table is recommended.

Learn how to use the empirical rule to determine the percentage of American women with shoe sizes no more than 6.87. Detailed step-by-step solution with the mean and standard deviation provided. Suitable for advanced high school students studying statistics.

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