Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.478.47 and a standard deviation of 1.51.5.  Using the empirical rule, what percentage of American women have shoe sizes that are at least 11.4711.47?  Please do not round your answer.

To solve this problem using the empirical rule (68-95-99.7 rule), let's break it down step by step:

1. **Understand the empirical rule**: The empirical rule states that for a bell-shaped (normal) distribution:
   - About 68% of the data falls within 1 standard deviation of the mean.
   - About 95% falls within 2 standard deviations.
   - About 99.7% falls within 3 standard deviations.

2. **Given values**:
   - Mean (\(\mu\)) = 8.47
   - Standard deviation (\(\sigma\)) = 1.5
   - We want to find the percentage of women with shoe sizes at least 11.47.

3. **Calculate how many standard deviations away from the mean 11.47 is**:
   \[
   z = \frac{x - \mu}{\sigma} = \frac{11.47 - 8.47}{1.5} = \frac{3}{1.5} = 2
   \]
   This means 11.47 is 2 standard deviations above the mean.

4. **Using the empirical rule**:
   - We know that approximately 95% of the data falls within 2 standard deviations (between \(\mu - 2\sigma\) and \(\mu + 2\sigma\)).
   - This implies that 5% of the data lies outside this range (both tails of the distribution).
   - Since the normal distribution is symmetric, half of this 5% will be in each tail. Thus, 2.5% of the data lies above 11.47.

Therefore, according to the empirical rule, the percentage of American women who have shoe sizes at least 11.47 is **2.5%**.

Would you like further details, or do you have any other questions?

Here are 5 related questions that could further expand on this topic:
1. What is the percentage of American women with shoe sizes less than 5.47?
2. How would you calculate the percentage of women with shoe sizes between 7 and 10 using the empirical rule?
3. What is the z-score for a shoe size of 9.47?
4. How does the empirical rule change if the distribution is not bell-shaped?
5. What is the probability of a randomly selected American woman having a shoe size within one standard deviation of the mean?

**Tip:** The empirical rule is a quick way to estimate probabilities and understand data distributions, but for more precise calculations, especially with non-symmetric distributions, using exact z-scores and normal distribution tables is advisable.

Discover how to calculate the percentage of American women who have shoe sizes at least 11.47 using the empirical rule. Understand the normal distribution, standard deviations, and z-scores involved in this statistical problem. Suitable for advanced high school students and beyond.

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