Suppose that shoe sizes of america women have a bell-shaped distribution with a mean of 8.47 and a standard deviation of 1.5. Using the empirical rule, what percentage of American women have shoe sizes that are between 5.47 and 11.47?

The empirical rule, also known as the **68-95-99.7 rule**, states that for a bell-shaped (normal) distribution:

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

In this case:
- The mean (\(\mu\)) is 8.47.
- The standard deviation (\(\sigma\)) is 1.5.

You are asked to find the percentage of women whose shoe sizes are between 5.47 and 11.47. Let's calculate:

1. The lower bound, 5.47, is exactly **2 standard deviations below** the mean:
   \[
   8.47 - 2 \times 1.5 = 5.47
   \]

2. The upper bound, 11.47, is exactly **2 standard deviations above** the mean:
   \[
   8.47 + 2 \times 1.5 = 11.47
   \]

Using the empirical rule, we know that **95%** of the data lies within 2 standard deviations of the mean. 

Thus, **95%** of American women have shoe sizes between 5.47 and 11.47.

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:
1. What percentage of women have shoe sizes between 6.97 and 9.97?
2. How would the percentage change if the range was within 1 standard deviation of the mean?
3. What is the percentage of women with shoe sizes greater than 11.47?
4. How would the empirical rule apply if the standard deviation changed to 2?
5. Can the empirical rule be used for any distribution, or only normal distributions?

**Tip:** The empirical rule only applies to bell-shaped, normal distributions. For other types of distributions, different methods like Chebyshev’s theorem may be used.

Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.47 and a standard deviation of 1.5. Using the empirical rule, what percentage of American women have shoe sizes that are between 5.47 and 11.47?

Find out what percentage of American women have shoe sizes between 5.47 and 11.47 using the empirical rule. With a mean of 8.47 and a standard deviation of 1.5, this statistical problem applies the 68-95-99.7 rule to a normal distribution. Perfect for students in Grades 9-12.

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