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Let's solve the problem using the **Empirical Rule**, which applies to normal distributions:

1. **Empirical Rule Basics**:
   - 68% of the data lies within 1 standard deviation of the mean.
   - 95% lies within 2 standard deviations.
   - 99.7% lies within 3 standard deviations.

Here, the mean (\( \mu \)) is 66, and the standard deviation (\( \sigma \)) is 7. Now let’s break down each graph.

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### 1. First Graph:
- Shaded area between 66 and 73.
- This range represents from the mean (66) to 1 standard deviation above the mean (66 + 7 = 73).
- By the Empirical Rule, 68% of the data lies within 1 standard deviation of the mean (both above and below), so half of this (34%) lies between 66 and 73.

**Answer for the first graph**: 34%

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### 2. Second Graph:
- Shaded area from 59 to 73.
- This range covers 1 standard deviation below the mean (59 = 66 - 7) to 1 standard deviation above the mean (73 = 66 + 7).
- This encompasses the entire 68% of the data within 1 standard deviation of the mean.

**Answer for the second graph**: 68%

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### 3. Third Graph:
- Shaded area from 52 to 66.
- This range represents from 2 standard deviations below the mean (52 = 66 - 14) up to the mean.
- By the Empirical Rule, 95% of the data lies within 2 standard deviations (both above and below). Half of this (47.5%) lies from 52 to 66.

**Answer for the third graph**: 47.5%

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Do you need more details or have any questions about this?

Here are 5 related questions to deepen understanding:
1. What percentage of the data lies between 59 and 80 based on the Empirical Rule?
2. How would the percentages change if the standard deviation was 5 instead of 7?
3. What is the probability that a score is above 73 based on the normal distribution?
4. How does the Empirical Rule apply to distributions that are not perfectly normal?
5. How can the z-score formula be used to find more precise probabilities?

**Tip**: The Empirical Rule is only an approximation and works best for perfectly normal distributions!

A set of exam scores are approximately normally distributed with a mean = 66 and standard deviation = 7. Use the Empirical Rule to determine the area of the shaded regions shown.

Solve a problem where exam scores are normally distributed with a mean of 66 and a standard deviation of 7. Using the Empirical Rule, determine the percentages of data within 1 and 2 standard deviations from the mean. Suitable for high school students.

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