https://file.aiquickdraw.com/mathbot/file/1728308359042c6p0q0m8.jpg

Let's start by analyzing the two questions based on the provided images.

### Question 1
We are given a bell curve representing a normal distribution of test scores, with key values marked on the horizontal axis (test scores). We need to use the **Empirical Rule** (68-95-99.7 Rule), which is typically applied to normal distributions. The Empirical Rule tells us:
- 68% of the data lies within 1 standard deviation (σ) of the mean.
- 95% of the data lies within 2 standard deviations of the mean.
- 99.7% of the data lies within 3 standard deviations of the mean.

#### To answer the questions:
- **a. Mean**: The mean (μ) is located at the center of the distribution, which corresponds to the score of **66** (since the peak of the bell curve is at 66).
  
- **b. Standard Deviation**: Using the Empirical Rule, we can calculate the standard deviation (σ). From the graph, the values at 1 standard deviation above and below the mean are **59** and **73**, respectively. So, the standard deviation is:
  
  \[
  \sigma = 73 - 66 = 7
  \]

- **c.** The percentage of scores between 59 and 73 corresponds to **1 standard deviation** on either side of the mean. According to the Empirical Rule, **68%** of the data falls within this range.

- **d.** The percentage of scores between 52 and 80 corresponds to **2 standard deviations** on either side of the mean. According to the Empirical Rule, **95%** of the data falls within this range.

- **e.** The percentage of scores between 45 and 87 corresponds to **3 standard deviations** on either side of the mean. According to the Empirical Rule, **99.7%** of the data falls within this range.

- **f.** The percentage of scores between 66 and 73 corresponds to **half of 1 standard deviation**. Since 68% of the scores are within 1 standard deviation, 34% of the data is between the mean and 1 standard deviation above the mean, so **34%**.

- **g.** The percentage of scores less than 66 corresponds to **half of the distribution**, since 66 is the mean. So, **50%** of the scores are less than 66.

- **h.** The percentage of scores less than 73 corresponds to **84%**. This is because 68% of the data is between 59 and 73, and half of the remaining 32% (or 16%) is below 59. Adding the two, \( 50\% + 34\% = 84\%\).

### Question 2
This question also deals with a normal distribution, where the values are labeled from **81 to 99**.

- **a. Mean**: The mean (μ) is located at the center of the distribution, which is **90**.

- **b. Standard Deviation**: The values at 1 standard deviation (σ) from the mean are **84** and **96**. Hence, the standard deviation is:
  
  \[
  \sigma = 96 - 90 = 6
  \]

- **c. 68% of the data values** are between **1 standard deviation** from the mean, so the range is **84** and **96**.

- **d. 95% of the data values** are between **2 standard deviations**, so the range is **78** and **102** (though the graph only goes to 99, ideally it extends to 102).

- **e. 99.7% of the data values** are between **3 standard deviations**, so the range is **72** and **108** (again, even though 108 is beyond the graph, this is the theoretical range).

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Let me know if you need further details or clarifications!

### Follow-up questions:
1. How would you calculate the z-score for a test score of 80?
2. What does it mean if a test score falls outside of 3 standard deviations?
3. Can the Empirical Rule be applied to skewed distributions?
4. How does the Empirical Rule help in assessing probabilities for normal distributions?
5. How can we use the mean and standard deviation to find the percentage of data above a specific value?

**Tip:** The Empirical Rule only applies to normal distributions, so always ensure the data is approximately normal before using this rule.

The user provided a normal distribution graph along with several questions requiring the use of the Empirical Rule to determine the mean, standard deviation, and percentages of test scores within various ranges.

Learn how to apply the Empirical Rule to a normal distribution to calculate the mean, standard deviation, and percentages of data within 1, 2, and 3 standard deviations. This example includes test scores ranging from 45 to 87 and explores concepts like standard deviation and mean.

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