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This question refers to the normal distribution of exam scores using the **empirical rule**. Let's break it down:

Given:
- Mean (\(\mu\)) = 84
- Standard deviation (\(\sigma\)) = 6

The **empirical rule** (68-95-99.7 rule) states:
- 68% of data falls within 1 standard deviation (\(\mu \pm \sigma\)).
- 95% of data falls within 2 standard deviations (\(\mu \pm 2\sigma\)).
- 99.7% of data falls within 3 standard deviations (\(\mu \pm 3\sigma\)).

### Question 1: "95% of the data values lie between ___ and ___."
- 95% is within 2 standard deviations of the mean, so:
  \[
  84 - 2(6) = 84 - 12 = 72
  \]
  \[
  84 + 2(6) = 84 + 12 = 96
  \]
  **Answer: 72 and 96**

### Question 2: "% of exam scores between 78 and 84."
- The range \(78\) to \(84\) is within 1 standard deviation below the mean:
  \[
  84 - 6 = 78
  \]
  - According to the empirical rule, 34% of the data falls between \(\mu - \sigma\) and \(\mu\).
  **Answer: 34%**

### Question 3: "% of exam scores between 84 and 96."
- This is within 1 standard deviation above the mean:
  \[
  84 + 6 = 96
  \]
  - Similarly, 34% of the data falls between \(\mu\) and \(\mu + \sigma\).
  **Answer: 34%**

### Question 4: "% of exam scores between 90 and 96."
- This range is between \(84 + 1\sigma\) and \(84 + 2\sigma\):
  \[
  84 + 6 = 90 \quad \text{and} \quad 84 + 12 = 96
  \]
  - According to the empirical rule, 13.5% of the data lies between \(\mu + \sigma\) and \(\mu + 2\sigma\).
  **Answer: 13.5%**

### Question 5: "% of exam scores are greater than or equal to 78."
- This includes all data above \(84 - 1\sigma = 78\).
  - Since 68% lies within 1 standard deviation, 32% lies outside of this. Half of that is below, and the other half is above, so:
  \[
  100\% - 16\% = 84\%
  \]
  **Answer: 84%**

Would you like any further clarification?

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### Related Questions:
1. What percentage of scores fall between 72 and 90?
2. How does the empirical rule apply to non-normal distributions?
3. How do z-scores relate to the empirical rule?
4. What is the significance of 99.7% of the data in the empirical rule?
5. How can the empirical rule help in making predictions about outliers?

**Tip:** Always double-check if data is normally distributed before applying the empirical rule.

For this question, use the empirical rule. A set of exam scores is normally distributed with a mean of 84 and a standard deviation of 6. 95% of the data values lie between ___ and ___. ___% of the exam scores are between 78 and 84. ___% of the exam scores are between 84 and 96. ___% of the exam scores are between 90 and 96. ___% of the exam scores are greater than or equal to 78.

Learn how to apply the empirical rule to a normally distributed set of exam scores with a mean of 84 and a standard deviation of 6. This guide covers calculating the range for 95% of data values and the percentage of scores within specific intervals, using the 68-95-99.7 rule.

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