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?

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.