Let's break this problem down into its parts, using the Empirical Rule (also known as the 68-95-99.7 Rule), which applies to symmetric, bell-shaped distributions.

Part A:

We are asked to find the values within 3 standard deviations of the mean.

• Mean (μ) = 96
• Standard deviation (σ) = 13
• 3 standard deviations means we calculate the range as follows:

$\text{Lower bound} = \mu - 3\sigma = 96 - 3(13) = 96 - 39 = 57$ $\text{Upper bound} = \mu + 3\sigma = 96 + 3(13) = 96 + 39 = 135$

Thus, the values within 3 standard deviations are between 57 and 135.

Part B:

According to the Empirical Rule:

• Within 3 standard deviations of the mean, approximately 99.7% of the values fall within this range.

Part C:

Since 99.7% of the values are within 3 standard deviations of the mean, the percentage of values more than 3 standard deviations away from the mean is:

$100\% - 99.7\% = 0.3\%$

Thus, approximately 0.3% of the values are more than 3 standard deviations away from the mean.

Summary of answers:

• A: Values between 57 and 135.
• B: Approximately 99.7% of values fall within 3 standard deviations.
• C: Approximately 0.3% of values are more than 3 standard deviations away.

Would you like further clarification on any part?

Here are 5 relative questions to expand understanding:

1. What happens to the range of values if the standard deviation changes?
2. How would the answers change if we were considering 2 standard deviations instead of 3?
3. What is the significance of the empirical rule for a non-symmetric distribution?
4. Can we use this rule for skewed distributions?
5. How would we calculate the percentage for a non-integer number of standard deviations?

Tip: When dealing with normal distributions, knowing the empirical rule (68-95-99.7) allows you to quickly estimate the spread of data without complex calculations.