The 0.68-0.95-0.997 rule (also known as the empirical rule) applies to normally distributed data and helps approximate probabilities for a normal distribution. According to this rule:

• 68% of the data falls within 1 standard deviation of the mean.
• 95% of the data falls within 2 standard deviations of the mean.
• 99.7% of the data falls within 3 standard deviations of the mean.

Given:

• Mean ($\mu$) = 33
• Standard deviation ($\sigma$) = 9

We need to find the probability that $X$ lies between 6 and 42.

Step 1: Calculate how many standard deviations the endpoints are from the mean.

• $X = 6$: $z = \frac{6 - 33}{9} = \frac{-27}{9} = -3$ This means 6 is 3 standard deviations below the mean.

• $X = 42$: $z = \frac{42 - 33}{9} = \frac{9}{9} = 1$ This means 42 is 1 standard deviation above the mean.

Step 2: Apply the empirical rule.

• From the rule, we know that:
  • About 68% of the data lies between $\mu - 1\sigma$ and $\mu + 1\sigma$ (i.e., between 24 and 42).
  • About 95% of the data lies between $\mu - 2\sigma$ and $\mu + 2\sigma$ (i.e., between 15 and 51).
  • About 99.7% of the data lies between $\mu - 3\sigma$ and $\mu + 3\sigma$ (i.e., between 6 and 60).

Step 3: Interpret the intervals.

Since 6 is 3 standard deviations below the mean and 42 is 1 standard deviation above the mean:

• The empirical rule tells us that approximately 99.7% of the data falls between $\mu - 3\sigma$ and $\mu + 3\sigma$, and 68% of the data falls within $\mu - 1\sigma$ and $\mu + 1\sigma$.

Therefore, the probability that $X$ lies between 6 and 42 can be approximated as:

$P(6 \leq X \leq 42) \approx 0.84$

Thus, the probability is approximately 0.84.

Would you like to go over the calculation or details of the empirical rule?

Here are 5 follow-up questions:

1. What is the probability that $X$ is between 24 and 42?
2. How can we calculate the exact probability without using the empirical rule?
3. What does it mean for a distribution to be "normally distributed"?
4. How would the probability change if the standard deviation were smaller?
5. What is the probability that $X$ is less than 6?

Tip: The empirical rule is a fast approximation tool, but using z-scores and a standard normal distribution table gives more precise probabilities.