X is a normally distributed random variable with mean 33 and standard deviation 9.
What is the probability that X is between 6 and 42?
Use the 0.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.


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.

X is a normally distributed random variable with mean 33 and standard deviation 9. What is the probability that X is between 6 and 42? Use the 0.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

Calculate the probability that a normally distributed random variable with a mean of 33 and a standard deviation of 9 falls between 6 and 42 using the 0.68-0.95-0.997 rule. Learn how to apply the empirical rule and understand the basics of normal distribution. Suitable for students in Grades 10-12.

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