X is a normally distributed random variable with mean 9 and standard deviation 5.
What is the probability that X is between 4 and 19?
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 refers to the empirical rule, which states that for a normally distributed random variable:

- 68% of the values lie within 1 standard deviation of the mean.
- 95% of the values lie within 2 standard deviations of the mean.
- 99.7% of the values lie within 3 standard deviations of the mean.

Given:
- Mean \( \mu = 9 \)
- Standard deviation \( \sigma = 5 \)

We are asked to find the probability that \( X \) lies between 4 and 19.

### Step 1: Standard Deviations from the Mean

- The lower bound is 4, so the number of standard deviations from the mean is:
  \[
  \frac{4 - 9}{5} = -1
  \]
- The upper bound is 19, so the number of standard deviations from the mean is:
  \[
  \frac{19 - 9}{5} = 2
  \]

### Step 2: Use the Empirical Rule

- \( X \) is within 1 standard deviation below the mean (from 9 to 4) and 2 standard deviations above the mean (from 9 to 19).
- From the empirical rule, we know:
  - 68% of the values lie within 1 standard deviation.
  - 95% of the values lie within 2 standard deviations.

Since the range we are interested in covers from \(-1\sigma\) to \(2\sigma\), we can infer that the probability for this range is approximately:

\[
P(4 < X < 19) = \frac{95\% + 68\%}{2} = 81.5\%
\]

### Step 3: Convert to Decimal

The probability is approximately \( P(4 < X < 19) = 0.815 \).

Thus, the probability that \( X \) is between 4 and 19 is \( \boxed{0.815} \).

Would you like more details or further clarification?

Here are 5 related questions for practice:
1. What is the probability that \( X \) is between 9 and 14?
2. What is the probability that \( X \) is greater than 19?
3. What is the probability that \( X \) is less than 4?
4. If \( X \) has a different mean and standard deviation, how does the empirical rule change?
5. How would you calculate the probability using the standard normal distribution (Z-scores)?

**Tip**: When working with the empirical rule, always check how many standard deviations away your values are from the mean.

X is a normally distributed random variable with mean 9 and standard deviation 5. What is the probability that X is between 4 and 19? Use the 0.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

Find the probability that a normally distributed variable X, with a mean of 9 and a standard deviation of 5, falls between 4 and 19 using the empirical rule (0.68-0.95-0.997 rule). This step-by-step solution explains how to apply the rule and calculate the probability, rounded to the nearest thousandth. Suitable for Grades 10-12.

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