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.