To solve this problem, we can use the 68-95-99.7 rule, also known as the empirical rule, which applies to normal distributions.

Step 1: Understand the Problem

• Mean (μ): 25
• Standard Deviation (σ): 8
• We want to find the probability that $X$ is between 1 and 17.

Step 2: Calculate the Z-scores

The Z-score for a value $X$ in a normal distribution is given by: $Z = \frac{X - \mu}{\sigma}$

• For $X = 1$: $Z_1 = \frac{1 - 25}{8} = \frac{-24}{8} = -3$
• For $X = 17$: $Z_2 = \frac{17 - 25}{8} = \frac{-8}{8} = -1$

Step 3: Apply the Empirical Rule

• Z = -1: This corresponds to 1 standard deviation below the mean. The empirical rule tells us that about 68% of the data lies within 1 standard deviation of the mean, so from $Z = -1$ to $Z = 1$, the probability is 68%. The area to the left of $Z = -1$ is therefore $\frac{1 - 0.68}{2} = 0.16$ or 16%.

• Z = -3: This corresponds to 3 standard deviations below the mean. According to the empirical rule, 99.7% of the data lies within 3 standard deviations of the mean. Therefore, the area to the left of $Z = -3$ is $\frac{1 - 0.997}{2} = 0.0015$ or 0.15%.

Step 4: Find the Probability Between Z = -3 and Z = -1

To find the probability that $X$ is between 1 and 17, subtract the area to the left of $Z = -3$ from the area to the left of $Z = -1$: $P(-3 < Z < -1) = P(Z < -1) - P(Z < -3) = 0.16 - 0.0015 = 0.1585$

Final Answer

The probability that $X$ is between 1 and 17 is approximately 0.159 (rounded to the nearest thousandth).

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is the probability that $X$ is between 25 and 41?
2. What is the probability that $X$ is greater than 33?
3. If the mean changes to 30 but the standard deviation remains the same, what is the probability that $X$ is between 10 and 22?
4. Calculate the Z-score for $X = 9$ with the same mean and standard deviation.
5. What is the probability that $X$ is less than 9?

Tip: The empirical rule is a quick way to estimate probabilities in a normal distribution, but for more precise calculations, use Z-tables or statistical software.