To solve the problem using the 68-95-99.7 rule, we'll follow a similar process as before.

Step 1: Understand the Problem

• Mean (μ): 92
• Standard Deviation (σ): 11
• We want to find the probability that $X$ is between 59 and 70.

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 $X = 59$: $Z_1 = \frac{59 - 92}{11} = \frac{-33}{11} = -3$
• For $X = 70$: $Z_2 = \frac{70 - 92}{11} = \frac{-22}{11} \approx -2$

Step 3: Apply the Empirical Rule

• Z = -2: This corresponds to 2 standard deviations below the mean. The empirical rule tells us that approximately 95% of the data lies within 2 standard deviations of the mean. Therefore, the area to the left of $Z = -2$ is $\frac{1 - 0.95}{2} = 0.025$ or 2.5%.

• 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 = -2

To find the probability that $X$ is between 59 and 70, subtract the area to the left of $Z = -3$ from the area to the left of $Z = -2$: $P(-3 < Z < -2) = P(Z < -2) - P(Z < -3) = 0.025 - 0.0015 = 0.0235$

Final Answer

The probability that $X$ is between 59 and 70 is approximately 0.024 (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 70 and 92?
2. What is the probability that $X$ is greater than 103?
3. If the standard deviation changes to 8 but the mean remains the same, what is the probability that $X$ is between 64 and 81?
4. Calculate the Z-score for $X = 81$ with the same mean and standard deviation.
5. What is the probability that $X$ is less than 70?

Tip: Remember, Z-scores help you understand how far a value is from the mean in terms of standard deviations. This is very useful in probability calculations involving normal distributions.