1. **State the problem:** We have a normally distributed data set with 12,658 scores. We want to find how many scores are expected to be more than two standard deviations away from the mean. 2. **Recall the empirical rule:** In a normal distribution, about 95% of data lies within two standard deviations of the mean. Therefore, about 5% lies outside this range (both tails combined). 3. **Calculate the proportion outside two standard deviations:** $$\text{Proportion outside} = 1 - 0.95 = 0.05$$ 4. **Calculate the number of scores outside two standard deviations:** $$\text{Number} = 0.05 \times 12658 = 632.9$$ 5. **Interpretation:** Approximately 633 scores are expected to be more than two standard deviations away from the mean. --- 1. **State the problem:** Given a normal distribution with mean $\mu = 23.6$ and standard deviation $\sigma = 4.4$, determine if the score $X = 33.1$ is an outlier. 2. **Calculate the z-score:** $$z = \frac{X - \mu}{\sigma} = \frac{33.1 - 23.6}{4.4} = \frac{9.5}{4.4} \approx 2.16$$ 3. **Interpret the z-score:** A common rule is that scores with $|z| > 3$ are considered outliers. 4. **Conclusion:** Since $|2.16| < 3$, the score 33.1 is not considered an outlier.