To solve these problems, we’ll use the formula for finding a z-score:

$z = \frac{X - \mu}{\sigma}$

where:

• $X$ is the score,
• $\mu$ is the mean of the distribution, and
• $\sigma$ is the standard deviation.

The problem states that the standard deviation, $\sigma$, is 10. Since we’re calculating scores relative to the mean, $\mu$, we assume the mean is 0.

Let’s calculate each case:

1. A score that is 20 points above the mean: $z = \frac{20}{10} = 2$

2. A score that is 10 points below the mean: $z = \frac{-10}{10} = -1$

3. A score that is 15 points above the mean: $z = \frac{15}{10} = 1.5$

4. A score that is 30 points below the mean: $z = \frac{-30}{10} = -3$

Summary of z-scores:

• 20 points above the mean: $z = 2$
• 10 points below the mean: $z = -1$
• 15 points above the mean: $z = 1.5$
• 30 points below the mean: $z = -3$

Would you like further details on z-scores or additional examples?

Here are five follow-up questions for further exploration:

1. How does the standard deviation affect the z-score calculation?
2. What does a positive z-score indicate about a score's position relative to the mean?
3. What does a negative z-score represent in a normal distribution?
4. How would you calculate a z-score if the standard deviation changed to 5?
5. What would be the z-scores for scores exactly at the mean?

Tip: Remember, z-scores allow comparison across different datasets by standardizing scores.