To solve these problems, we need to find the z-scores based on the standard deviation given.

Given Information

• The standard deviation ($\sigma$) is 10.
• The formula for z-score is: $z = \frac{X - \mu}{\sigma}$ where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Let's solve each part:

1. A score that is 20 points above the mean:

   • $X - \mu = 20$
   • $z = \frac{20}{10} = 2$
   • Answer: $z = 2$

2. A score that is 10 points below the mean:

   • $X - \mu = -10$
   • $z = \frac{-10}{10} = -1$
   • Answer: $z = -1$

3. A score that is 15 points above the mean:

   • $X - \mu = 15$
   • $z = \frac{15}{10} = 1.5$
   • Answer: $z = 1.5$

4. A score that is 30 points below the mean:

   • $X - \mu = -30$
   • $z = \frac{-30}{10} = -3$
   • Answer: $z = -3$

Would you like further details on z-score calculations or have additional questions?

Here are some related questions to expand understanding:

1. What is the significance of a z-score in a normal distribution?
2. How can z-scores help in comparing scores from different distributions?
3. What does a positive or negative z-score indicate?
4. How would the z-scores change if the standard deviation were different?
5. How are z-scores used in determining probability?

Tip: Understanding z-scores can help interpret how far a data point is from the mean in terms of standard deviations, useful for understanding data spread and making comparisons.