https://file.aiquickdraw.com/mathbot/file/1730983839897nuuz6ucx.jpg

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.

A normal distribution of scores has a standard deviation of 10. Find the z-scores corresponding to each of the following values:
- A score that is 20 points above the mean.
- A score that is 10 points below the mean.
- A score that is 15 points above the mean.
- A score that is 30 points below the mean.

Learn how to calculate z-scores for scores relative to the mean in a normal distribution with a standard deviation of 10. Find z-scores for values 20 points above the mean, 10 points below the mean, 15 points above the mean, and 30 points below the mean. Suitable for high school students.

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