1. A normal distribution of scores has a standard deviation of 10. Find the z-scores 
corresponding to each of the following values:

a) A score that is 20 points above the mean.
b) A score that is 10 points below the mean.
c) A score that is 15 points above the mean. 
d) A score that is 30 points below the mean

To find the \( z \)-scores for each given value, we use the formula for the \( z \)-score:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \( x \) is the raw score,
- \( \mu \) is the mean of the distribution,
- \( \sigma \) is the standard deviation.

Since we are not given the mean (\( \mu \)), but only relative distances from it, we can directly compute the \( z \)-scores using the standard deviation \( \sigma = 10 \). Let’s solve each part:

### a) A score that is 20 points above the mean:
\[
z = \frac{20}{10} = 2
\]

### b) A score that is 10 points below the mean:
\[
z = \frac{-10}{10} = -1
\]

### c) A score that is 15 points above the mean:
\[
z = \frac{15}{10} = 1.5
\]

### d) A score that is 30 points below the mean:
\[
z = \frac{-30}{10} = -3
\]

Thus, the \( z \)-scores are:
- a) \( z = 2 \)
- b) \( z = -1 \)
- c) \( z = 1.5 \)
- d) \( z = -3 \)

Would you like more details on how these values were computed, or do you have any questions?

### Related Questions:
1. How do you interpret a \( z \)-score in a normal distribution?
2. What does it mean if a \( z \)-score is positive or negative?
3. How can the \( z \)-score be used to find probabilities in a normal distribution?
4. What is the empirical rule in relation to standard deviation and \( z \)-scores?
5. How can we convert a \( z \)-score back to a raw score?

**Tip:** A \( z \)-score of 0 represents a value that is exactly equal to the mean.

A normal distribution of scores has a standard deviation of 10. Find the z-scores corresponding to each of the following values:

a) A score that is 20 points above the mean.
b) A score that is 10 points below the mean.
c) A score that is 15 points above the mean.
d) A score that is 30 points below the mean.

This problem focuses on calculating z-scores for values that are above or below the mean in a normal distribution with a standard deviation of 10. It covers key statistical concepts and is suitable for students in Grades 9-12.

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