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.