To find the z-scores for each of these cases, we use the formula:

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

where:

• $X$ is the score,
• $\mu$ is the mean (not given directly here, but it’s assumed that "above" or "below the mean" indicates deviations from it),
• $\sigma$ is the standard deviation, which is given as 4.

Let’s calculate each case:

1. A score that is 4 points above the mean: $z = \frac{4}{4} = 1$

2. A score that is 6 points below the mean: $z = \frac{-6}{4} = -1.5$

3. A score that is 12 points above the mean: $z = \frac{12}{4} = 3$

4. A score that is 12 points below the mean: $z = \frac{-12}{4} = -3$

Each z-score represents how many standard deviations each score is from the mean. Would you like additional details on interpreting these z-scores or information on the corresponding percentages?

Here are five related questions to delve deeper:

1. How do we interpret positive and negative z-scores in a normal distribution?
2. What percentage of scores fall within one standard deviation of the mean in a normal distribution?
3. How would the z-scores change if the standard deviation was 8 instead of 4?
4. Why is the z-score useful for comparing scores from different distributions?
5. How can we calculate the probability associated with each z-score?

Tip: When working with z-scores, remember they indicate how far and in what direction (above or below) a value is from the mean in terms of standard deviations.