To solve for the z-score, we will use the formula:

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

where:

• $X$ is the value,
• $\mu$ is the mean,
• $\sigma$ is the standard deviation.

Now, let's go step by step for each question:

1. Given $X = 55$, $s = 5$, and $\mu = 60$:

Using the formula: $z = \frac{55 - 60}{5} = \frac{-5}{5} = -1$ The z-score for this case is $z = -1$.

2. Given $X = 75$, $\sigma = 10$, and $\mu = 55$:

Using the formula: $z = \frac{75 - 55}{10} = \frac{20}{10} = 2$ The z-score for this case is $z = 2$.

3. Given $X = 100$, $\sigma = 4$, and $\mu = 135$:

Using the formula: $z = \frac{100 - 135}{4} = \frac{-35}{4} = -8.75$ The z-score for this case is $z = -8.75$.

Let me know if you need any further clarification or detailed explanation! Also, here are some related questions for you:

1. How do z-scores help in interpreting data?
2. Can the z-score be negative or greater than 3? What does that indicate?
3. How do you calculate the z-score when the standard deviation is given as a sample standard deviation instead of the population standard deviation?
4. What is the interpretation of a z-score of 0?
5. How do z-scores relate to probability in a normal distribution?

Tip: Z-scores are particularly useful in understanding how far away a particular data point is from the mean, which is important in various statistical analyses such as hypothesis testing.