To find the $X$ value corresponding to $z = 0.75$ for each distribution, we use the following formula:

$X = \mu + z \cdot \sigma$

Where:

• $\mu$ is the mean,
• $\sigma$ is the standard deviation,
• $z$ is the z-score (which is 0.75 in this case).

1. For $\mu = 90$ and $\sigma = 4$:

$X = 90 + (0.75 \cdot 4) = 90 + 3 = 93$

2. For $\mu = 90$ and $\sigma = 8$:

$X = 90 + (0.75 \cdot 8) = 90 + 6 = 96$

3. For $\mu = 90$ and $\sigma = 12$:

$X = 90 + (0.75 \cdot 12) = 90 + 9 = 99$

4. For $\mu = 90$ and $\sigma = 20$:

$X = 90 + (0.75 \cdot 20) = 90 + 15 = 105$

Summary of $X$-values:

1. $\mu = 90$, $\sigma = 4$: $X = 93$
2. $\mu = 90$, $\sigma = 8$: $X = 96$
3. $\mu = 90$, $\sigma = 12$: $X = 99$
4. $\mu = 90$, $\sigma = 20$: $X = 105$

Would you like further explanation or details on this?

Here are 5 related questions to consider:

1. How would the value of $X$ change if $z$ was negative?
2. What is the interpretation of a z-score in a normal distribution?
3. Can you calculate the $X$-value if $z$ is 1.25 for $\mu = 90$ and $\sigma = 12$?
4. What is the significance of the standard deviation in a normal distribution?
5. How would you find $z$-values for a given $X$?

Tip: The z-score tells you how many standard deviations a data point is from the mean.