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The image contains a problem on calculating z-scores for a normally distributed random variable \( X \sim N(5, 1.2) \). I'll walk you through how to solve each part.

### a. What is the z-score associated with \( X = 7 \)?

To calculate the z-score for a value \( x \), use the formula:
\[
z = \frac{x - \mu}{\sigma}
\]
where:
- \( \mu \) is the mean (here \( \mu = 5 \)),
- \( \sigma \) is the standard deviation (here \( \sigma = 1.2 \)),
- \( x \) is the given value (here \( x = 7 \)).

Substituting the values:
\[
z = \frac{7 - 5}{1.2} = \frac{2}{1.2} \approx 1.667
\]
So, the z-score is approximately \( 1.667 \).

---

### b. What value is associated with \( z = 1.1 \)?

To find the value \( x \) associated with a given z-score, use the inverse of the z-score formula:
\[
x = \mu + z \cdot \sigma
\]
Substitute the given values \( z = 1.1 \), \( \mu = 5 \), and \( \sigma = 1.2 \):
\[
x = 5 + 1.1 \times 1.2 = 5 + 1.32 = 6.32
\]
So, the value associated with \( z = 1.1 \) is \( x = 6.32 \).

---

### c. What value is 2 standard deviations to the right of the mean?

A value 2 standard deviations to the right of the mean can be calculated as:
\[
x = \mu + 2 \cdot \sigma
\]
Substitute \( \mu = 5 \) and \( \sigma = 1.2 \):
\[
x = 5 + 2 \times 1.2 = 5 + 2.4 = 7.4
\]
So, the value is \( x = 7.4 \).

---

Let me know if you'd like more details or have any questions!

Here are 5 related questions:
1. How would the z-score formula change for a distribution with different mean and standard deviation?
2. How do you interpret a negative z-score?
3. What is the probability of observing a value within 1 standard deviation from the mean in a normal distribution?
4. How would you calculate the value associated with \( z = -2 \)?
5. How is the empirical rule related to z-scores in a normal distribution?

**Tip**: When calculating z-scores, always ensure the standard deviation and mean match the context of the problem to avoid calculation errors.

Suppose X ~ N(5, 1.2). What is the z-score associated with X = 7? What value is associated with z = 1.1? What value is 2 standard deviations to the right of the mean?

This example covers the calculation of z-scores for a normal distribution N(5, 1.2). It walks through finding the z-score for X = 7, determining the value for z = 1.1, and calculating the value two standard deviations to the right of the mean.

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