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The image shows a normal distribution problem asking to calculate Z-scores and related values.

The distribution is given by:
\[
X \sim N(8, 1.8)
\]
Where 8 is the mean (\(\mu\)) and 1.8 is the standard deviation (\(\sigma\)).

The three sub-questions are:

### Part a:
What is the Z-score associated with \(x = 6\)?

The formula for calculating the Z-score is:
\[
z = \frac{x - \mu}{\sigma}
\]
Substituting the values:
\[
z = \frac{6 - 8}{1.8} = \frac{-2}{1.8} = -1.111
\]
So, the Z-score is approximately \(-1.111\).

### Part b:
What value is associated with \(z = 1.4\)?

To find the corresponding value for a given Z-score, we rearrange the Z-score formula:
\[
x = z\sigma + \mu
\]
Substituting the values:
\[
x = (1.4)(1.8) + 8 = 2.52 + 8 = 10.52
\]
So, the value associated with \(z = 1.4\) is approximately \(10.52\).

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

This means \(z = 2\). Using the same formula as in part b:
\[
x = (2)(1.8) + 8 = 3.6 + 8 = 11.6
\]
So, the value 2 standard deviations to the right of the mean is \(11.6\).

Let me know if you'd like further details or clarification.

Here are five related questions to deepen understanding:
1. How would the Z-score change if the standard deviation were larger?
2. What does a Z-score of 0 represent in a normal distribution?
3. How do you interpret a Z-score of \(-2.5\)?
4. How would you calculate the probability that a value is below a certain Z-score?
5. How would the graph of the distribution change if the mean were increased?

**Tip:** The Z-score tells you how many standard deviations a data point is from the mean, making it useful for comparing values from different normal distributions.

The query asks for calculating Z-scores and related values for a normal distribution X ~ N(8, 1.8), with three sub-questions: calculating the Z-score for x=6, finding the value associated with z=1.4, and determining the value two standard deviations to the right of the mean.

Explore the detailed solution for calculating Z-scores and values associated with standard deviations in the normal distribution X ~ N(8, 1.8). This includes solving for the Z-score at x=6, the value for z=1.4, and determining the value two standard deviations to the right of the mean.

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