To find the $z$-score, we use the formula:

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

where:

• $x = 13.5$
• $\mu = 9$ (mean)
• $\sigma = 1.5$ (standard deviation)

Substitute these values into the formula:

$z = \frac{13.5 - 9}{1.5} = \frac{4.5}{1.5} = 3$

So, the $z$-score is 3.

Interpretation

The $z$-score of 3 means that $x = 13.5$ is 3 standard deviations to the right of the mean.

Final Answer

The $z$-score when $x = 13.5$ is 3. The mean is 9. This $z$-score tells you that $x = 13.5$ is 3 standard deviations to the right of the mean.

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Do you want more details or have any questions?

Here are 5 related questions:

1. What is the probability of $X$ being greater than 13.5 in this distribution?
2. How would the $z$-score change if the standard deviation were increased to 2?
3. What does a negative $z$-score indicate in terms of position relative to the mean?
4. How would you interpret a $z$-score of 0?
5. If the mean were 12, what would the $z$-score be for ( x = 13