To solve these problems, we need to use the z-score formula, which is:

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

Where:

• $z$ is the z-score,
• $x$ is the value of the variable,
• $\mu$ is the mean, and
• $\sigma$ is the standard deviation.

In this case, the mean $\mu = 24$ and the standard deviation $\sigma = 5$.

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a) Determine the $z$-score for $x = 27$

Using the formula:

$z = \frac{{27 - 24}}{{5}} = \frac{3}{5} = 0.60$

So, $z = 0.60$.

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b) Determine the $z$-score for $x = 22$

$z = \frac{{22 - 24}}{{5}} = \frac{-2}{5} = -0.40$

So, $z = -0.40$.

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c) What value of $x$ has a $z$-score of 1.2?

Here, we solve for $x$ using the z-score formula:

$1.2 = \frac{{x - 24}}{{5}}$

Multiplying both sides by 5:

$x - 24 = 6$

So, $x = 30$.

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d) What value of $x$ has a $z$-score of 0?

If the z-score is 0, the value of $x$ is equal to the mean, since:

$z = \frac{{x - 24}}{{5}} = 0$

So, $x = 24$.

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e) What value of $x$ has a $z$-score of 0? (This part repeats d)

Again, $x = 24$, as explained in part d.

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Let me know if you need further details or clarifications!

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5 Related Questions:

1. How is the z-score used to find probabilities in a normal distribution?
2. Can you calculate the probability that $x$ falls between two values given their z-scores?
3. What does a negative z-score signify in terms of data position?
4. How do you interpret a z-score of exactly 1 or -1?
5. How would the z-scores change if the standard deviation was smaller or larger?

Tip:

A z-score of 0 means the data point is exactly at the mean of the distribution!