To solve this, we use the z-score formula:

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

where:

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

In this case, $\mu = 21$ and $\sigma = 5$.

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Solution:

Part (a): Determine the z-score for $x = 26$.

$z = \frac{26 - 21}{5} = \frac{5}{5} = 1.0$

Answer: $z = 1.0$

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Part (b): Determine the z-score for $x = 17$.

$z = \frac{17 - 21}{5} = \frac{-4}{5} = -0.8$

Answer: $z = -0.8$

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Part (c): What value of $x$ has a $z$-score of $1.8$?

Rearrange the formula to solve for $x$: $x = z \cdot \sigma + \mu$ $x = 1.8 \cdot 5 + 21 = 9 + 21 = 30$

Answer: $x = 30$

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Part (d): What value of $x$ has a $z$-score of $-0.4$?

$x = -0.4 \cdot 5 + 21 = -2 + 21 = 19$

Answer: $x = 19$

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Part (e): What value of $x$ has a $z$-score of $0$?

$x = 0 \cdot 5 + 21 = 21$

Answer: $x = 21$

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Would you like further details or have any questions?

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Related Questions:

1. What is the z-score formula and how is it derived?
2. How do we interpret positive and negative z-scores?
3. If the mean changes to 25 and standard deviation to 6, how would the z-scores change?
4. How do z-scores relate to percentiles in a normal distribution?
5. What are the uses of z-scores in real-world applications?

Tip:

The z-score provides a way to measure how far any value is from the mean, making it easier to compare different data points on the same scale.