To solve this, we'll calculate the z-scores for each test score using the formula:

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

Where:

• $x$ is the test score,
• $\mu = 1487$ is the mean,
• $\sigma = 317$ is the standard deviation.

Step 1: Calculate z-scores for each value

1. For $x = 1890$:

$z = \frac{1890 - 1487}{317} = \frac{403}{317} \approx 1.27$

2. For $x = 1240$:

$z = \frac{1240 - 1487}{317} = \frac{-247}{317} \approx -0.78$

3. For $x = 2180$:

$z = \frac{2180 - 1487}{317} = \frac{693}{317} \approx 2.19$

4. For $x = 1390$:

$z = \frac{1390 - 1487}{317} = \frac{-97}{317} \approx -0.31$

Step 2: Determine whether the values are unusual

In general, a z-score is considered unusual if:

$|z| > 2$

• For $x = 1890$, $z \approx 1.27$: Not unusual.
• For $x = 1240$, $z \approx -0.78$: Not unusual.
• For $x = 2180$, $z \approx 2.19$: Unusual ($|z| > 2$).
• For $x = 1390$, $z \approx -0.31$: Not unusual.

Conclusion:

The test score of $2180$ is unusual because its z-score is greater than 2. None of the other scores are unusual.

Would you like me to provide further explanations or visualizations? Here are related questions:

1. How do we interpret the z-score for normal distributions?
2. What is the probability associated with each z-score?
3. Why is a z-score above 2 considered unusual?
4. How do standard deviations affect the spread of a dataset?
5. Can we visualize the distribution of the data and z-scores?

Tip: Z-scores are a powerful way to compare data points across distributions with different means and standard deviations.