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### Part (a) - Find the z-score for an ACT score of 19.
We are given:
- Mean ACT score: \( \mu = 22.2 \)
- Standard deviation for the ACT: \( \sigma = 5.3 \)
- ACT score: \( X = 19 \)

The formula for the z-score is:
\[
z = \frac{X - \mu}{\sigma}
\]
Substitute the values:
\[
z = \frac{19 - 22.2}{5.3} = \frac{-3.2}{5.3} \approx -0.60
\]

The z-score for an ACT score of 19 is approximately **-0.60**.

---

### Part (d) - Emma's SAT score had a z-score of -1.4. What was her SAT score?

We are given:
- Mean SAT score: \( \mu = 515 \)
- Standard deviation for the SAT: \( \sigma = 117 \)
- z-score: \( z = -1.4 \)

To find Emma's SAT score, we use the formula for z-score rearranged to solve for \( X \):
\[
X = \mu + z \cdot \sigma
\]
Substitute the values:
\[
X = 515 + (-1.4) \cdot 117 = 515 - 163.8 = 351.2
\]

Emma's SAT score is approximately **351**.

---

Would you like more details or further explanation on any of these steps? Here are some related questions you might consider:

1. How do we interpret z-scores in terms of standard deviations?
2. How would the z-score change if Emma's SAT score was higher than 515?
3. What is the significance of a negative z-score in standardized tests?
4. How would you calculate the ACT score for a given z-score?
5. Can you explain how the bell curve affects these calculations?

**Tip**: The z-score tells you how many standard deviations a data point is from the mean, which helps compare scores across different tests!

In a particular year, the mean score on the ACT test was 22.2 and the standard deviation was 5.3. The mean score on the SAT mathematics test was 515 and the standard deviation was 117. Find the z-score for an ACT score of 19 and calculate Emma's SAT score if her z-score was -1.4.

This problem explores how to calculate the z-score for an ACT score of 19 given a mean score of 22.2 and standard deviation of 5.3. Additionally, it shows how to determine Emma's SAT score from her z-score of -1.4, with the SAT mean score of 515 and a standard deviation of 117.

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