The mean finish time for a yearly amateur auto race was 185.84 minutes with a standard deviation of 0.377 minute. The winning​ car, driven by Alex​, finished in 184.98 minutes. The previous​ year's race had a mean finishing time of 111.6 with a standard deviation of 0.146 minute. The winning car that​ year, driven by Sally​, finished in 111.24 minutes. Find their respective​ z-scores. Who had the more convincing​ victory?

To determine who had the more convincing victory, we need to calculate the **z-scores** for both Alex's and Sally's finishing times in their respective races.

### Formula for Z-Score:
\[
z = \frac{x - \mu}{\sigma}
\]
Where:
- \(x\) is the individual score (finishing time in this case),
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.

#### 1. Alex's Z-Score:
For Alex, we have:
- \(x = 184.98\) minutes,
- \(\mu = 185.84\) minutes,
- \(\sigma = 0.377\) minute.

Now, calculating Alex's z-score:
\[
z_A = \frac{184.98 - 185.84}{0.377} = \frac{-0.86}{0.377} \approx -2.28
\]

#### 2. Sally's Z-Score:
For Sally, we have:
- \(x = 111.24\) minutes,
- \(\mu = 111.6\) minutes,
- \(\sigma = 0.146\) minute.

Now, calculating Sally's z-score:
\[
z_S = \frac{111.24 - 111.6}{0.146} = \frac{-0.36}{0.146} \approx -2.47
\]

### Interpretation:
- **Z-scores** indicate how many standard deviations away a score is from the mean.
- The **lower the z-score**, the further away from the mean (in this case, faster relative to the average) the individual is.

Since **Sally's z-score (-2.47)** is lower than **Alex's z-score (-2.28)**, Sally's performance was more statistically exceptional, meaning Sally had the more convincing victory.

---

Would you like any further details or have any questions?

### Related Questions:
1. What does a negative z-score signify in this context?
2. How do z-scores help in comparing performances across different distributions?
3. Can we use z-scores to compare athletes from different sports or competitions?
4. What would a z-score of 0 indicate in a race?
5. How does the standard deviation affect the interpretation of z-scores?

### Tip:
Z-scores are particularly useful when comparing performances across different events with varying averages and spreads, as they standardize different datasets.

Learn how to calculate z-scores to compare the winning times of Alex and Sally in two different amateur auto races. This problem explains key statistical concepts such as z-scores and standard deviation, helping students understand how to evaluate performance in different contexts. Suitable for Grades 10-12.

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