The first Stats exam had a mean of 65 and a standard deviation of 15 ​points; the second had a mean of 80 and a standard deviation of 10 points. Derrick scored an 80 on both tests. Julie scored a 70 on the first test and a 90 on the second. They both totaled 160 points on the two​ exams, but Julie claims that her total is better. Explain. 
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer boxes within your choice. 
​(Round to three decimal places as​ needed.)
A.
​Julie's claim is correct.​ Derrick's z-scores are 
  
enter your response here for the first test and 
  
enter your response here for the second test.​ Julie's z-scores are 
  
enter your response here for the first test and 
  
enter your response here for the second test.​ Derrick's total is 
  
enter your response here​, which is less than​ Julie's total, 
  
enter your response here.
B.
​Julie's claim is incorrect.​ Derrick's z-scores are 
  
enter your response here for the first test and 
  
enter your response here for the second test.​ Julie's z-scores are 
  
enter your response here for the first test and 
  
enter your response here for the second test.​ Derrick's total is 
  
enter your response here​, which is more than​ Julie's total, 
  
enter your response here.
C.
​Julie's claim is incorrect. They both totaled 160 points on the two exams so neither student did better than the other. 

To analyze Julie's claim, we need to compare the relative performance of both Derrick and Julie in terms of **z-scores** on each exam. Z-scores help to understand how well a student performed relative to the average performance of the class.

### Z-Score Formula:
The z-score is calculated by:
\[
z = \frac{X - \mu}{\sigma}
\]
Where:
- \( X \) is the score,
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation.

---

### Derrick's Z-Scores:
For the **first test** (mean = 65, standard deviation = 15):
\[
z_1(\text{Derrick}) = \frac{80 - 65}{15} = \frac{15}{15} = 1.000
\]
For the **second test** (mean = 80, standard deviation = 10):
\[
z_2(\text{Derrick}) = \frac{80 - 80}{10} = \frac{0}{10} = 0.000
\]

### Julie's Z-Scores:
For the **first test**:
\[
z_1(\text{Julie}) = \frac{70 - 65}{15} = \frac{5}{15} = 0.333
\]
For the **second test**:
\[
z_2(\text{Julie}) = \frac{90 - 80}{10} = \frac{10}{10} = 1.000
\]

---

### Total Z-Scores:
To compare their overall performance, we sum the z-scores for both tests.

- **Derrick's total z-score**:
\[
z_{\text{total}}(\text{Derrick}) = 1.000 + 0.000 = 1.000
\]
- **Julie's total z-score**:
\[
z_{\text{total}}(\text{Julie}) = 0.333 + 1.000 = 1.333
\]

### Conclusion:
Since Julie's total z-score is higher than Derrick's, her claim is **correct**—she performed better relative to the class averages across both tests.

### Final Answer: 
**A. Julie's claim is correct. Derrick's z-scores are 1.000 for the first test and 0.000 for the second test. Julie's z-scores are 0.333 for the first test and 1.000 for the second test. Derrick's total is 1.000, which is less than Julie's total, 1.333.**

---

Do you have any questions or need more details?

---

Here are five related questions to expand on this:

1. How would their performance be different if the standard deviation on the second test was 5 instead of 10?
2. What does a z-score of 0 indicate about Derrick's performance on the second test?
3. How can z-scores be used to compare performance across different exams or subjects?
4. How would the z-scores change if Julie scored 85 on both tests?
5. What would be the impact on the total z-scores if the first test had a lower mean?

### Tip:
Z-scores are a useful tool to compare performances across different tests, especially when the means and standard deviations vary.

The first Stats exam had a mean of 65 and a standard deviation of 15 ​points; the second had a mean of 80 and a standard deviation of 10 points. Derrick scored an 80 on both tests. Julie scored a 70 on the first test and a 90 on the second. They both totaled 160 points on the two​ exams, but Julie claims that her total is better. Explain.

Explore how to compare student performance on two statistics exams using z-scores. Learn how Derrick and Julie's scores reflect their performance relative to class averages, with detailed calculations for each test. Suitable for students in Grades 9-12.

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