Question
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Part 1
Anna scored an 89 on an exam with muequals68 points and sigmaequals10 points. Damian scored a 95 on an exam with muequals76 points and sigmaequals12 points. If both exams had normally distributed scores, what was the z-score for each student? Who did better relative to the population on their exam? Explain.
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Part 1
Select the correct choice below and fill in the answer boxes within your choice.
(Type integers or decimals rounded to the nearest tenth as needed.)
A.
Anna's z-score is
enter your response here, and Damian's z-score is
enter your response here. Damian did better relative to the population for his exam because his z-score is higher than Anna's z-score.
B.
Anna's z-score is
enter your response here, and Damian's z-score is
enter your response here. Anna did better relative to the population for her exam because her z-score is higher than Damian's z-score.
C.
Anna's z-score is
enter your response here, and Damian's z-score is
enter your response here. Damian did better relative to the population for his exam because his raw score is higher than Anna's raw score.
Part 1
Anna scored an 89 on an exam with muequals68 points and sigmaequals10 points. Damian scored a 95 on an exam with muequals76 points and sigmaequals12 points. If both exams had normally distributed scores, what was the z-score for each student? Who did better relative to the population on their exam? Explain.
Question content area bottom
Part 1
Select the correct choice below and fill in the answer boxes within your choice.
(Type integers or decimals rounded to the nearest tenth as needed.)
A.
Anna's z-score is
enter your response here, and Damian's z-score is
enter your response here. Damian did better relative to the population for his exam because his z-score is higher than Anna's z-score.
B.
Anna's z-score is
enter your response here, and Damian's z-score is
enter your response here. Anna did better relative to the population for her exam because her z-score is higher than Damian's z-score.
C.
Anna's z-score is
enter your response here, and Damian's z-score is
enter your response here. Damian did better relative to the population for his exam because his raw score is higher than Anna's raw score.
Answers
Answered by
GPT-4o mini
To find the z-scores for both Anna and Damian, we can use the formula for the z-score, which is given by:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the raw score,
- \(\mu\) is the mean (average) score of the population,
- \(\sigma\) is the standard deviation of the population.
**For Anna:**
- \(X = 89\)
- \(\mu = 68\)
- \(\sigma = 10\)
Calculating Anna's z-score:
\[
z_A = \frac{89 - 68}{10} = \frac{21}{10} = 2.1
\]
**For Damian:**
- \(X = 95\)
- \(\mu = 76\)
- \(\sigma = 12\)
Calculating Damian's z-score:
\[
z_D = \frac{95 - 76}{12} = \frac{19}{12} \approx 1.5833 \quad (rounds to 1.6)
\]
Now to summarize:
- Anna's z-score is **2.1**.
- Damian's z-score is **1.6**.
To determine who did better relative to the population on their exam, we can compare their z-scores. A higher z-score indicates a score further above the average, showing better performance relative to peers.
Since Anna's z-score (2.1) is greater than Damian's z-score (1.6), we can conclude that Anna performed better relative to her population on the exam.
Putting this all together, the correct choice is:
**A. Anna's z-score is 2.1, and Damian's z-score is 1.6. Anna did better relative to the population for her exam because her z-score is higher than Damian's z-score.**
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the raw score,
- \(\mu\) is the mean (average) score of the population,
- \(\sigma\) is the standard deviation of the population.
**For Anna:**
- \(X = 89\)
- \(\mu = 68\)
- \(\sigma = 10\)
Calculating Anna's z-score:
\[
z_A = \frac{89 - 68}{10} = \frac{21}{10} = 2.1
\]
**For Damian:**
- \(X = 95\)
- \(\mu = 76\)
- \(\sigma = 12\)
Calculating Damian's z-score:
\[
z_D = \frac{95 - 76}{12} = \frac{19}{12} \approx 1.5833 \quad (rounds to 1.6)
\]
Now to summarize:
- Anna's z-score is **2.1**.
- Damian's z-score is **1.6**.
To determine who did better relative to the population on their exam, we can compare their z-scores. A higher z-score indicates a score further above the average, showing better performance relative to peers.
Since Anna's z-score (2.1) is greater than Damian's z-score (1.6), we can conclude that Anna performed better relative to her population on the exam.
Putting this all together, the correct choice is:
**A. Anna's z-score is 2.1, and Damian's z-score is 1.6. Anna did better relative to the population for her exam because her z-score is higher than Damian's z-score.**
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