Math Problem Statement
The first Stats exam had a mean of 60 and a standard deviation of 10 points; the second had a mean of 85 and a standard deviation of 5 points. Derrick scored an 85 on both tests. Julie scored a 70 on the first test and a 100 on the second. They both totaled 170 points on the two exams, but Julie claims that her total is better. Explain.
Solution
This question asks about comparing z-scores for two students based on their scores in two exams. Let’s break it down step by step:
Key details:
- Exam 1: Mean = 60, Standard deviation = 10.
- Exam 2: Mean = 85, Standard deviation = 5.
- Derrick's scores: 85 on both exams.
- Julie's scores: 70 on the first exam, 100 on the second exam.
- Both students totalled 170 points across both exams.
Z-score formula:
Where:
- is the score.
- is the mean.
- is the standard deviation.
Step-by-step calculations:
Derrick's z-scores:
- First test: Derrick scored 85.
- Second test: Derrick scored 85 again.
Julie's z-scores:
- First test: Julie scored 70.
- Second test: Julie scored 100.
Total z-scores:
- Derrick's total z-score:
- Julie's total z-score:
Thus, Julie's claim is correct, as her total z-score (4.000) is higher than Derrick's (2.500).
The correct answer is:
- B. Julie's claim is correct.
Would you like further details or have any questions? Here are 5 related questions to expand your understanding:
- How do z-scores help compare scores from different distributions?
- What would happen to the z-scores if the standard deviations were reversed between the two tests?
- Can you explain how the z-score reflects performance relative to the class average?
- How would you interpret a negative z-score?
- What is the significance of having a z-score of 0?
Tip: Z-scores allow comparison between scores from different tests or datasets by standardizing them based on the mean and standard deviation.
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Math Problem Analysis
Mathematical Concepts
Statistics
Z-Scores
Standard Deviation
Mean
Formulas
Z = (X - μ) / σ
Theorems
Z-Score Comparison
Suitable Grade Level
Grades 9-12