Andy took a physics test and scored a 35. The class average was 26 and the standard deviation was 5. Trevor took a chemistry test and scored an 82. The class average was 70 and the standard deviation was 8. Who performed better? What is the rationale for your answer, i.e., how do you know this?
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The percentage of students above Andy's score of 35 = appr 3.59%
The percentage of students above Trevor's score of 82 = appr 6.68%
So what is your verdict?
To determine who performed better, we can compare the scores of Andy and Trevor using z-scores. Z-scores allow us to standardize the scores and compare them relative to their respective distributions.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
- x is the individual score,
- μ is the mean (class average),
- σ is the standard deviation.
For Andy's physics test:
z = (35 - 26) / 5 = 1.8
For Trevor's chemistry test:
z = (82 - 70) / 8 = 1.5
The z-score is a measure of the number of standard deviations an individual's score is above or below the mean. The higher the z-score, the better the individual performed relative to the class.
Since Andy's z-score is 1.8 and Trevor's z-score is 1.5, Andy performed better. A z-score of 1.8 for Andy means his score is 1.8 standard deviations above the mean, while a z-score of 1.5 for Trevor means his score is 1.5 standard deviations above the mean.
Therefore, based on z-scores, Andy performed better on his physics test compared to Trevor on his chemistry test.
To determine who performed better between Andy and Trevor, we need to compare their scores relative to their respective class averages and standard deviations. We can use standard scores, also known as z-scores, to make this comparison.
The formula to calculate the z-score is:
z = (x - μ) / σ
where:
- z is the z-score,
- x is the individual score,
- μ is the class average, and
- σ is the standard deviation.
Let's calculate the z-scores for both Andy and Trevor:
For Andy:
z_andy = (35 - 26) / 5
= 9 / 5
= 1.8
For Trevor:
z_trevor = (82 - 70) / 8
= 12 / 8
= 1.5
The z-score measures how many standard deviations an individual score is from the mean. A higher z-score indicates a better performance relative to the class. Therefore, in this case, Andy performed better, as his z-score of 1.8 is higher than Trevor's z-score of 1.5.
The rationale for this answer is based on the fact that z-scores standardize the scores by converting them to a common scale. It allows us to compare scores from different distributions by considering their deviation from the mean of their respective populations. In this case, Andy's physics score was 1.8 standard deviations above the mean for his class, while Trevor's chemistry score was 1.5 standard deviations above the mean for his class.