Final answer:
To compare students' performances relative to their own class, we calculate each student's z-score. Rachel's z-score is 1.2, Becca's is 2, and Matt's is 1.625, indicating how many standard deviations above the mean each student scored. By examining the z-scores, we can gauge their performance compared to their respective class peers.
Step-by-step explanation:
The question posed by the student requires an understanding of several statistical concepts, particularly how to compare different sets of scores that have different means and standard deviations from one another. To determine who performed better relative to their class, we calculate the z-score for each student. The z-score tells us how many standard deviations a particular score is from the mean.
For Rachel, with a class mean score of 74 and a standard deviation of 5, her score of 80 results in a z-score of (80-74)/5 = 1.2. This means Rachel's score is 1.2 standard deviations above the class mean.
For Becca, a mean of 47 and a standard deviation of 2, with a score of 51, results in a z-score of (51-47)/2 = 2. This indicates that Becca's score is 2 standard deviations above her class mean. Becca appears to have done relatively better compared to her class than Rachel did.
Finally, Matt with a class mean of 70 and a standard deviation of 8, scoring 83, has a z-score of (83-70)/8 = 1.625, which places his score 1.625 standard deviations above the class mean. When comparing z-scores, we can see who performed better relative to their own class's distribution.