Final answer:
For an exam score of 53, the highest relative comparison to classmates occurs with a mean of 50 and a standard deviation of 1, yielding a z-score of 3, which indicates the score is 3 standard deviations above the mean.
Step-by-step explanation:
If you earned a score of X=53 on an exam, the highest relative score compared to other people in the class would be achieved using the parameters where the mean (average) is lowest and the standard deviation (SD) is highest. To determine the highest relative score, we calculate the z-score, which is the number of standard deviations a data point (your score) is from the mean. The formula for a z-score is z = (X - µ) / σ, where X is your score, µ is the mean, and σ is the standard deviation.
- For a mean of 56 and SD of 3: z = (53 - 56) / 3 = -1
- For a mean of 56 and SD of 1: z = (53 - 56) / 1 = -3
- For a mean of 50 and SD of 1: z = (53 - 50) / 1 = 3
- For a mean of 50 and SD of 3: z = (53 - 50) / 3 = 1
The z-score is highest for option c, where the mean is 50 and the SD is 1. Thus, option c would give you the highest relative score as 3 standard deviations above the mean is a greater achievement compared to being 1 standard deviation above (or below) the mean in the other scenarios.