On an exam with a mean of M = 78, you obtain a score of X = 84. (30 pts. total) a. Would you prefer a standard deviation of s = 2 or s = 10 (5 pt.)? Explain your answer (10 pt.). (Hint: Sketch each distribution and find the location of your score.) b. If your score were X = 72, would you prefer s = 2 or s = 10 (5 pt.)? Explain your answer (10 pt.).

A smaller standard deviation means that scores are more tightly clustered around the mean. In this case, with a standard deviation of 2, most scores would fall within a narrow range around the mean of 78. Since your score of 84 is relatively close to the mean, it would be considered high compared to the rest of the scores. However, it may not be significantly higher than other scores due to the tight clustering.

2. Standard deviation of s = 10:

A larger standard deviation means that scores are more spread out from the mean. In this case, with a standard deviation of 10, scores would be more widely dispersed around the mean of 78. Your score of 84 would still be considered high, but it would stand out more prominently from the rest of the scores due to the greater spread.

Given that you obtained a score higher than the mean, a larger standard deviation (s = 10) would make your score stand out more, potentially giving you a greater sense of achievement compared to other students. Therefore, you would likely prefer a standard deviation of s = 10.

b. For a score of X = 72:

1. Standard deviation of s = 2:

With a smaller standard deviation of 2, most scores would still be tightly clustered around the mean of 78. Your score of 72 would now be relatively low compared to the mean and the rest of the scores, indicating below-average performance. However, the tight clustering might make it harder to distinguish between different levels of performance.

2. Standard deviation of s = 10:

With a larger standard deviation of 10, scores would be more spread out from the mean. Your score of 72 would still be below the mean but may not stand out as much as it would with a smaller standard deviation. Other scores could vary widely, making it more challenging to determine your relative performance compared to other students.

Given that you obtained a score lower than the mean, a smaller standard deviation (s = 2) would provide a clearer indication of your below-average performance compared to other students. Therefore, you would likely prefer a standard deviation of s = 2.