Last week, Sarah had exams in Math and in Spanish. On the Math exam, the mean was 30 with a standard deviation of 5, and Sarah had a score of 45. On the Spanish exam, the mean was 60 with a standard deviation of 6, and Sarah had a score of 65. In which exam did Sara have a better standing?

Question

Last week, Sarah had exams in Math and in Spanish. On the Math exam, the mean was 30 with a standard deviation of 5, and Sarah had a score of 45. On the Spanish exam, the mean was 60 with a standard deviation of 6, and Sarah had a score of 65. In which exam did Sara have a better standing?

BlackTom AI · Accepted Answer

First, let's restate the scenario and the question to ground our analysis. Sarah took two exams: Math and Spanish. For Math, the mean was 30 with a standard deviation of 5, and Sarah scored 45. For Spanish, the mean was 60 with a standard deviation of 6, and Sarah scored 65. The question asks in which exam did Sarah have a better standing compared to her peers.
Option 1: "the Math exam". To evaluate standing, we compare Sarah's score to the distribution of scores via a z-score. In Math, z = (45 - 30) / 5 = 3. This means Sarah scored 3 standard deviations above the mean, which is exceptionally high relative to her peers.
Option 2: "the Spanish exam". Here, z = (65 - 60) / 6 ≈ 0.83. This indicates Sarah did a bit better than average but not nearly as outstanding as in Math, since 0.83 standard deviations above the mean is a more modest advantage.
Option 3: "impossible to determine given the information provided". The needed information to compare standing is exactly the mean, standard deviation, and Sarah’s score for each exam, which we have. Therefore a comparison using z-scores is feasible and informative. This option ignores the core data provided and overstates uncertainty.
Option 4: "Sarah's standing in the two exams was the same". The calculations show a substantial difference in z-scores between the two exams (3 vs about 0.83), indicating the standings are not the same. Treating the two distributions as equivalent would be a misinterpretation of the relative positions of her score.
In summary, the key idea is to translate each score into a relative position within its own distribution. A z-score of 3 places Sarah far above most peers on the Math exam, while a z-score of about 0.83 places her above average but well below the Math-level standout. This helps distinguish why one standing is comparatively stronger than the other.

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