The 2013 SAT Verbal test had a mean score of 496 and a standard deviation of 115. ACT Critical Reading scores have a mean of 21.4 and a standard deviation of 6.2. Rose took both tests and scored 520 on the SAT and 24 on the ACT.

Determine which test Rose performed better on, relative to other test-takers.

The student's question is addressed by calculating and comparing Z-scores for SAT and ACT scores, demonstrating how values from different distributions can be compared through standard deviations from their means.

Explanation:

The student's question involves comparing SAT and ACT scores using the concept of Z-scores, a fundamental statistical tool. Z-scores allow for the comparison of scores from different distributions by indicating how many standard deviations an observation is from the mean. To calculate a Z-score, the formula Z = (X - μ) / σ is used, where X is the score of interest, μ is the mean, and σ is the standard deviation.

For Rose's SAT verbal score of 520, with a mean of 496 and standard deviation of 115, her Z-score is calculated as follows: Z = (520 - 496) / 115 ≈ 0.209, indicating she scored roughly 0.21 standard deviations above the mean. For her ACT score of 24, with a mean of 21.4 and a standard deviation of 6.2, her Z-score is Z = (24 - 21.4) / 6.2 ≈ 0.419, meaning she scored about 0.42 standard deviations above the ACT mean. This mathematical technique is crucial in understanding and comparing standardized test scores across different scales.