Question

Carl scored 32 on the ACT mathematics Test and 730 on the mathematics portion of the SAT. If the ACT Math Test had a mean score of 21.0 with a standard deviation of 5.3, and the mathematics section of the SAT had a mean score of 516 with a standard deviation of 116, on which exam did Carl earn a better score with respect to his peers?

Answer 1

Answer:

Carl's ACT grade had a higher z-score, which means that he earned a better score with respect to his peers on the ACT test.

Step-by-step explanation:

Z-score

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

On which exam did Carl earn a better score with respect to his peers?

On whichever exam he had the higher z-score.

SAT

Scored 730, mean 516, standard deviation 116. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{730 - 516}{116}$

$Z = 1.84$

ACT

Scored 32, mean 21, standard deviation 5.3. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{32 - 21}{5.3}$

$Z = 2.08$

Carl's ACT grade had a higher z-score, which means that he earned a better score with respect to his peers on the ACT test.