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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{730 - 516}{116}[/tex]
[tex]Z = 1.84[/tex]
ACT
Scored 32, mean 21, standard deviation 5.3. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{32 - 21}{5.3}[/tex]
[tex]Z = 2.08[/tex]
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.