Question

In 2017 the SAT had a mean score of 1060 with standard deviation of 195 (max score of 1600) while the ACT had a mean of 20.9 with standard deviation of 5.6 (max composite score of 36). Reed and Maria apply to the same college. Maria took the SAT and received a score of 1270 while Reed took the ACT and received a score of 27. Which student performed better in relationship to their peers

Answer 1

Due to the higher z-score, Reed performed better in relationship to their peers

Explanation:

Z-score:

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

`Z = (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.

In this question:

Whoever had the higher z-score performed better in relation to their peers.

Maria:

Took the SAT, grade 1270, so
`X = 1270`

Mean score of 1060 with standard deviation of 195 (max score of 1600). This means that
`\mu = 1060, \sigma = 295`. So

`Z = (X - \mu)/(\sigma)`

`Z = (1270 - 1060)/(295)`

`Z = 0.71`

Reed:

Took the ACT, score of 27, so
`X = 27`

Mean of 20.9 with standard deviation of 5.6, which means that
`\mu = 20.9, \sigma = 5.6`. So

`Z = (X - \mu)/(\sigma)`

`Z = (27 - 20.9)/(5.6)`

`Z = 1.09`

Due to the higher z-score, Reed performed better in relationship to their peers