Question

In a certain year, when she was a high school senior, Idonna scored 671 on the mathematics part of the SAT. The distribution of SAT math scores in that year was Normal with mean 509 and standard deviation 115 . Jonathan took the ACT and scored 24 on the mathematics portion. ACT math scores for the same year were Normally distributed with mean 21.1 and standard deviation 5.3 . Find the standardized scores (±0.01) for both students. Assuming that both tests measure the same kind of ability, who had the higher score?

Idonna's standardized score is ____ Jonathan's standardized score is _____
A.) Idonna's score is higher than Jonathan's
B.) Idonna's score is less than Jonathan's
C.) Scores are equal

Answer 1

Idonna's standardized score is 1.41.

Jonathan's standardized score is 0.55.

A.) Idonna's score is higher than Jonathan's

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.

Idonna scored 671 on the mathematics part of the SAT. The distribution of SAT math scores in that year was Normal with mean 509 and standard deviation 115.

This means that her standardized score is Z when
`X = 671, \mu = 509, \sigma = 115`. So

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

`Z = (671 - 509)/(115)`

`Z = 1.41`

Idonna's standardized score is 1.41.

Jonathan took the ACT and scored 24 on the mathematics portion. ACT math scores for the same year were Normally distributed with mean 21.1 and standard deviation 5.3 .

This means that his standardized score is Z when
`X = 24, \mu = 21.1, \sigma = 5.3`

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

`Z = (24 - 21.1)/(5.3)`

`Z = 0.55`

Jonathan's standardized score is 0.55.

Due to the higher z-score, Iddona's has a higher score.