Question

In a particular year, the mean score on the ACT test was 25 and the standard deviation was 5.9. The mean score on the SAT mathematics test was 480 and the standard deviation was 130. The distributions of both scores are approximately bell-shaped. Round the answers to at least two decimal places.

Required:
Find the z-score for an ACT score of 26.

Answer 1

The z-score for an ACT score of 26 is 0.17.

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 a particular year, the mean score on the ACT test was 25 and the standard deviation was 5.9.

This means that
`\mu = 25, \sigma = 5.9`

Find the z-score for an ACT score of 26.

This is Z when X = 26. So

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

`Z = (26 - 25)/(5.9)`

`Z = 0.17`

The z-score for an ACT score of 26 is 0.17.