## Related questions with answers

In 1998 Ben took both the SAT (Scholastic Aptitude Test) and the ACT (American College Test). On the mathematics section of the SAT, he earned a score of 624. On the mathematics section of the ACT, he earned a score of 31. For the SAT the mean was 512 and the standard deviation was 112. For the ACT the mean was 21 and the standard deviation was 5.

Explain how you could translate ACT scores such as 15, 20, 25, and 30 into equivalent SAT scores if you know the mean and standard deviation of each exam.

Solution

VerifiedFor this example we'll use the means and standard deviations given on the header of exercise 56.

First we express the ACT scores in terms of their standard deviation from the mean (we call this a $\textit{z-score}$). A z-score is obtained by the formula: $Z(x) = \frac{x-\overline x}{\sigma}$. Hence:

$Z(15) = \frac{15-21}{5} = -1.2$

$Z(20) = \frac{20-21}{5} = -0.2$

$Z(25) = \frac{25-21}{5} = 0.8$

$Z(30) = \frac{30-21}{5} = 1.8$

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