Question

a student took both the sat and the act, and wants to send the stronger score to colleges. the student scored 1000 on the sat, in a year that had a mean score of 1102 with a standard deviation of 184. the student scored 19 on the act, in a year that had a mean of 20.8 and a standard deviation of 5.8. using z - scores, determine which score he should send to the colleges. he should send the score to colleges because comparing the two z - score indicates that is the mean.

Explanation:

Step1: Calculate SAT z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the score, $\mu$ is the mean, and $\sigma$ is the standard deviation. For SAT, $x = 1000$, $\mu=1102$, $\sigma = 184$. So $z_{SAT}=\frac{1000 - 1102}{184}=\frac{- 102}{184}\approx - 0.554$.

Step2: Calculate ACT z - score

For ACT, $x = 19$, $\mu = 20.8$, $\sigma=5.8$. So $z_{ACT}=\frac{19 - 20.8}{5.8}=\frac{-1.8}{5.8}\approx - 0.31$.

Step3: Compare z - scores

Since $- 0.31>-0.554$ (the closer a z - score is to 0, the better relative to the mean), the ACT z - score is higher.

Answer:

He should send the ACT score to colleges because comparing the two z - score indicates that it is closer to the mean.