Question

1. sat versus act eleanor scores 680 on the sat mathematics test. the distribution of sat math scores is symmetric and single - peaked with mean 500 and standard deviation 100. gerald takes the american college testing (act) mathematics test and scores 29. act scores also follow a symmetric, single - peaked distribution - but with mean 21 and standard deviation 5. find the standardized scores for both students. assuming that both tests measure the same kind of ability, who has the higher score?

Explanation:

Step1: Recall the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Calculate Eleanor's z - score

For Eleanor who took the SAT, $x = 680$, $\mu=500$, and $\sigma = 100$.
$z_{Eleanor}=\frac{680 - 500}{100}=\frac{180}{100}=1.8$

Step3: Calculate Gerald's z - score

For Gerald who took the ACT, $x = 29$, $\mu = 21$, and $\sigma=5$.
$z_{Gerald}=\frac{29 - 21}{5}=\frac{8}{5}=1.6$

Step4: Compare the z - scores

Since $1.8>1.6$, Eleanor has a higher standardized score.

Answer:

Eleanor's standardized score is $1.8$ and Gerald's standardized score is $1.6$. Eleanor has the higher score.