Chapter 11.1, Problem 34E

To determine

To explain: the conversion of ACT scores into corresponding SAT scores.

Answer to Problem 34E

The process of the conversion of ACT scores into corresponding SAT scores is explained.

Explanation of Solution

Calculation:

To convert ACT score into the corresponding SAT score, take the following steps.

1. Using the mean and standard deviation of the ACT convert score into the corresponding $z-score$ using the formula $z=\frac{x-\mu }{\sigma }$ .
2. In the next step, use this $z-score$ and the mean and standard deviation of the SAT to find the corresponding SAT score by substituting these values in the formula boxed below.

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ \Rightarrow 2\sigma =x-\mu \\ \Rightarrow x=\mu +2\sigma \end{array}$

Suppose a person scored $29$ on the mathematics section of ACT, and that ACT mean is $21.0$ and standard deviation is $5.3$ . That is $\mu =21.0$ and $\sigma =5.3$ .

So, find the $z-score$ corresponding to an $x-value$ of $29$ .

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{29-21}{5.3}\\ \approx 1.5\end{array}$

Given that for SAT mean is $514$ and standard deviation is $118$ . That is $\mu =514$ and $\sigma =118$

Substituting, these values in

  $\begin{array}{l}x=\mu +2\sigma \\ =514+\left(1.5\right)118\\ =691\end{array}$

Hence, $691$ is the corresponding SAT score.

Conclusion:

Hence, the process of the conversion of ACT scores into corresponding SAT scores is explained.