Question

In Exercises 9 –12, consider a value to be significantly low if its z score is less than or equal to -2 or consider the value to be significantly high if its z score is greater than or equal to 2.

ACT The ACT test is used to assess readiness for college. In a recent year, the mean ACT score was 21.1 and the standard deviation was 5.1. Identify the ACT scores that are significantly low or significantly high

Short answer

ACT scores equal to or below 10.9 are significantly low, and ACT scores equal to or above 31.3 are significantly high.

Step-by-step solution

Given information

The mean ACT score is given as 51.1.

The standard deviation of ACT scores is given as 5.1.

Z-score

Thez-scoreof a value from a sample is computed using the following expression:

$z=\frac{x-\overline{x}}{s}$

A value whose score is less than or equal to -2 is considered to be significantly low.

A value whose score is greater than or equal to 2 is considered to be significantly high.

Determine whether the ACT score is significantly low or high

The following calculation is done to compute the values of scores that are significantly low:

$$\begin{gathered} z⩽-2 \\ \frac{x-\overline{x}}{s}⩽-2 \\ \frac{x-21.1}{5.1}⩽-2 \\ x⩽10.9 \end{gathered}$$

Therefore, ACT scores equal to or below 10.9 aresignificantly low.

The following calculation is done to compute the values of scores that are significantly high:

$$\begin{gathered} Z⩾2 \\ \frac{x-\overline{x}}{s}⩾2 \\ \frac{x-21.1}{5.1}⩾2 \\ x⩾31.3 \end{gathered}$$

Therefore, ACT scores equal to or above 31.3 aresignificantly high.