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.

MCAT In a recent year, scores on the Medical College Admission Test (MCAT) had a mean of 25.2 and a standard deviation of 6.4. Identify the MCAT scores that are significantly low or significantly high.

Short answer

MCAT scores equal to or below 12.4 are significantly low, and MCAT scores equal to or above 38.0 are significantly high.

Step-by-step solution

Given information

The mean MCAT score is equal to 25.2.

The standard deviation of MCAT scores is equal to 6.4.

Formula of z-score

Thez-scoreof observation gives the distance of the observation from the mean in terms of standard deviations. Its value is calculated using the expression stated below:

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

When the value of the z-score is equal to or below -2, the corresponding observation is significantly low.

When the value of the z-score is equal to or above 2, the corresponding observation is significantly high.

Identify the MCAT scores that are significantly low or high

To obtain the significantly low values, the following computations need to be done:

$$\begin{gathered} z⩽-2 \\ \frac{x-\overline{x}}{s}⩽-2 \\ \frac{x-25.2}{6.4}⩽-2 \\ x⩽12.4 \end{gathered}$$

Therefore, MCAT scores equal to or below 12.4 aresignificantly low.

To obtain the significantly high values, the following computations are done:

$$\begin{gathered} z⩾2 \\ \frac{x-\overline{x}}{s}⩾2 \\ \frac{x-25.2}{6.4}⩾2 \\ x⩾38.0 \end{gathered}$$

Therefore, MCAT scores equal to or above 38.0 are significantly high.