Q.3.224 - Elementary Statistics

Question

SAT Scores. Each year, thousands of high school students bound for college take the Scholastic Assessment Test (SAT). This   test measures the verbal and mathematical abilities of prospective college stedents. Student scores are reported on a scale that ranges from a low of $200$ to a high of $800$ . Summary results for the scores are published by the College Entrance Examination Board in Collese Bound Seniors. In one high school graduating class, the mean SAT math score is $528$ with a standard deviation of $105$ ; the mean SAT verbal score is $475$ with a standard deviation of $98$ . A student in the graduating class scored $740$ on the SAT math and $715$ on the SAT verbal.

a. Under what conditions would it be reasonable to use z-scores to compare the standings of the student on the two tests relative to the other students in the graduating class?

b. Assuming that a comparison using z-scores is legitimate, relative to the other students in the graduating class, on which test did the student do better?

Step-by-Step Solution

Answer

a). The z-scores are utilized if the population units are evenly distributed.

b). When compared to the other scores, the student scored higher on the SAT verbal. 

Part (a) Step 1: Given Information

The condition under which it would be reasonable to use z- scores to compare the standings of the student on the two tests relative to the other students in the graduating class.

Part (a) Step 2: Explanation

The z-score is obtained by,

$z=\frac{\overline{x}-\mu }{\sigma }$

If the population units are regularly distributed, the z-scores are utilized.

The SAT scores are taken into account for thousands of observations, implying that the distribution is normal due to the high sample size.

Part (b) Step 1: Given Information

Each year, thousands of high school students bound for college take the Scholastic Assessment Test (SAT). This   test measures the verbal and mathematical abilities of prospective college stedents. 

Part (b) Step 2: Explanation

The z-score is obtained by,

$z=\frac{\overline{x}-\mu }{\sigma }$

Determine the SAT math score's z-score.

Substitute $740$ for $\overline{x},528$ for $\mu$ and $105$ for $\sigma$ in the formula $z=\frac{\overline{x}-\mu }{\sigma }$.

$z=\frac{740-528}{105}$

$=2.02$

Calculate the z- score for SAT verbal score.

Substitute $715$ for $\overline{x},475$ for $\mu$ and $98$ for $\sigma$ in the formula $z=\frac{\overline{x}-\mu }{\sigma }$.

$z=\frac{715-475}{98}$

$=2.45$

The maximum SAT verbal score is based on the two standard normal scores shown above.

As a result, when compared to the other scores, the student scored higher on the SAT verbal.