USING THE EMPIRICAL RULE

The scores for all high school seniors taking the verbal section of the Scholastic Aptitude Test (SAT) in a particular year had a mean of 490 and a standard deviation of 100. The distribution of SAT scores is bell-shaped.

A. What percentage of seniors scored between 390 and 590 on this SAT test?

B. One student scored 795 on this test. How did this student do compared to the rest of the scores?

C. A rather exclusive university only admits students who were among the highest 16% of the scores on this test. What score would a student need on this test to be qualified for admittance to this university?

SOLUTION

The data being described are the verbal SAT scores for all seniors taking the test one year. Since this is describing a population, we will denote the mean and standard deviation as m = 490 and s = 100, respectively. A bell shaped curve summarizing the percentages given by the empirical rule is below.

A. From the figure above, about 68% of seniors scored between 390 and 590 on this SAT test.

B. Since about 99.7% of the scores are between 190 and 790, a score of 795 is excellent. This is one of the highest scores on this test.

C. Since about 68% of the scores are between 390 and 590, this leaves 32% of the scores outside this interval. Since a bell-shaped curve is symmetric, one-half of the scores, or 16%, are on each end of the distribution. The figure below shows these percentages.

Since about 16% of the students scored above 590 on this SAT test, to be qualified for admittance to this university, a student would need to score 590 or above on this test.

COMMENTS ABOUT THE SOLUTION

- Before applying the empirical rule it is a good idea to identify the data being described, and the value of the mean and standard deviation.

- You should also sketch a graph summarizing the information provided by the empirical rule. If helpful, more than one graph may be needed to help find the desired solution.