Question

The grades on the last math exam had a mean of 72%. Assume the population of grades on math exams is known to be distributed normally, with a standard deviation of 5%. Using the empirical rule, approximately what percent of students earned a score between 72% and 87%?

A) About 34%
B) About 47.5%
C) About 68%
D) About 95%

Answer 1

Using the empirical rule and the fact that the data is normally distributed with a mean of 72% and a standard deviation of 5%, approximately 47.5% of students earned a score between 72% and 87% on the math exam.

Step-by-step explanation:

The question involves using the empirical rule, also known as the 68-95-99.7 rule, which applies to normally distributed data. This rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Given that the mean score is 72% and the standard deviation is 5%, a score of 87% is three standard deviations above the mean (72% + 3*5% = 87%). According to the empirical rule, 68% of the data falls between 67% and 77% (one standard deviation from the mean), 95% falls between 62% and 82% (two standard deviations from the mean), and 99.7% falls between 57% and 87% (three standard deviations).

So, to find the approximate percentage of students who earned a score between 72% and 87%, we need to calculate the area under the normal distribution curve from the mean to three standard deviations above the mean. This corresponds to approximately 50% (half of the 68%) plus 47.5% (half of the 95%), totaling approximately 97.5%. However, we only want the percentage from the mean to one standard deviation above the mean, which is approximately 34%. Since we are dealing with only half of the data from the mean to one standard deviation above, we have to subtract 50% (the other half) from 97.5%, giving us the answer of 47.5%.