About how many students took the test?

Question

The scores on a math test are normally distributed with a mean of 78 and a standard deviation of 6. The test scores range from 0 to 100. Nineteen students had test scores between 72 and 90. Estimate the number of students who took the test.

Answer 1

For this problem, you need to understand the 68-95-99.7 rule. This states that in a normal distribution, 68% of the data is within 1 standard deviation of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations. I suggest you look up a picture of this to visualize what this looks like on a curve.

72 is one standard deviation from the mean, and 90 is 2 standard deviations from the mean. Since 68% of the data is within one standard deviation, this covers 72 to 84. To get the remaining range of 84-90, you need to find the percentile of a score of 90. Divide 95% by 2 to get 47.5, and add this to 50% to find that the percentile is 97.5. The percentile of 72, on the other hand is 50 - 68%/2, or 16. So the total percentage in this range is 97.5-16, or 81.5.

If 19 students scored between a 72 and a 90, and this should represent 81.5% of the data, you can set up a ratio and solve for x.

19/x = 81.5/100

81.5x = 1900

x = 23 people took the test