A city school district has students take a mathematics test at the end of eighth

Question

A city school district has students take a mathematics test at the end of eighth grade to determine placement in high school math courses. The scores of this test are approximately normally distributed with a mean of 73 and a standard deviation of 9. The Empirical Rule states that in a normal distribution, 68% of values are within one standard deviation of the mean, 95% of values are within two standard deviations of the mean, and 99.7% of values are within three standard deviations of the mean. Using the Empirical Rule, complete the table to show the probability, as a percent, that a random student receives a score less than 55, between 55 and 82, and above 82. Express all answers to the tenths place.

Explanation / Answer

55 is 2 standard deviations below mean

So, probability of less than 55 = (1 - 0.95)/2 = 0.025

Probability of score between 55 and 73 = 0.95/2 = 0.475

Probability of score between 73 and 82(one standard deviation above mean) = 0.68/2 = 0.34

So, probability of score between 55 and 82 = 0.475+0.34 = 0.815

Probability of score below 82 = Probability of score below 73 + Probability of score between 73 and 82

= 0.5 + 0.34 = 0.84

So, probability of score above 82 = 1-0.84 = 0.16

Description Probability (%) less than 55 2.5 between 55 and 82 81.5 above 82 16

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