Question

The quantitative reasoning GRE scores are known to approximately have a Normal distribution with a mean of  = 151.3 points and a standard deviation of  = 8.7 points.

a. Use the Empirical Rule to specify the ranges into which 68%, 95%, and 99.7% of test takers fall. Include a picture to illustrate the ranges.

b. A graduate program in Public Policy Analysis admits only students with quantitative reasoning GRE scores in the top 30%. What is the lowest GRE score the program will accept?

c. Above what score do the top 1% of GRE scores fall?

Answer 1

a) According to the empirical rule about 68% of the data fall within one standard deviation from the mean.

- = 151.3 - 8.7 = 142.6

+ = 151.3 + 8.7 = 160

68% of test takers fall between 142.6 and 160.

According to the empirical rule about 95% of the data fall within two standard deviation from the mean.

- 2 = 151.3 - 2 * 8.7 = 133.9

+ 2 = 151.3 + 2 * 8.7 = 168.7

95% of test takers fall between 133.9 and 168.7.

According to the empirical rule about 99.7% of the data fall within three standard deviation from the mean.

- 3 = 151.3 - 3 * 8.7 = 125.2

+ 3 = 151.3 + 3 * 8.7 = 177.4

99.7% of test takers fall between 125.2 and 177.4.

b) P(X > x) = 0.3

Or, P((X - )/ > (x - )/) = 0.3

Or, P(Z > (x - 151.3)/8.7) = 0.3

Or, P(Z < (x - 151.3)/8.7) = 0.7

Or, (x - 151.3)/8.7 = 0.52

Or, x = 0.52 * 8.7 + 151.3

Or, x = 155.824

c) P(X > x) = 0.01

Or, P((X - )/ > (x - )/) = 0.01

Or, P(Z > (x - 151.3)/8.7) = 0.01

Or, P(Z < (x - 151.3)/8.7) = 0.99

Or, (x - 151.3)/8.7 = 2.33

Or, x = 2.33 * 8.7 + 151.3

Or, x = 171.571