Question

The Graduate Record Examination (GRE) is a standardized test commonly taken by graduate school applicants in the United States. The total score is comprised of three compo- nents: Quantitative Reasoning, Verbal Reasoning, and Analytical Writing. The first two components are scored from 130 - 170. The mean score for Verbal Reasoning section for all test takers was 151 with a standard deviation of 7, and the mean score for the Quantitative Reasoning was 153 with a standard deviation of 7.67. Suppose that both distributions are nearly normal. (a) A student scores 160 on the Verbal Reasoning section and 157 on the Quantitative Reasoning section. Relative to the scores of other students, which section did the student perform better on? (b) Calculate the student’s percentile scores for the two sections. What percent of test takers per- formed better on the Verbal Reasoning section? (c) Computethescoreofastudentwhoscoredinthe80thpercentileontheQuantitativeReasoning section. (d) Compute the score of a student who scored worse than 70% of the test takers on the Verbal Reasoning section.

Answer 1

Let Q, V and A denote the random variable denoting the score obtained by the students in Quantitative Reasoning, Verbal Reasoning, and Analytical Writing respectively.

Given:

Assuming normality, since the variables Q and V follow different distributions, standardizing it would bring them down to the same graph with zero mean and unit variance:

For

Hence, we may convert both the variables into standard Z score:

By definition of standard normal Z score:

(a) For a  student who scores 160 on the Verbal Reasoning section and 157 on the Quantitative Reasoning section:

b. To compute the percentiles, which are nothing but the area to the left of the Z score in the standard normal curve. From standard normal table,

About 90% of the students score less than this student on Verbal reasoning session while about 70% of the students score less than his score on Quantitative reasoning.

Hence, we may say that the student scored better in Quantitative reasoning.

(c) The score of a student who scored in the 80th percentile on the Quantitative Reasoning section:

Looking for the area 0.80 in the normal table,

We get Z = 0.84; i.e

= 159.44

(d) Score of a student who scored worse than 70% of the test takers on the Verbal Reasoning section:

i.e. score of a student who scored better than only (100 - 70)% = 30% of the students

Looking for the area 0.30:

We get Z = -0.52

= 147.36