Question

A college senior who took the Graduate Record Examination exam scored 540 on the Verbal Reasoning section and 690 on the Quantitative Reasoning section. The mean score for Verbal Reasoning section was 456 with a standard deviation of 123, and the mean score for the Quantitative Reasoning was 441 with a standard deviation of 153. Suppose that both distributions are nearly normal. Round calculated answers to 4 decimal places unless directed otherwise.

1.Write down the short-hand for these two normal distributions.

The Verbal Reasoning section has a distribution N( , )

The Quantitative Reasoning section has a distribution of N( , )

2. What is her Z score on the Verbal Reasoning section?

3. What is her Z score on the Quantitative Reasoning section?

4. Relative to others, which section did she do better on?

A. She did the same on both sections
B. Verbal Reasoning
C. Quantitative Reasoning

5. What is her percentile score on the Verbal Reasoning section? Round to nearest whole number.

6. What is her percentile score on the Quantitative Reasoning section? Round to nearest whole number.

7. What percent of the test takers did better than she did on the Verbal Reasoning section?   %

8. What percent of the test takers did better than she did on the Quantitative Reasoning section?   %

9. What is the score of a student who scored in the 64th64th percentile on the Quantitative Reasoning section? Round to the nearest integer.

10. What is the score of a student who scored worse than 76% of the test takers in the Verbal Reasoning section? Round to the nearest integer.

Answer 1

1.
Verbal reasoning section : N(456, 123)
Quantitative reasoning section: N(441,153)

For Normal Distribution,

2) For verbal reasoning,

Z = (540-456)/123 = 0.6829

3) For quantitative reasoning,

Z = (690-441)/153 =1.6274

4) she do better on Reasoning section

5) For verbal reasoning,

Z = (540-456)/123 = 0.6829

P-value corresponding to Z= -0.6829 is 75.27 percentile. Rounding to nearest whole number, we get 75 percentile

6) For quantitative reasoning,

Z = (690-441)/153 =1.6274

P-value corresponding to Z=1.6274 is 94.82 percentile. Rounding to nearest whole number, we get 95 percentile

7) 100-75.27= 24.73

24.73 percentage of candidates did better on verbal reasoning.

8) 100-94.82=5.18

5.18 percentage of candidates did better on quantitative reasoning

9) Z-value corresponding to 64th percentile is 0.36

0.36 = (X-441)/153

55.08 = X - 441

X = 441+55.08 = 496.08 ~496

10) Z-value corresponding to 46th percentile is 0.71

0.71 = (X-456)/123

87.33 = X - 456

X = 456+87.33 = 543.33 ~ 543