Question 3.3

3.3 GRE scores, Part I. Sophia who took the Graduate Record Examination (GRE) scored 160 on the Verbal Reasoning section and 157 on the Quantitative Reasoning section. 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.

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

Verbal Reasoning section Normal distribution short hand is

\[ N ( \mu = 151, \sigma = 7 ) \]

Quantitative Reasoning section Normal distribution short hand is b \[ N ( \mu = 153, \sigma = 7.67 ) \]

  1. What is Sophia’s Z-score on the Verbal Reasoning section? On the Quantitative Reasoning section? Draw a standard normal distribution curve and mark these two Z-scores.

\[ Zscore = \frac{(Obs - \mu)}{\sigma} \]

Sophia’s Zscore on Verbal:

(160-151)/7
## [1] 1.285714

Sophia’s Zscore on Quantitative:

(157-153)/7.67
## [1] 0.5215124
x   <- seq(-3,3,length=1000)
y   <- dnorm(x,mean=0, sd=1)
h=plot(x,y, type="l", lwd=1)
abline(v=.52, lwd=1)
text(.52,0,"Z=0.52",srt=0.2,pos=3)
abline(v=1.29, lwd=1)
text(1.29,0,"Z=1.29",srt=0.2,pos=3)

  1. What do these Z-scores tell you?

She scored 1.29 standard deviations above the mean on the Verbal Reasoning section (90.15 Percentile)and 0.52 standard deviations above the mean on the Quantitative Reasoning section (69.85 Percentile).

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

She did better in the Verbal Reasoning section

  1. Find her percentile scores for the two exams. Using the Zscore table. She was in the 90th Percentile in Verbal Reasonining and 70th Percentile in the Quantitative Reasoning

  2. What percent of the test takers did better than her on the Verbal Reasoning section? On the Quantitative Reasoning section?

100-90 = 10% of test takers did better than her on the Verbal Reasoning

100-70 = 30% of test takers did better than her on the Quantitative Reasoning

  1. Explain why simply comparing her raw scores from the two sections would lead to the incorrect conclusion that she did better on the Quantitative Reasoning section.

We cannot compare the raw scores since they are on different scales. Comparing her percentile scores is more appropriate when comparing her performance to others.

  1. If the distributions of the scores on these exams are not nearly normal, would your answers to parts (b) - (f) change? Explain your reasoning.

Answer to part (b) would not change as Z-scores can be calculated for distributions that are not normal. However, we could not answer parts (d)-(f) since we cannot use the normal probability table to calculate probabilities and percentiles without a normal model.