Question

3. It is frequently difficult for graduate schools to compare students from different undergraduate programs since universities have vastly different scales of assessment. Some schools, for example, are known to in ate grades while others maintain more rigorous grading standards. Say that you are a graduate school admissions officer charged with selecting those applicants that have the best undergraduate grades relative to their peers. You must decide between two students - one from Princeton and one from Temple. The Princeton student has a 3.9 grade point average (GPA), and the Temple student has a 3.4 average (GPA). At Princeton, the mean GPA is 3.65 with a standard deviation of .3, while at Temple the mean GPA is 3.15 with a standard deviation of .27. At both schools, GPAs are Normally distributed. Answer the following questions, and show how you arrived at these conclusions. (14 points)

a. If you are interested in selecting the student with the better record compared to the other students at his/her university, which student do you select and why? Make sure to back up your decision with evidence produced by your quantitative toolkit so that you're able to compare both students despite the fact that their schools are quite different.

b. What proportion of Temple students have a higher GPA than this Temple student (who is applying to graduate school in the above problem)?

Answer 1

At Princeton, the mean GPA is 3.65 with a standard deviation of .3

Let X be the GPA of any given Princeton student. X has a normal distribution with mean and standard deviation

while at Temple the mean GPA is 3.15 with a standard deviation of .27.

Let Y be the GPA of any given Temple student. Y has a normal distribution with mean and standard deviation

a) The Princeton student has a 3.9 grade point average (GPA), and the Temple student has a 3.4 average (GPA).

We are interested in selecting the student with the better record compared to the other students at his/her university. We will use z scores to get the record of each of these students compared to the other students at his/her university.

The z score of the Princeton student with a GPA 3.9 is

The z score of the Temple student with a GPA 3.4 is

We can see that the Princeton student is 0.83 standard deviations above the average GPA at Princeton and the Temple student is 0.93 standard deviations above the average GPA at Temple. Hence we can say that Temple student has a better record compared to the other students at his/her university.

ans: We will select the student from Temple. This student has a better record compared to the other students at his/her university, as indicated by the higher z score for the Temple student when compared to the z score of Princeton student.

b. What proportion of Temple students have a higher GPA than this Temple student (who is applying to graduate school in the above problem)?

The proportion of Temple students have a higher GPA than this Temple student is same as the probability that a randomly selected Temple student has a GPA higher than 3.4

ans: The proportion of Temple students have a higher GPA than this Temple student is 0.1762