Introduction

SAT is one of the tests that many US universities use as an admission requirement. TheChronicle of Higher Education(2003, January 29) summarized an article that appeared onThe American Prospectwebsite titled "College Try: Why Universities Should Stop Encouraging Applicants to Take the SATs Over and Over Again." This paper argues that current college admission policies that permit applicants to take the SAT exam multiple times and then use the highest score for admission consideration favor students from families with higher incomes (who can afford to take the exam many times). The author proposes two alternatives that he believes would be fairer than using the highest score. They are:

Alternative 1: Use the average of all test scores Alternative 2: Use only the most recent score

In this activity, you will investigate the differences between the three possibilities by looking at the sampling distributions of three statistics for the cases of a test taker who takes the exam twice and a test taker who takes the exam five times:

max= maximum score

mean= average score

recent= most recent score

Scenario

An individual's scores on the SAT exam will fluctuate between test administrations. Suppose that a particular student's "true ability" is reflected by an SAT score of 1200 but, due to chance fluctuations, the test score on any particular administration of the exam can be considered a random variable that has a distribution that is approximately normal with mean 1200 and standard deviation 30. If we select a sample from this normal distribution, the resulting set of observations can be viewed as a collection of test scores that might have been obtained by this student.

Assume that you are writing a recommendation for the admission criteria of a selective university. You will submit your simulations, analysis, and recommendation to the university administrators for consideration in one zipped file.

Requirements

Part I

Using simulation, you will generate samples of two test scores,score1,andscore2for 1,000 SAT test takers. Then you will compute the values ofmax,mean, andrecentfor each pair of scores. The resulting values ofmax,mean, andrecentwill be used to construct approximations to the sampling distributions of the three statistics, which will be referred to asmax2, mean2,andrecent2.

a.Using a tool of choice (Excel or R), obtain 1,000 sets of five "test scores" by generating observations from a normal distribution with mean 1200 and standard deviation 30. Explain how you generated the pairs.

b.Calculate the values ofmax2, mean2,andrecent2. Construct density diagrams for each.

Part II

Using simulation, you will generate samples of five test scores,score1, score2, score3, score4,andscore5for 1,000 SAT test takers. Then you will compute the values ofmax,mean, andrecentfor each quintuplet of scores. The resulting values ofmax5,mean5, andrecent5will be used to construct approximations to the sampling distributions of the three statistics.

a.Using a tool of choice, obtain 1,000 sets of five "test scores" by generating observations from a normal distribution with mean 1200 and standard deviation 30. Describe how you generated the quintuplets.

b.Calculate the values ofmax5, mean5,andrecent5. Construct density diagrams for each.

Part III

a.Explain how the sampling distributions ofmean2andmean5compare to what is expected based on the general properties of thex.

b.Based on the three distributions from Part I, for a two-time test taker, describe the advantage of using the maximum score compared to using either the average score or the most recent score.

c.Explain how the approximate sampling distributions of the maximum score for two-time and for five-time test takers compare.

d.Explain if a student who takes the exam five times has a big advantage over a student of equal ability who only takes the exam twice if the maximum score is used for college admission decisions.

e.Justify why you would recommend using the maximum test score, the average test score, or the most recent test score in making admission decisions.