Question

A school wishes to accept 2000 students for their freshman class, and they expect 20,000 applications. In order to make their admissions decisions very easy, the only criterion they will use is SAT score. So, their goal is to accept a student if and only if their SAT score is in the top 10%. However, because their computer system is so old, the applications only come in one at a time, and they must decide whether to accept or reject before moving on to the next application. Assuming that SAT scores are normally distributed with a mean of 1000 and a standard deviation of 200, how should they set the score threshold to end up with as close to 2000 students as possible? Give your answer first symbolically (in terms of a pdf, cdf, etc), then use a normal distribution table1 to provide a numerical answer.

Answer 1

456

Explanation:

Let X be the SATscore scored by the students

Given that X is normal (1000,200)

By converting into standard normal variate we can say that

`z=(x-1000)/(200)` is N(0,1)

To find the top 10% we consider the 90th percentile for z score

Z 90th percentile = 1.28

`X= 200+1.28(200)\\= 200+256\\=456`

i.e. only students who scored 456 or above only should be considered.