Question

Suppose that combined verbal and math SAT scores follow a normal distribution with a mean 896 and standard deviation 174. Suppose further that Peter finds out he scored in the top 2.5 percentile (97.5% of students scored below him). Determine how high Peter's score must have been.

Answer 1

Let x* denote Peter's score. Then

P(X > x*) = 0.025

P((X - 896)/174 > (x* - 896)/174) = 0.025

P(Z > z*) = 1 - P(Z < z*) = 0.025

P(Z < z*) = 0.975

where Z follows the standard normal distribution (mean 0 and std dev 1).

Using the inverse CDF, we find

P(Z < z*) = 0.975  ==>  z* = 1.96

Then solve for x*:

(x* - 896)/174 = 1.96  ==>  x* = 1237.04

so Peter's score is roughly 1237.