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.

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 * = 1.96

Then solve for x*:

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

so Peter's score is roughly 1237.