Question

The national average SAT score is 1028. The standard deviation is 92. What is the 78th percentile score?

Answer 1

You're looking for the score
`k` such that


`\mathbb P(X\le k)=0.78`

First transform
`X` to the standard normal random variable.


`\mathbb P(X\le k)=\mathbb P\left((X-1028)/(92)\le(k-1028)/(92)\right)=\mathbb P(Z\le \hat k)=0.78`

Now 0.78 corresponds to a z-score of approximately
`\hat k=0.7722`, which means


`(k-1028)/(92)=\hat k\implies k\approx92(0.7722)+1028=1099.04\approx1099`