Question

In a certain school district in a large metropolitan area, the SAT scores over that past five years are normally distributed with a mean of 1538. Furthermore, P30 is 1384. What is the P99P99 score for this population?

Answer 1

Solution:

Given, the Normal distribution with,

Mean = 1538

Let be the standard deviation.

Also given , 30th percentile P30 is 1384

By definition of percentile,

P(X < P₃₀) = 30%

P(X < 1384) = 0.30

For z , P(Z < z) = 0.30

But from z table , P(Z < -0.524) = 0.30

so , z = -0.524

Using z score formula , P₃₀ = + (z * )

1384 = 1538 + (-0.524 * )

= 293.8931

Now , we find 99th percentile P₉₉

P(Z < z) = 99%

P(Z < z) = 0.99

But from z table ,

P(Z < 2.326) = 0.99

So , z = 2.326

Now , using z score formula,

P₉₉ =   + (z * ) = 1538 + (2.326 * 293.8931) = 2221.59541985 nearly 2222

P₉₉ =  2221.59541985 nearly 2222