Question

A college-entrance exam is designed so that scores are normally distributed with a mean of 500 and a standard deviation of 100. What percent of exam scores are between 400 and 600?

__% of exam scores are between 400 and 600.

Answer 1

68% of exam scores lie within one std. dev. of the mean

Explanation:

Because the standard deviation is 100, one standard deviation above the mean comes out to 600. Likewise, one std. dev. below the mean comes out to 400. By the Empirical Rule, 68% of exam scores lie within one std. dev. of the mean.

Answer 2

`P(400<x<600) = 68.3\%`

Explanation:

We know that the average
`\mu` is:

`\mu=500`

The standard deviation
`\sigma` is:

`\sigma=100`

The Z-score is:

`Z=(x-\mu)/(\sigma)`

We seek to find

P(400<x<600)

This is:

`P(400<x<600)=P((400-500)/(100)<(x-\mu)/(\sigma)<(600-500)/(100))\\\\P(400<x<600)=P(-1<Z<1)`

Looking for the value of z in a normal table we have to:

`P(Z>1) =0.1587\\\\P(Z>-1) = 0.8413`

So

`P(-1<Z<1)=P(Z>-1) - P(Z>1) =0.8413-0.1587=0.6826\\\\P(400<x<600) = 68.3\%`