Question

The GRE is an entrance exam that most students are required to take upon entering graduate school. In 2014, the combined scores for the Verbal and Quantitative sections were approximately normally distributed with a mean of 310 and a standard deviation of 12.

What percent of scores were between 286 and 322? Round your answer to the nearest whole number.

Answer 1

82

Explanation:

Answer 2

`\mathbb P(286<X<322)=\mathbb P\left((286-310)/(12)<(X-310)/(12)<(322-310)/(12)\right)`

`\mathbb P(286<X<322)=\mathbb P(-2<Z<1)`

The empirical rule says that approximately 68% of any normal distributed data set lies within one standard deviation of the mean.


`\mathbb P(-2<Z<1)=\mathbb P(-2<Z<-1)+\mathbb P(-1<Z<1)`

`\implies\mathbb P(-2<Z<1)=\mathbb P(-2<Z<-1)+0.68`

The same rule states that about 95% lies within two standard deviations of the mean. Using this rule, and the fact that any normal distribution is symmetric about its mean, you have


`\mathbb P(-2<Z<2)=\mathbb P(-2<Z<-1)+\mathbb P(-1<Z<1)+\mathbb P(1<Z<2)`

`0.95=2\mathbb P(-2<Z<-1)+0.68`

`\implies\mathbb P(-2<Z<-1)=0.135`


`\implies\mathbb P(-2<Z<1)=0.135+0.68=0.815\approx82\%`