[tex]\mathbb P(286<X<322)=\mathbb P\left(\dfrac{286-310}{12}<\dfrac{X-310}{12}<\dfrac{322-310}{12}\right)[/tex]
[tex]\mathbb P(286<X<322)=\mathbb P(-2<Z<1)[/tex]
The empirical rule says that approximately 68% of any normal distributed data set lies within one standard deviation of the mean.
[tex]\mathbb P(-2<Z<1)=\mathbb P(-2<Z<-1)+\mathbb P(-1<Z<1)[/tex]
[tex]\implies\mathbb P(-2<Z<1)=\mathbb P(-2<Z<-1)+0.68[/tex]
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
[tex]\mathbb P(-2<Z<2)=\mathbb P(-2<Z<-1)+\mathbb P(-1<Z<1)+\mathbb P(1<Z<2)[/tex]
[tex]0.95=2\mathbb P(-2<Z<-1)+0.68[/tex]
[tex]\implies\mathbb P(-2<Z<-1)=0.135[/tex]
[tex]\implies\mathbb P(-2<Z<1)=0.135+0.68=0.815\approx82\%[/tex]
