Answer to Question #251639 in Statistics and Probability for kevu

Question

At a local high school, GPA's are normally distributed with a mean of 2.5 and standard deviation of 0.6. What percentage of students at the high school have a GPA higher than 3.0?

Answer 1

Let $X$ be a random variable representing the the GPA's of students then, $X$~$N(\mu, \sigma^2)$ where $\mu=2.5$ and $\sigma^2=(0.6)^2.$ Therefore, $X$~$N(2.5,0.6^2)$.

In order to determine the percentage of students who have a GPA higher than 3.0, we first find $p(X\gt3.0)$.

Now,

$p(X\gt3)=p((X-\mu)/\sigma\gt(3-\mu)/\sigma)=p(Z\gt(3-2.5)/0.6)=p(Z\gt0.83)$

This can also be written as,

$p(Z\gt 0.83)=1-p(Z\lt0.83)=1-0.79673=0.20327$

The probability that students at the high school have a GPA higher than 3.0 is 0.20(2 decimal places).

Therefore, percentage of students who have a GPA higher than 3.0 is 0.20*100%=20%