After a math exam, Professor Lee marked and recorded down the grades of 100 students randomly selected from the entire group of 1000 students. It shows that their average grade is 65.
He has reasons to assume the grades are normally distributed with standard deviation equal to 15. Now he wants to test if the grades of all students follow a normal distribution whose mean is 64.5.
1.

Determine the null hypothesis:


Null Hypothesis: ${\mathrm{\mu}}_{0}=64.5$ (the actual mean).

2.

Substitute the information into the formula:


$z=\frac{\left(6564.5\right)}{\left(\frac{15}{\sqrt{100}}\right)}$ = 0.333333


$p\mathrm{value}=\mathrm{Probability}\left(\rightZ>0.333333)=\mathrm{Probability}\left(Z<0.333333\right)+\mathrm{Probability}\left(Z>0.333333\right)$= 0.738883 $Z\u02dc\mathrm{Normal}\left(0,1\right)$.


This statistical test does not provide enough evidence to conclude that the null hypothesis is false, so we fail to reject the null hypothesis.
