## Exam 1

1. #### Question

A machine fills milk into $500$ml packages. It is suspected that the machine is not working correctly and that the amount of milk filled differs from the setpoint ${\mu }_{0}=500$. A sample of $226$ packages filled by the machine are collected. The sample mean $\stackrel{‾}{y}$ is equal to $517.2$ and the sample variance ${s}_{n-1}^{2}$ is equal to $262.56$.
Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the absolute value of the $t$ test statistic?

#### Solution

The $t$ test statistic is calculated by:
 $\begin{array}{ccc}\multicolumn{1}{c}{t}& =\hfill & \frac{\stackrel{‾}{y}-{\mu }_{0}}{\sqrt{\frac{{s}_{n-1}^{2}}{n}}}=\frac{517.2-500}{\sqrt{\frac{262.56}{226}}}=15.958.\hfill \end{array}$

The absolute value of the $t$ test statistic is thus equal to $15.958$.