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 ${\mathit{\mu }}_0=500$. A sample of $247$ packages filled by the machine are collected. The sample mean $\bar{y}$ is equal to $521.3$ and the sample variance $s_{n-1}^2$ is equal to $527.08$.
   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:
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{t}& =\hfill & \frac{\bar{y}-{\mathit{\mu }}_0}{\sqrt{\frac{s_{n-1}^2}{n}}}=\frac{521.3-500}{\sqrt{\frac{527.08}{247}}}=14.581.\hfill \end{array}}$

   
   The absolute value of the $t$ test statistic is thus equal to $14.581$.