## Exam 1

1. #### Question

A machine fills milk into 500ml 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 $\bar{y}$ is equal to 517.2 and the sample variance $s^2_{n-1}$ 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{aligned} t = \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{517.2 - 500}{\sqrt{\frac{262.56}{226}}} = 15.958. \end{aligned} The absolute value of the t-test statistic is thus equal to 15.958.