Critical Region

The test procedure partitions the sample space into an acceptance region and a rejection/critical region for $H_0$

When the $H_0$ is true, the probability of obtaining a value of test statistics that the corresponding random samples fall inside the critical region is called the size of the critical region or the level of significance of the test

• the level of evidence against the null
  General procedure:
• Assume $H_0$ is true, so that the observations follow the distribution described by $H_0$
• Partition the sample space into a critical/rejection $C$ and its complement $C^c=\overline{C}$
• decide whether to reject or accept $H_0$ based on the data and $C/\overline{C}$

For discrete distributions, we may specify $C$ with a maximum size of test to be $\alpha$ instead of a specific value, if $C=\{X\le 14\}:$

We can then alter the critical region to adjust $\alpha$ and $\beta$ : $C=\{X\le 15\}\to \alpha =0.043,\beta =0.0509$

• the larger $C$, the larger $\alpha$, and smaller $\beta$

Test function