Type I and Type II Errors

When we perform a hypothesis test, we make one of two decisions: reject $H_0,$ or fail to reject $H_0.$ Since this decision is based on a random sample, our conclusion can be wrong in two distinct ways.

A Type I error occurs when we reject $H_0$ even though $H_0$ is actually true.

A Type II error occurs when we fail to reject $H_0$ even though $H_0$ is actually false.

The four possible outcomes of a test are summarized below:

$$\begin{array}{c|cc} & H_0 \text{ true} & H_0 \text{ false} \\ \hline \text{Reject } H_0 & \textbf{Type I error} & \text{Correct} \\ \text{Fail to reject } H_0 & \text{Correct} & \textbf{Type II error} \end{array}$$

For example, in a courtroom where $H_0:$ "the defendant is innocent," convicting an innocent defendant is a Type I error, while acquitting a guilty defendant is a Type II error.