The level of significance, sometimes referred to as **size**, of a test is the probability of making a [[Type I Error]], denoted $\alpha$. For the simple null hypothesis $H_0: \theta = \theta_0$ $\begin{align} \alpha &= P(\text{Type I Error}) \\ &= P(\text{Reject } H_0; \ \theta_0) \end{align}$ where the notation $P(\text{Reject } H_0; \ \theta_0)$ indicates that we're assuming $\theta = \theta_0$ (or in other words that the null hypothesis is true). For a composite null hypothesis like $H_0: \theta \le \theta_0$, we must consider that there are many potential values of $\theta$. For this, we must consider the maximum probability of a Type I error under all possible values of $\theta$. $\begin{align} \alpha &= P(\text{Type I Error}) \\ &= max \ P(\text{Reject } H_0; \ \theta_0) \end{align}$