We can formulate a hypothesis test about the parameter $\theta$. Therefore we construct a “null hypothesis” $H_0$ and an “alternative hypothesis” $H_1$, which are two case statements, about what $\theta$ is. Accordingly $\Theta_0, \Theta_1$ constitute two partitions (disjoint sets) of the possible [[Sample Space]] of $\Theta$. $ \begin{cases} H_0: \theta \in \Theta_0\\ H_1: \theta \in \Theta_1 \end{cases} $ **Interpretation of null hypothesis:** We frame the $H_0$ to be our status quo of knowledge, which we want to reject. The possible outcomes of a test are.. - Reject $H_0$ - Fail to reject $H_0$ >[!note:] >Since we only work with samples, and not with the actual population data, we can never prove $H_0$ or $H_1$. We only reject and never accept a hypothesis. ## Types of Hypotheses - *Simple hypothesis:* When $\Theta_k$ takes only a single number. $ \Theta_k= \{ \theta_k\} $ - *Composite hypothesis:* When $\Theta_k$ takes more than a single number (i.e. set of numbers, ranges). $ \Theta_k=\{\theta:\theta> \theta_k \} ,\quad \Theta_k=\{\theta:\theta \not = \theta_k \} $ ## Symmetry of Tests - *Two-sided test:* When $H_0$ is a simple hypothesis and $H_1$ is a composite hypothesis. $ \begin{cases} H_0:\theta = \theta_0 \\ H_1:\theta \not = \theta_0 \\ \end{cases} $ - *One-sided test:* When both $H_0, H_1$ are composite hypothesis. $ \begin{cases} H_0:\theta \le \theta_0 \\ H_1:\theta > \theta_0 \\ \end{cases} \quad \begin{cases} H_0:\theta \ge \theta_0 \\ H_1:\theta < \theta_0 \\ \end{cases} $ ## Number of Samples - *One-sample test:* Hypothesis test, where an unknown parameter $\theta$ is compared to a fixed reference $\theta_0$. - *Two-sample test:* Hypothesis test, where two unknown parameters are compared to each other (e.g. averages of treatment and control group). $ \begin{cases} H_0:\theta_{\text{drug}} > \theta_{\text{placebo}} \\ H_1:\theta_{\text{drug}} = \theta_{\text{placebo}} \\ \end{cases} $