[[Hypothesis Tests]] describe the entire process of inferring something about a population. One part of it is the statistical test, which is a quantitative decision rule that outputs either $\{0,1\}$. It does not depend on the true unknown $\theta$, and can thus be calculated from the data. **Example:** We want to test if people have a tendency to kiss from one side or not. The collected observations are $X_1, \dots ,X_n \stackrel{iid}{\sim} \text{Ber}(p)$ from [[Bernoulli Distribution|Bernoulli]], where $p$ is the unknown parameter. - Formulating hypothesis: $ \begin{cases} H_0:p = 0.5 \\ H_1:p \not = 0.5 \\ \end{cases} $ - Conducting statistical test: $ \psi=\mathbf 1 \{\underbrace{\lvert \bar X_n -0.5\rvert}_{T_n} >c \} $ The statistical test $\psi$ is an indicator variable $\mathbf 1$. It returns either $\{0,1\}$, dependent if the test-statistic $T_n$ is above or below the critical value $c$. - If $\psi =0$ then $H_0$ is not rejected (it points to $H_0$). - If $\psi = 1$ then $H_0$ is rejected (it points to $H_1$).