Hypothesis Testing
🚧 Work in progress — this is an unfinished draft and may change.
Brief introduction to hypothesis testing.
Hypothesis
$ \hspace{3mm} H_0 : \theta = \theta_0 $
$ \hspace{3mm} H_1: \theta \neq **theta_0** $
Test Statistic
$ \hspace{3mm} T = g(X_1, X_2, \dots, X_n) $
$\hspace{3mm} \text{where} \; 333X_1, X_2, \dots, X_n$
P-Value
$ \hspace{3mm} p = P(T \geq t_{\text{obs}} \, \vert \, H_0) $
Decision Rule
$ \hspace{3mm} H_0 \iff p \leq \alpha $
$ \hspace{3mm} H_0 \iff T \in C_{\alpha} $
Significance Level
\[\hspace{3mm}\]Error Types
\[\hspace{3mm}\]Power
\[\hspace{3mm}\]Z-Test Example
\[\hspace{3mm}\]E-Processes
Optimal Power
It is desireable to maximize power.