Hypothesis Testing

Definition

Types of Errors

	$H_0$ is true	$H_1$ is true
Not reeject $H_0$	$1-\alpha$	type 2 error ($\beta$)
Reject $H_0$	type 1 error ($\alpha$)	$1-\beta$

Let the set $C$ where $H_0$ is rejected be a rejection region and a statistic $T$ for testing be a test statistic, then the size $\alpha$ is the maximum of probability of type 1 error a test of size $\alpha$ is $\max P_{H_0}(T\in C)\le \alpha$ a power of the test is $P_{H_1}(T\in C)=1-\beta$ the probability of the type 2 error is $P_{H_1}(T\notin C)=\beta$

p-value

The probability of obtaining test results at least as extreme as the result of actually observed under $H_0$