Framework

1. State $H_0$ (null) and $H_1$ (alternative)
2. Choose a test statistic $T$ whose distribution is known under $H_0$
3. Compute $p = P(T \geq t_{\text{obs}} \mid H_0)$
4. Reject $H_0$ if $p < \alpha$ (significance level, typically 0.05)

Type I and Type II errors

	$H_0$ true	$H_0$ false
Reject $H_0$	Type I error (rate $\alpha$)	Correct (power $1-\beta$)
Fail to reject	Correct	Type II error (rate $\beta$)

Power $= 1 - \beta$ increases with sample size $n$, effect size, and $\alpha$.

The $p$-value misconception

A $p$-value is not the probability that $H_0$ is true given the data. It is $P(\text{data} \mid H_0)$, not $P(H_0 \mid \text{data})$. Confusing the two is a common error. Bayesian testing avoids it by computing the posterior directly.