Type I and Type II Errors, and Power

Two ways to be wrong

A significance test makes a decision from a sample, so it can err in two directions. A Type I error is rejecting $H_0$ when $H_0$ is actually true — a false alarm. A Type II error is failing to reject $H_0$ when $H_a$ is actually true — a missed signal. The probability of a Type I error is exactly the significance level $\alpha$: at $\alpha=0{,}05$, one true null in twenty will be wrongly rejected. The probability of a Type II error is called $\beta$, and it depends on how far the truth is from $H_0$.

Power

Power is the probability that the test correctly rejects $H_0$ when a particular alternative is true: $\text{power}=1-\beta$. A test with power $0{,}80$ will detect that particular effect four times in five. Four things raise power: a larger sample size, a larger $\alpha$ (at the cost of more Type I errors), a larger true effect (further from $H_0$), and less variability in the data (or a paired/blocked design that removes it). Example: a factory tests $H_0:\mu=500$ ml against $H_a:\mu\lt500$ with $n=20$ and $\alpha=0{,}05$; if the true mean is $495$ ml the power is $0{,}55$, but with $n=50$ it climbs to $0{,}91$.

Writing about errors in context

The exam asks three things: describe each error in the words of the problem, name a consequence of each, and say which is more serious here. Structure: "A Type I error would mean concluding that [$H_a$ in context] when in fact [$H_0$ in context]; the consequence is that [what happens next]." No symbols, no "rejecting a true null" — the context is the whole answer.