Summary
- A gap between a sample and a claim proves nothing on its own — sampling
noise guarantees some gap. The test asks whether the gap is bigger than
noise.
- \(\displaystyle Z_{calc} = \frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}\) when
\(\sigma\) is known; \(\displaystyle t_{calc} = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}\)
with \(df = n-1\) when it is not.
- Reject when the statistic beats the critical value, or equivalently when the
\(p\)-value falls below \(\alpha\). The two rules never disagree.
- A confidence interval is the set of null values a test would not reject.
Interval and test are the same statement.
- A \(p\)-value is \(P(\text{data} \mid H_0)\), never \(P(H_0 \mid \text{data})\).
- The equals sign always lives in \(H_0\). “Can you reject the claim?” puts the
claim in \(H_0\); “is there evidence supporting the claim?” puts it in \(H_A\).
- A Type I error rejects a true null, with probability \(\alpha\). A Type
II error fails to reject a false one, with probability \(\beta\). Lowering
one raises the other; only better data lowers both.
- Power \(= 1 - \beta\) is the chance of detecting an effect that is really
there. It is not chosen but inherited — from the sample size, the size of the
effect and the noise. A trial with 64% power misses a real effect more than
once in three.
qnorm() and qt() replace every printed table you own.