Section 19.2

Hypothesis Tests and Sample Sizes

Type I and II errors, power, the sample size needed for a given precision or power.

Toolkit

A hypothesis test fixes a rejection region $R$ under $H_0$ ; the resulting error probabilities drive sample-size choices.

Type I error: $\alpha = \mathbb{P}(\text{reject } H_0 \mid H_0) = \mathbb{P}(T \in R \mid H_0)$ , with $T$ ’s null distribution computed exactly (e.g. binomial tail sums) when not relying on asymptotics.
Type II error: $\beta = \mathbb{P}(\text{fail to reject } H_0 \mid H_1) = \mathbb{P}(T \notin R \mid H_1)$ ; power $=1-\beta$ . First find the critical value from $\alpha$ under $H_0$ , then recompute the relevant tail probability under $H_1$ ’s distribution.
For a mean/proportion with $\hat\theta \approx \mathcal{N}(\theta, \sigma^2/n)$ , a two-sided $(1-\alpha)$ confidence interval or test has half-width $z_{\alpha/2}\sigma/\sqrt n$ .
To guarantee half-width $\le E$ : solve $n \ge \left(\dfrac{z_{\alpha/2}\,\sigma}{E}\right)^2$ ; for a proportion use $\sigma^2 = p(1-p)$ (or its worst case $1/4$ ).
To guarantee power $1-\beta$ against a specific alternative, add the corresponding $z_\beta$ term to the critical-value equation before solving for $n$ .