Hypothesis Testing

Sizes: P_H(X ≤ 2) = 2/20 = 0.1 and P_H(X ≥ 19) = 0.1. Powers: P_K(X ≥ 19) = 39/210 ≈ 0.186 and $P_K(X\le 2)=3/210\approx \mathrm{0.014.}\psi$ rejects where the likelihood ratio is largest — the Neyman–Pearson direction.

Type I = false rejection of a true null; its probability is the significance level $\alpha$. Accepting a false null is Type II, with probability $\beta =1-$ power. The 0.05 threshold is a convention, not a definition.

(1) is the Neyman–Pearson lemma. (2) A larger $\alpha$ admits more rejection regions, and the optimum over a larger class is no worse. (3) is the classic misreading — the p-value is computed assuming $H_0$ and is not a posterior probability of $H_0.(4)$ is the Karlin–Rubin theorem.

With m = 7, n = 9 and C(16,7) = 11440 arrangements: P(R = 2k) = 2·C(m−1,k−1)·C(n−1,k−1)/C(16,7) gives P(R=6) = 2·15·28/11440 = 21/286; P(R = 2k+1) = [C(m−1,k)C(n−1,k−1) + C(m−1,k−1)C(n−1,k)]/C(16,7) gives P(R=9) = (15·56 + 20·70)/11440 = 28/143.

Width ∝ $1/\sqrt{n}$, so halving it needs $\sqrt{n}$ to double, i.e. n to quadruple — the square-root law that makes extra precision expensive.

The three statements are the fundamental sampling theorem for normal data. The independence in (3) is in fact a characterisation of the normal (Geary–Lukacs): in no other distribution are the sample mean and variance independent. So (4) fails outright; for large n the t-statistic is only approximately normal via the CLT.