Solution. (e) Null hypothesis. Take this first, since the other four are defined against it. \(H_0\) is the statement being tested — conventionally the statement of no effect, no difference, no association. It is given the benefit of the doubt: the test asks whether the data are embarrassing for \(H_0\), and only a sufficiently embarrassing sample leads to its rejection. It is never proved; it is either rejected or not rejected.

(a) Critical region. The set of values of the test statistic for which \(H_0\) is rejected. It is fixed before the data are seen, by choosing a significance level \(\alpha \) and then finding the values of the statistic whose total probability under \(H_0\) is at most \(\alpha \). Its complement is the acceptance region. For the sign test with \(n=10\) at \(\alpha =0.05\) two-sided, for instance, the critical region is \(N_+\leq 1\) or \(N_+\geq 9\).

(b) Type I error. Rejecting \(H_0\) when it is in fact true — a false positive. Its probability is \[P(\text {Type I error})=P\left (\text {statistic falls in the critical region} \mid H_0 \text { true}\right )=\alpha .\] This is the one error rate the experimenter controls directly, by choosing \(\alpha \) and constructing the critical region to match.

(c) Type II error. Failing to reject \(H_0\) when it is false — a false negative. Its probability is written \(\beta \). Unlike \(\alpha \) it is not a single number: \(\beta \) depends on how false \(H_0\) is, so there is a different \(\beta \) for every value the parameter might really take. A small departure from \(H_0\) is hard to detect and gives a large \(\beta \); a gross one gives a small \(\beta \).

(d) Power. The probability of correctly rejecting a false \(H_0\), \[\text {power}=1-\beta =P\left (\text {reject } H_0 \mid H_0 \text { false}\right ),\] again a function of the true parameter value rather than one number. Power rises with the sample size, with the size of the real effect, and with \(\alpha \).

Solution. (a). A parametric test assumes the population belongs to a family of distributions indexed by a finite number of parameters — most often the normal family \(N\left (\mu ,\sigma ^{2}\right )\) — and the hypothesis is a statement about those parameters, such as \(H_0:\mu =\mu _0\). The null distribution of the test statistic is derived from that assumed form, so if the assumption fails the stated error rate is not the real one.

A non-parametric test assumes no such family. It typically requires only that the distribution be continuous, and the hypothesis concerns the distribution itself or a feature of it that needs no parametric form, such as its median. The statistic is built from signs, ranks or order statistics, whose null distribution is the same whatever the underlying continuous distribution happens to be — which is why these tests are also called distribution-free.

(b). Two reasons.

Power. When the parametric assumption is actually true, the parametric test is more efficient: it uses the numerical values, while a rank test uses only their order. The \(t\)-test on normal data beats the sign test comfortably, so with genuinely normal data the non-parametric test wastes information and needs a larger sample for the same power.

Scope. Parametric theory is far more fully developed. Estimation, confidence intervals, regression, factorial designs, covariates and mixed models all have standard parametric treatments; the non-parametric versions are fewer, more awkward, and in some cases do not exist. A researcher with a complicated design often has no distribution-free option at all.