AHL 4.18 (HL)—Advanced hypothesis testing

Choose the test from the parameter and information available: use a normal mean test when $\sigma$ is known, a t-test when $\sigma$ is unknown, a binomial test for a proportion, a Poisson test for a rate/mean, or a technology test of $H_0:\rho=0$ for bivariate normal data.

Samples may be paired or unpaired; matched pairs become one sample of differences. Normal, Poisson and binomial tests use the tail named by $H_1$; discrete critical regions maximize Type I error probability while keeping it below $\alpha$. t-test critical regions need not be calculated manually.

Error interpretation

A Type I error rejects a true $H_0$ and has probability controlled by the critical region. A Type II error fails to reject a false $H_0$; calculate its probability under the stated alternative parameter by finding the chance of landing outside the rejection region.

Do not select a test by distribution name alone: identify the population parameter, known or unknown variance, pairing, assumptions and direction. Poisson and binomial hypothesis tests are one-tailed in this syllabus.