Hypothesis testing

Decide whether sample evidence is sufficiently unusual under a stated claim.

Work through twelve lessons on hypotheses, tail probabilities, critical regions and complete tests. Finish with a documented investigation and independent mixed practice.

Hypotheses and test statisticsDistinguish a population probability from a sample count, write hypotheses in context and state a binomial model under the null hypothesis.
Choosing one or two tailsTranslate a claim into a one-sided or two-sided alternative, choose the correct tail and reverse direction when the counted event is complemented.
Significance and tail evidenceUnderstand why a hypothesis test uses an observed-or-more-extreme tail, interpret a significance threshold and avoid confusing evidence with the probability a claim is true.
Upper critical regionsBuild an upper critical region for a binomial hypothesis test, prove its cutoff with adjacent tail probabilities and distinguish a critical value from the whole rejection region.
Lower critical regionsConstruct a lower binomial critical region, check the largest allowed cutoff and recognise when no supported count is sufficiently unusual.
Actual significance levelCalculate the null probability of a critical region, distinguish actual and nominal significance and handle independent repetitions without treating the level as a probability that a hypothesis is true.
Two-tailed critical regionsConstruct both binomial rejection tails, halve the nominal level, distinguish at-most and explicitly closest-tail instructions and calculate the actual total.
Complete one-tailed hypothesis testsCarry out a full one-sided binomial test from contextual hypotheses through a null-model tail probability to a justified conclusion.
Complete two-tailed hypothesis testsComplete a two-sided binomial test with explicit equal-tail boundaries, inclusive probabilities and a careful contextual conclusion.
Hypothesis-test conclusions and limitationsWrite justified contextual conclusions, distinguish statistical evidence from proof or causation and evaluate sampling and repeated-testing limitations.
A hypothesis-testing investigationPlan and document a binary-data investigation, audit a clearly synthetic example and evaluate sampling, seasonality and dependence before drawing a conclusion.
Hypothesis testing: mixed practiceIndependent mixed problems on hypotheses, exact binomial tails, critical regions, actual levels, repeated tests and justified contextual conclusions.