A-level · Hypothesis testing

Critical regions and actual significance levelExam-style questions

The critical-region method for a one-tailed test: the critical value is the boundary and the critical region is the set of outcomes that would lead to rejecting H0. How to find it (the smallest c with P(X ≥ c) ≤ α for an upper test, the largest c with P(X ≤ c) ≤ α for a lower test), why the discreteness of the binomial means the actual significance level is usually below the nominal α, and how to use the region to conclude.

Start practising Read the lesson

12 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
A council claims that $60\%$ of the households in a town put out a recycling bin each week. A campaigner believes that the true proportion is lower than this.

The campaigner observes a random sample of $20$ households in the town and records the number, $X$ , that put out a recycling bin in a particular week.
(a) Using a $5\%$ level of significance, find the critical region for a test of the campaigner's belief. State your hypotheses clearly. (3 marks)
(b) State the actual significance level of the test, and explain why it is not exactly $5\%$ . (2 marks)
(c) In the sample, $8$ households put out a recycling bin. State the conclusion of the test in context. (2 marks)

The campaigner then observes a second random sample of $150$ households in the town, of which $77$ put out a recycling bin.
(d) Assuming that the council's claim is correct, use a suitable approximation to estimate the probability that $77$ or fewer households in a random sample of $150$ put out a recycling bin. Justify the use of your approximation, and comment on what your answer suggests about the council's claim. (5 marks)
Level 5 · 12 marksTry it