A-level · Hypothesis testing

One-tailed tests (the probability method)Exam-style questions

Carrying out a one-tailed binomial hypothesis test by the probability (p-value) method: assume H0, model X ~ B(n, p0), find the probability of a result at least as extreme as the one observed in the direction of H1 — P(X ≥ x) for an upper test or P(X ≤ x) for a lower test — compare it with the significance level, and write a full conclusion in context.

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11 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 treatment succeeds for $40\%$ of patients. In a trial of $25$ patients it succeeds for $6$ . Test, at the $5\%$ level, whether the success rate has fallen ( $X \sim B(25, 0.4)$ ).
Level 4 · 5 marksTry it
2
A claim states $p = 0.2$ . In a sample of $30$ , the value $X = 11$ is observed. Test $H_0: p = 0.2$ against $H_1: p > 0.2$ at the $5\%$ level.
Level 4 · 4 marksTry it
3
A fair coin is suspected of favouring tails. In $15$ tosses it shows $3$ heads. Test $H_0: p = 0.5$ against $H_1: p < 0.5$ at the $5\%$ level, where $p = P(\text{head})$ .
Level 4 · 4 marksTry it
4
A test of $H_0: p = 0.45$ against $H_1: p < 0.45$ uses $X \sim B(20, 0.45)$ and observes $X = 4$ . Test at the $5\%$ level.
Level 4 · 4 marksTry it
5
Seeds of a certain plant germinate with probability $0.3$ . A gardener plants $20$ seeds in a new compost, and claims the compost increases the germination rate. Let $X$ be the number of seeds that germinate, and assume $X \sim B(20, p)$ .
(a) State two conditions needed for a binomial model to be suitable here. (2 marks)
(b) Assuming $p = 0.3$ , find (i) $P(X = 4)$ and (ii) $P(X \ge 7)$ . (4 marks)
(c) Using a $5\%$ level of significance, find the critical region for a test of the gardener's claim, and state the actual significance level of the test. (5 marks)
(d) In fact $11$ of the $20$ seeds germinate. State the conclusion of the test in context. (2 marks)
Level 4 · 13 marksTry it

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