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

DifficultyAllLevel 3Level 4Level 5
  1. 1
    A test of H0:p=0.3H_0: p = 0.3 against H1:p>0.3H_1: p > 0.3 uses X∼B(20,0.3)X \sim B(20, 0.3) and observes X=10X = 10. Using a 5%5\% significance level, carry out the test.
    Level 3 · 4 marksTry it
  2. 2
    At a market stall, customers are offered a taste of a cheese before they decide whether to buy any.

    Over a long period, Lena has found that 40%40\% of the customers who taste this cheese then buy some.

    Lena changes the way the cheese is displayed on her stall and claims that the proportion of customers who taste the cheese and then buy some has increased.

    A random sample of 3030 customers who taste the cheese after the change is taken and 1818 of them buy some cheese.

    Use a suitable test to assess Lena's claim.

    You should
    • state your hypotheses clearly
    • use a 5%5\% level of significance
    (4 marks)
    Level 3 · 4 marksTry it
  3. 3
    A treatment succeeds for 40%40\% of patients. In a trial of 2525 patients it succeeds for 66. Test, at the 5%5\% level, whether the success rate has fallen (X∼B(25,0.4)X \sim B(25, 0.4)).
    Level 4 · 5 marksTry it
  4. 4
    A claim states p=0.2p = 0.2. In a sample of 3030, the value X=11X = 11 is observed. Test H0:p=0.2H_0: p = 0.2 against H1:p>0.2H_1: p > 0.2 at the 5%5\% level.
    Level 4 · 4 marksTry it
  5. 5
    A fair coin is suspected of favouring tails. In 1515 tosses it shows 33 heads. Test H0:p=0.5H_0: p = 0.5 against H1:p<0.5H_1: p < 0.5 at the 5%5\% level, where p=P(head)p = P(\text{head}).
    Level 4 · 4 marksTry it

6 more questions on one-tailed tests (the probability method)

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