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Hypothesis testing with the binomial questions

A claim about a proportion meets some data. The test assesses whether the observed result would be sufficiently unlikely if the null hypothesis were true. Assume the claim, compute how unlikely the observed data would be under it, and compare that with a threshold agreed in advance.

7 original questions · 29 marks · the hypothesis testing with the binomial notes · Statistics

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• 3 marks · define · warm-up

  Define the null hypothesis, the alternative hypothesis and the significance level of a hypothesis test.

  Worked answer

  The null hypothesis H₀ states the assumed value of the population parameter, the value taken as true while the test runs. The alternative H₁ states the change being tested for. The significance level is the probability threshold, so evidence rarer than that threshold under H₀ counts as grounds for rejection. B1 B1 B1, one for each of the three definitions. Both hypotheses must name a parameter, never an observed count.

• 3 marks · warm-up

  A die is suspected of being biased towards sixes. It is rolled 30 times. Write down suitable hypotheses in terms of p, and state the distribution of the number of sixes X under H₀.

  Worked answer

  H₀: p = 1/6 and H₁: p > 1/6, where p is the probability that a single roll gives a six. Under H₀, X ~ B(30, 1/6). B1 B1 for the two hypotheses and B1 for the distribution. 'Biased towards sixes' points one way only, so the alternative is one-tailed with a strict inequality.

• 5 marks

  A spinner is claimed to land on red with probability 0.3. In 25 independent spins it lands on red 12 times. Test, at the 5% significance level, whether the spinner favours red more often than claimed. State your hypotheses clearly.

  Worked answer

  H₀: p = 0.3 and H₁: p > 0.3, where p is the probability of red on one spin. Under H₀, X ~ B(25, 0.3). The probability of a result at least as extreme as the one observed is P(X ≥ 12) = 1 − P(X ≤ 11) = 0.0442. Since 0.0442 < 0.05, reject H₀. There is significant evidence at the 5% level that the spinner lands on red more often than claimed. B1 hypotheses, M1 the binomial model, M1 the correct tail, A1 0.0442, A1 comparison and conclusion in context. Working out P(X ≤ 12) on the calculator instead of 1 − P(X ≤ 11) gives 0.9827 and a conclusion pointing the wrong way.

• 4 marks

  Under H₀, X ~ B(20, 0.4), and the alternative hypothesis is H₁: p > 0.4. Find the critical region for a test at the 5% significance level, and state the actual significance level of the test.

  Worked answer

  Work down from the top of the distribution. P(X ≥ 12) = 0.0565, which is bigger than 0.05, so 12 cannot be in the region. P(X ≥ 13) = 0.0210, which is smaller, so 13 can. The critical region is X ≥ 13, and the actual significance level is 0.0210, or 2.10%. M1 for attempting a tail, A1 for 0.0565, A1 for the region, B1 for the actual level. Because X is discrete the advertised 5% is almost never achievable exactly, and the actual level is the true tail probability of whatever region you end up with.

• 3 marks

  For a test of H₀: p = 0.4 against H₁: p > 0.4 based on X ~ B(20, 0.4), the critical region is X ≥ 13. State the conclusion when a value of 12 is observed, and explain why this conclusion is not the same as proving that p = 0.4.

  Worked answer

  12 lies outside the critical region, so do not reject H₀. There is insufficient evidence at the 5% level that p has risen above 0.4. That is a failure to disprove, not a proof. The data are consistent with p = 0.4 and equally consistent with p = 0.45 or p = 0.5, none of which the test has ruled out. M1 for comparing with the region, A1 for the conclusion in context, B1 for the explanation. A hypothesis test never proves the null, and writing 'so p = 0.4' throws away the last mark.

• 6 marks · stretch

  A researcher believes the proportion p of a population holding a certain view differs from 0.25. A random sample of 30 people is taken and X, the number holding the view, is recorded. Find the critical region for a two-tailed test of H₀: p = 0.25 at the 5% significance level, and find the actual significance level of the test.

  Worked answer

  Under H₀, X ~ B(30, 0.25), and a two-tailed 5% test puts at most 2.5% in each tail. Lower end: P(X ≤ 2) = 0.0106, which is under 0.025, while P(X ≤ 3) = 0.0374, which is over, so the lower region is X ≤ 2. Upper end: P(X ≥ 13) = 0.0216, under 0.025, while P(X ≥ 12) = 0.0507, over, so the upper region is X ≥ 13. The critical region is X ≤ 2 or X ≥ 13, and the actual significance level is 0.0106 + 0.0216 = 0.0322, or 3.22%. B1 for splitting the 5% into two 2.5% halves, M1 A1 for the lower tail, M1 A1 for the upper tail, A1 for the actual level. Two errors dominate here. Comparing each tail with 0.05 rather than 0.025 makes the region far too big, and quoting 5% as the actual level ignores the whole point of the last part.

• 5 marks · stretch

  A machine is claimed to produce faulty items with probability 0.3. An engineer believes the proportion has fallen. A random sample of 30 items is inspected. Find the critical region for a test of H₀: p = 0.3 against H₁: p < 0.3 at the 1% significance level, and write down the probability of a Type I error for this test.

  Worked answer

  Under H₀, X ~ B(30, 0.3), and the alternative points downwards so the region sits in the lower tail. P(X ≤ 3) = 0.00932, which is under 0.01, while P(X ≤ 4) = 0.0302, which is over. The critical region is therefore X ≤ 3. A Type I error is rejecting H₀ when it is true, which happens exactly when X lands in the critical region while p really is 0.3, so its probability is 0.00932. M1 for working in the lower tail, A1 for 0.00932, A1 for 0.0302, A1 for the region, B1 for the Type I probability. The probability of a Type I error and the actual significance level are the same number, and questions ask for it under either name.

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