Exam-style questionsBinomial Hypothesis Testing questions with mark schemes
Easiest first. Try each question on paper, then open its mark scheme: M marks are for method, A marks for accurate answers that follow from the method, B marks for independent results.
Question 1easy5 marksYear 12
In a large town, 30% of adults own an electric bicycle. A cycle shop owner believes the proportion is higher in her village. In a random sample of 20 adults from the village, 10 own an electric bicycle.
(a) Test, at the 5% level of significance, whether there is evidence to support the owner's belief. State your hypotheses clearly. (5)
Mark scheme and answer
(a) \(H_0: p = 0.3\), \(H_1: p \gt 0.3\) B1
\(X\sim\mathrm{B}(20,\ 0.3)\) M1
\(P(X\ge10) = 1 - P(X\le9) = 0.04796\) A1
\(0.04796 \lt 0.05\), reject \(H_0\) dM1
There is evidence that the proportion of adults in the village owning an electric bicycle is greater than 0.3 A1
Answer: P(X ≥ 10) = 0.04796 < 0.05: reject H0
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 2easy4 marksYear 12
A bakery claims that 60% of its customers buy bread. After a price increase, the manager suspects this proportion has fallen. In a random sample of 15 customers, 6 buy bread.
(a) Test the manager's suspicion at the 10% level of significance. (4)
Mark scheme and answer
(a) \(H_0:p=0.6,\ H_1:p\lt 0.6\) B1
\(X\sim\mathrm{B}(15,0.6)\), \(P(X\le6) = 0.09505\) M1A1
\(0.09505 \lt 0.1\), reject H0: evidence the proportion buying bread has fallen A1
Answer: P(X ≤ 6) = 0.09505; reject H0
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 3easy4 marksYear 12
In a hypothesis test of \(H_0: p = 0.2\) against \(H_1: p \gt 0.2\), using \(X \sim \mathrm{B}(18,\ 0.2)\) at the 5% level, the critical region is \(X \ge 8\).
(a) Explain what is meant by a critical region. (1)
(b) Verify the critical region, showing the relevant probabilities. (2)
(c) Write down the acceptance region. (1)
Mark scheme and answer
(a) The set of values of the test statistic for which the null hypothesis would be rejected B1
(b) \(P(X\ge8) = 0.01628 \le 0.05\) M1
\(P(X\ge7) = 0.05127 \gt 0.05\) A1
(c) \(X \le 7\) B1
Answer: (a) values leading to rejection of H0; (b) P(X ≥ 8) = 0.01628; (c) X ≤ 7
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 4medium5 marksYear 12
A spinner is designed so that the probability of landing on red is 0.4. Jake thinks the spinner lands on red less often than this. He spins it 25 times and records the number of reds, \(X\).
(a) Using a 5% significance level, find the critical region for a test of Jake's belief. (3)
(b) State the actual significance level of the test. (1)
(c) Jake gets 6 reds. State his conclusion. (1)
Mark scheme and answer
(a) \(H_0:p=0.4,\ H_1:p\lt 0.4\); \(X\sim\mathrm{B}(25,0.4)\) M1
\(P(X\le5) = 0.02936\), \(P(X\le6) = 0.07357\) A1
Critical region \(X\le5\) A1
(b) 0.02936 (i.e. 2.94%) B1
(c) 6 is not in the critical region: insufficient evidence that the spinner lands on red less often B1ft
Answer: (a) X ≤ 5; (b) 0.02936; (c) do not reject H0
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 5medium6 marksYear 12
A biologist claims that 25% of the frogs in a lake carry a certain fungus. A random sample of 30 frogs is tested to see whether the proportion is different from 25%.
(a) Using a 5% level of significance, find the critical region for this test. The probability in each tail should be as close as possible to, but not exceeding, 0.025. (5)
(b) Find the actual significance level of the test. (1)
Mark scheme and answer
(a) \(H_0:p=0.25,\ H_1:p\ne0.25\); \(X\sim\mathrm{B}(30,0.25)\) B1M1
\(P(X\le2) = 0.01060\) (and \(P(X\le3) = 0.03745\)) A1
\(P(X\ge13) = 0.02159\) (and \(P(X\ge12) = 0.05066\)) A1
\(X\le2\) or \(X\ge13\) A1
(b) \(0.01060 + 0.02159 = 0.03219\) B1
Answer: (a) X ≤ 2 or X ≥ 13; (b) 0.03219
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 6medium5 marksYear 12
A coin is thought to be biased. It is tossed 40 times and lands heads 27 times.
(a) Test, at the 5% level of significance, whether the coin is biased. (5)
Mark scheme and answer
(a) \(H_0:p=0.5,\ H_1:p\ne0.5\) B1
\(X\sim\mathrm{B}(40,0.5)\) M1
\(P(X\ge27) = 0.01924\) A1
Compare with 0.025 (or \(2\times0.01924=0.0385\) with 0.05) M1
\(0.01924 \lt 0.025\), reject H0: evidence the coin is biased A1
Answer: P(X ≥ 27) = 0.01924 < 0.025: reject H0
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 7medium4 marksYear 12
Historically 15% of students at a college walk to college. The college introduces a "walk to learn" scheme. Afterwards, in a random sample of 12 students, 5 walk to college.
(a) Find the \(p\)-value for a test of whether the proportion who walk has increased. (2)
(b) Hence state whether there is evidence of an increase at the 5% level and at the 1% level. (2)
Mark scheme and answer
(a) \(X\sim\mathrm{B}(12,0.15)\) M1
\(P(X\ge5) = 0.02392\) A1
(b) 5%: \(0.0239 \lt 0.05\), evidence of increase B1
1%: 0.0239 > 0.01, insufficient evidence of increase B1
Answer: (a) 0.02392; (b) significant at 5%, not significant at 1%
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 8medium5 marksYear 12
A factory knows that 10% of its phone cases have a cosmetic flaw. After a change of supplier, the quality manager thinks the proportion has increased. She takes a random sample of 50 cases.
(a) Find the critical region for a test at the 1% level of significance. (4)
(b) Given that 10 cases in the sample are flawed, state the conclusion of the test. (1)
Mark scheme and answer
(a) \(H_0:p=0.1,\ H_1:p\gt 0.1\) B1
\(P(X\ge11) = 0.009355\), \(P(X\ge10) = 0.02454\) M1A1
\(X\ge11\) A1
(b) 10 is not in the critical region: insufficient evidence at the 1% level that the proportion of flawed cases has increased B1ft
Answer: (a) X ≥ 11; (b) do not reject H0
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 9medium7 marksYear 12
A mobile game states that 20% of its reward chests contain a rare item. A player thinks this value is wrong. She opens 25 chests, which may be regarded as a random sample, and finds a rare item in 1 of them.
(a) Test, at the 5% level of significance, whether the proportion of chests containing a rare item differs from 20%. (5)
(b) Find the critical region for this test, with the probability in each tail not exceeding 0.025, and confirm it is consistent with your conclusion. (2)
Mark scheme and answer
(a) \(H_0:p=0.2,\ H_1:p\ne0.2\) B1
\(X\sim\mathrm{B}(25,0.2)\), \(P(X\le1) = 0.02739\) M1A1
Compare with 0.025 M1
> 0.025: do not reject H0; insufficient evidence that the proportion differs from 20% A1
(b) \(P(X\le0) = 0.003778\), \(P(X\ge10) = 0.01733\) M1
\(X\le0\) or \(X\ge10\); 1 is not in the critical region, consistent with (a) A1
Answer: (a) P(X ≤ 1) = 0.02739 > 0.025: do not reject H0; (b) X ≤ 0 or X ≥ 10
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 10medium3 marksYear 12
Nadia tests whether a new advert has increased the proportion of shoppers who recognise a brand from 45%. In a random sample of 16 shoppers, 12 recognise the brand. She calculates \(P(X\ge12) = 0.01494\) and writes: "This is less than 5%, so the null hypothesis is false and the advert definitely works."
(a) State the hypotheses Nadia should have used. (1)
(b) Criticise her conclusion and write a correct conclusion. (2)
Mark scheme and answer
(a) \(H_0:p=0.45,\ H_1:p\gt 0.45\) B1
(b) A hypothesis test cannot prove \(H_0\) false / the advert works; there is only evidence at the 5% level B1
Correct: there is evidence that the proportion of shoppers who recognise the brand has increased B1
Answer: (a) H0: p = 0.45, H1: p > 0.45; (b) evidence, not proof
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 11hard6 marksYear 12
A supermarket finds that 35% of its customers use self-checkout. A new layout is trialled in one store, and the manager wants to test whether the proportion using self-checkout has changed. A random sample of 20 customers is observed.
(a) Find the critical region for a two-tailed test at the 10% level. The probability in each tail should be as close as possible to, but not exceeding, 0.05. (4)
(b) State the actual significance level. (1)
(c) In the sample, 12 customers use self-checkout. State the conclusion. (1)
Mark scheme and answer
(a) \(X\sim\mathrm{B}(20,0.35)\) M1
\(P(X\le3) = 0.04438\), \(P(X\le4) = 0.1182\) A1
\(P(X\ge12) = 0.01958\), \(P(X\ge11) = 0.05317\) A1
\(X\le3\) or \(X\ge12\) A1
(b) 0.06395 B1
(c) 12 is in the critical region: evidence the proportion has changed B1ft
Answer: (a) X ≤ 3 or X ≥ 12; (b) 0.06395; (c) evidence of change
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 12hard3 marksYear 12
A test of \(H_0: p = 0.3\) against \(H_1: p \lt 0.3\) is to be carried out at the 5% level of significance, using a random sample of size \(n\).
(a) Find the smallest value of \(n\) for which it is possible to reject \(H_0\). (3)
Mark scheme and answer
(a) Need \(P(X=0) = 0.7^n \le 0.05\) M1
\(n \ge \dfrac{\ln 0.05}{\ln 0.7} = 8.399\) dM1
\(n = 9\) A1
(Trial method accepted: \(0.7^8 = 0.0576 \gt 0.05\), \(0.7^9 = 0.0404 \lt 0.05\) M1 dM1 A1)
Answer: n = 9
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 13hard5 marksYear 12
Nationally, 12% of households have a heat pump. A councillor believes the proportion in her district is higher. She asks the first 30 people who attend an eco-fair in the district, and 8 have a heat pump.
(a) Assuming the sample is random, test at the 5% level whether the councillor's belief is supported. (4)
(b) Explain why the conclusion may not be valid. (1)
Mark scheme and answer
(a) \(H_0:p=0.12,\ H_1:p\gt 0.12\) B1
\(X\sim\mathrm{B}(30,0.12)\), \(P(X\ge8) = 0.02205\) M1A1
\(0.02205 \lt 0.05\): reject H0, evidence the proportion in the district is higher than 12% A1
(b) The sample is not random: people at an eco-fair are more likely to have a heat pump (biased opportunity sample) B1
Answer: (a) P(X ≥ 8) = 0.02205; reject H0; (b) biased sample
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 14hard5 marksYear 12
Maya wants to test whether a coin is fair using 10 tosses and a 5% significance level (two-tailed, probability in each tail not exceeding 0.025).
(a) Find the critical region. (3)
(b) Find the probability of incorrectly rejecting \(H_0\). (1)
(c) Explain why Maya might prefer to toss the coin more times. (1)
Mark scheme and answer
(a) \(X\sim\mathrm{B}(10,0.5)\) M1
\(P(X\le1) = 0.01074\), \(P(X\le2) = 0.05469\) A1
\(X\le1\) or \(X\ge9\) A1
(b) \(2\times0.01074 = 0.02148\) B1
(c) With only 10 tosses the actual significance level is much smaller than 5% and only extreme results are significant; a larger sample makes it easier to detect bias B1
Answer: (a) X ≤ 1 or X ≥ 9; (b) 0.02148; (c) more tosses: more sensitive test
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Question 15challenging8 marksYear 12
A garden centre sells packets of mixed tulip bulbs. The supplier states that 15% of bulbs produce red flowers. Each packet contains 20 bulbs.
(a) State two assumptions needed to model the number of red-flowering bulbs in a packet by \(\mathrm{B}(20,\ 0.15)\). (2)
(b) Find the probability that a packet contains exactly 3 red-flowering bulbs. (2)
(c) Oliver buys a packet and 7 bulbs produce red flowers. He claims the percentage of red bulbs is higher than stated. Test his claim at the 5% level. (4)
Mark scheme and answer
(a) Bulbs are independent of each other B1
Probability of red is constant (0.15) for each bulb B1
(b) \(\binom{20}{3}(0.15)^3(0.85)^{17}\) M1
\(= 0.2428\) A1
(c) \(H_0:p=0.15,\ H_1:p\gt 0.15\) B1
\(P(X\ge7) = 0.02194\) M1A1
\(0.02194 \lt 0.05\): reject H0; evidence that more than 15% of bulbs are red A1
Answer: (a) independence, constant probability; (b) 0.2428; (c) P(X ≥ 7) = 0.02194; reject H0
Board notes: Edexcel: critical regions are chosen so that the probability of rejection is as close as possible to, without exceeding, the significance level (in each tail for two-tailed tests, unless the question says otherwise). AQA and OCR use the same convention; OCR sometimes asks for the p-value comparison instead of a critical region. All boards expect a contextual, non-assertive conclusion.
Original questions in the style of Edexcel 9MA0, independently checked before publishing. Not past-paper questions. Covers: One-tailed test (upper), One-tailed test (lower), Language of hypothesis testing, Critical region (one-tailed), Critical region (two-tailed), Two-tailed test, p-values, Sample size, Test in context, Actual significance level, Binomial model and test.
A Level Math Revision