Practise Binomial hypothesis tests. 2 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
A seed company states that 75% of its seeds germinate. A gardener believes that seeds that have been stored for a year are less likely to germinate. A test of \(H_0: p = 0.75, \ H_1: p \lt 0.75\) is carried out at the 1% level of significance, where \(p\) is the proportion of seeds that germinate. In a random sample of 22 seeds, 10 germinate.
Using \(X \sim B(22, 0.75)\), find \(P(X \le 10)\) and state, with a reason, whether \(H_0\) is rejected.[3]
A1ft for a correct decision from a comparison with 0.01
Question 2Medium8 marks
A game designer claims that a spinner lands on red with probability \(\frac{1}{4}\). A player believes that it lands on red less often than this. In 40 spins, the spinner lands on red 4 times. The player carries out a test at the 5% level of significance and writes: \(H_0: p = 0.1\), \(H_1: p \lt 0.1\) \(P(X = 4) = 0.0113\), which is less than 0.05, so reject \(H_0\).
(a) Identify two errors in the player's working.[2]
(b) Carry out the test correctly.[4]
(c) Find the critical region for this test.[2]
Show the answer and mark scheme
(a)Answer: The hypotheses should use the claimed value \(p = 0.25\), not the sample proportion 0.1; the player should find \(P(X \le 4)\), not \(P(X = 4)\).
B1 for the hypotheses should be about the claimed population proportion: \(H_0: p = 0.25\), \(H_1: p \lt 0.25\) (0.1 is the sample proportion)
B1 for the probability should be \(P(X \le 4)\) (the observed value or a more extreme one), not \(P(X = 4)\)
Worked solution: The hypotheses must be about the population parameter under the designer's claim, \(p = 0.25\); 0.1 is just the sample proportion. And the test needs the probability of a result at least as extreme as the one observed, \(P(X \le 4)\), not the probability of exactly 4.
(b)Answer: \(H_0: p = 0.25\), \(H_1: p \lt 0.25\); \(X \sim \mathrm{B}(40, 0.25)\), \(P(X \le 4) = 0.0160 \lt 0.05\): reject \(H_0\). There is evidence that the spinner lands on red less often than \(\frac{1}{4}\).
B1 for \(H_0: p = 0.25\), \(H_1: p \lt 0.25\)
M1 for \(X \sim \mathrm{B}(40, 0.25)\) and \(P(X \le 4)\)
A1 for 0.0160 (awrt 0.016)
A1 for reject \(H_0\), with a conclusion in context: evidence that the spinner lands on red less often than the designer claims
Worked solution: \(H_0: p = 0.25\), \(H_1: p \lt 0.25\). Under \(H_0\), \(X \sim \mathrm{B}(40, 0.25)\). \(P(X \le 4) = 0.0160 \lt 0.05\), so reject \(H_0\): there is evidence that the spinner lands on red less often than \(\frac{1}{4}\) of the time.
(c)Answer: \(X \le 5\)
M1 for \(P(X \le 5) = 0.0433\) and \(P(X \le 6) = 0.0962\)
A1 for \(X \le 5\)
Worked solution: \(P(X \le 5) = 0.0433 \le 0.05\) and \(P(X \le 6) = 0.0962 \gt 0.05\), so the critical region is \(X \le 5\).
Question 3Hard6 marks
At a restaurant, 25% of customers buy a dessert. The owner believes that a new dessert menu has increased the proportion of customers who buy a dessert. A random sample of 27 customers is taken, and \(X\) is the number who buy a dessert.
(a) Find the critical region for a test of \(H_0: p = 0.25, \ H_1: p \gt 0.25\) at the 10% level of significance, and state the actual significance level.[4]
(b) In the sample, 13 buy a dessert. State, with a reason, the conclusion of the test in context.[2]
Show the answer and mark scheme
(a)Answer: \(X \ge 11\); actual significance level 0.0528.
M1 for \(X \sim B(27, 0.25)\)
A1 for \(P(X \ge 11) = 0.0528\) and \(P(X \ge 10) = 0.1133\)
A1 for \(X \ge 11\)
A1 for 0.0528
(b)Answer: 13 is in the critical region, so reject \(H_0\). There is sufficient evidence to suggest that the proportion of customers who buy a dessert has increased.
M1 for a correct decision from their critical region