A-level · Hypothesis testing

Critical regions and actual significance levelExam-style questions

The critical-region method for a one-tailed test: the critical value is the boundary and the critical region is the set of outcomes that would lead to rejecting H0. How to find it (the smallest c with P(X ≥ c) ≤ α for an upper test, the largest c with P(X ≤ c) ≤ α for a lower test), why the discreteness of the binomial means the actual significance level is usually below the nominal α, and how to use the region to conclude.

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12 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 survey carried out in a town in 2020 found that $30\%$ of adults regularly cycled to work. Since then a network of cycle lanes has opened in the town. A council officer believes that the proportion of adults in the town who regularly cycle to work has changed. To investigate, she asks $25$ adults leaving the town's leisure centre one Saturday morning whether they regularly cycle to work, and uses $X$ , the number who say yes, to carry out a hypothesis test.
(a) Explain why the officer should use a two-tailed test. (1 mark)
(b) Stating your hypotheses clearly, determine the critical region for the test at the $5\%$ significance level, showing the probabilities you use. Find the actual significance level of the test. (5 marks)
(c) In fact $12$ of the $25$ adults say that they regularly cycle to work. State, with a reason, the conclusion of the test in context. (2 marks)
(d) Explain why the way the officer chose her sample makes the conclusion of the test unreliable. (1 mark)
Level 4 · 9 marksTry it
2
Over a long period, $25\%$ of the visitors to a museum have bought something from the museum's gift shop.

The museum then moves its gift shop to the entrance hall.

Priya wants to find out whether the proportion of visitors who buy something from the gift shop is now different.

She plans to take a random sample of $30$ visitors to the museum and to carry out a hypothesis test at the $10\%$ significance level.
(a) State suitable null and alternative hypotheses for Priya's test.

(1 mark)
(b) Using a $10\%$ level of significance, find the critical region for Priya's test.

(5 marks)
(c) Of the $30$ visitors in Priya's sample, $12$ buy something from the gift shop.

State Priya's conclusion in context.

(1 mark)
Level 4 · 7 marksTry it
3
A company makes electric kettles. In the past, 4% of the kettles that it made were faulty. The company has recently changed the supplier of the heating elements used in its kettles, and the production manager wishes to test whether the proportion of kettles that are faulty has increased. She will check a random sample of 300 kettles made since the change, and carry out a hypothesis test at the 5% significance level, using the null hypothesis $p = 0.04$ , where $p$ is the proportion of all kettles made since the change that are faulty.
(a) Assume that the null hypothesis is true. Determine the probability that the manager's test will lead her to conclude that the proportion of faulty kettles has increased.

(5 marks)
(b) In the manager's sample, 17 of the kettles are faulty. Her assistant says

“If $p$ were still $0.04$ , we would expect only 12 faulty kettles in the sample. We found 17, so the sample shows that the proportion of faulty kettles has increased.”

Criticise the assistant's statement.

(1 mark)
Level 4 · 6 marksTry it
4
The random-number function on a calculator generates a single digit from 0 to 9. Each time the function is used, each of the ten digits should have the same probability of being generated, independently of the digits generated before. Dev suspects that his calculator generates the digit 0 less often than it should.

To test his suspicion, Dev will use the function 150 times and record the number of times, $X$ , that the digit 0 is generated. He will carry out the test at the 5% significance level.
(a) State suitable null and alternative hypotheses for Dev's test.

(2 marks)
(b) Determine the largest value of $X$ for which Dev would conclude that his calculator generates the digit 0 less often than it should.

(3 marks)

A second calculator of the same model is tested in exactly the same way, using the function 150 times and the 5% significance level.
(c) Given that the second calculator generates the digit 0 with probability 0.1, find the probability that the test leads to the conclusion that it generates the digit 0 less often than it should.

(1 mark)
Level 4 · 6 marksTry it
5
Three years ago, 30% of the adults in a large town said that they listened to the town's local radio station at least once a week. The station's new manager wishes to test whether this proportion has decreased.

Let $p$ denote the proportion of adults in the town who now listen to the station at least once a week. The manager's hypotheses are $\mathrm{H}_0: p = 0.3$ and $\mathrm{H}_1: p < 0.3$ , and he will carry out the test at the 1% significance level.

He will ask a random sample of 360 adults in the town, and $X$ will denote the number of them who say that they listen to the station at least once a week.

A normal random variable, $Y$ , is used to approximate the distribution of $X$ when $\mathrm{H}_0$ is true.
(a) Calculate $y$ , where $\mathrm{P}(Y < y) = 0.01$ .

(2 marks)
(b) Hence determine the largest value of $X$ for which the manager would reject $\mathrm{H}_0$ .

(3 marks)
Level 4 · 5 marksTry it