A-level · Hypothesis testing

Setting up a hypothesis testExam-style questions

The language and structure of a binomial hypothesis test: the population proportion p, the null hypothesis H0 : p = p0 and the alternative hypothesis H1, deciding between a one-tailed (p < p0 or p > p0) and a two-tailed (p ≠ p0) test from the wording, the test statistic X ~ B(n, p0) under H0, and what the significance level means.

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9 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Under an old production process, breakdowns occurred at random at a mean rate of $4$ per week. After a new process is introduced, the manager wishes to test whether the breakdown rate has decreased. In the first week under the new process there is $1$ breakdown.

The number of breakdowns per week is modelled by a Poisson distribution.
(a) State suitable null and alternative hypotheses. (2 marks)
(b) Test the manager's claim at the $5\%$ significance level. (3 marks)
(c) Find the critical region for this test, and state the actual significance level of a test based on that critical region. (3 marks)
(d) State, with a reason, what the conclusion of the test would have been if there had been no breakdowns in the week. (2 marks)
Level 5 · 10 marksTry it
2
A weather forecaster finds that her forecasts for the following day are correct on $70\%$ of days. She starts to use a new computer model, hoping that it will increase the proportion of days on which her forecast is correct. She will note the number of correct forecasts in a random sample of $20$ days, and use it in a test at the $5\%$ significance level.
(a) Explain why the test is one-tailed, and state suitable null and alternative hypotheses.

(2 marks)
(b) Use a binomial distribution to find the critical region for the test, and calculate the probability of a Type I error.

(5 marks)

Her forecast turns out to be correct on $18$ of the $20$ days.
(c) State the conclusion of the test, explaining your answer.

(2 marks)
Level 5 · 9 marksTry it