Hypothesis test for the mean of a Poisson distributionEdexcel A-Level Further Maths: Mind map

Exam questions on Hypothesis test for the mean of a Poisson distribution

1. A help desk historically receives calls at a mean rate of $3$ per hour, modelled by a Poisson distribution. After a publicity campaign, the manager records $7$ calls in a randomly chosen hour and wants to test, at the $5\%$ significance level, whether the mean rate of calls has increased.
   Write down the conclusion of the test, in context.2 marks
2. In a long-running textbook, typing errors occur at a mean rate of $1.2$ per page, modelled by a Poisson distribution. A new editor checks $5$ randomly chosen pages and finds $2$ errors in total. A test is carried out at the $5\%$ significance level to see whether the mean rate of errors has decreased.
   Find the critical region for this test, and state the actual significance level of the test.2 marks
3. A factory's machines have historically broken down at a mean rate of $4$ per week. During a randomly chosen two-week period there are $14$ breakdowns. The manager believes that the weekly rate has changed. Breakdowns are assumed to occur at random, independently and at a constant average rate, and the test is carried out at the $10\%$ significance level.
   State suitable hypotheses for the test, where $\lambda$ is the mean number of breakdowns per week, and the distribution of the number of breakdowns in the two-week period under $\mathrm{H}_0$.3 marks