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

1. 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.

   (a)

   Which of the following gives the correct hypotheses, where $\lambda$ is the mean number of calls per hour?

   [1 mark]

   • A$\mathrm{H}_0:\lambda=3,\ \mathrm{H}_1:\lambda\neq3$
   • B$\mathrm{H}_0:\lambda=3,\ \mathrm{H}_1:\lambda<3$
   • C$\mathrm{H}_0:\lambda=3,\ \mathrm{H}_1:\lambda>3$
   • D$\mathrm{H}_0:X=3,\ \mathrm{H}_1:X>3$

   (b)

   Under $\mathrm{H}_0$, find the probability of observing $7$ or more calls in the hour.

   [1 mark]

   • A$0.0119$
   • B$0.0335$
   • C$0.9665$
   • D$0.0839$

   (c)

   Write down the conclusion of the test, in context.

   [2 marks]

   Total for question 1: 4 marks

2. 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.

   (a)

   Let $X$ be the total number of errors on the $5$ pages. What is the distribution of $X$ under $\mathrm{H}_0$?

   [1 mark]

   • A$\mathrm{Po}(1.2)$
   • B$\mathrm{Po}(2)$
   • C$\mathrm{Po}(5)$
   • D$\mathrm{Po}(6)$

   (b)

   Find the probability of observing $2$ or fewer errors under $\mathrm{H}_0$.

   [1 mark]

   • A$0.0620$
   • B$0.0174$
   • C$0.1512$
   • D$0.9380$

   (c)

   Find the critical region for this test, and state the actual significance level of the test.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   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]

   (b)

   Carry out the test.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A company's mail server receives emails at a mean rate of $2.5$ per minute, modelled by a Poisson distribution. A technician suspects that the rate has changed and decides to count the number of emails $X$ received in a $4$-minute period, using a two-tailed test at the $10\%$ significance level with $5\%$ in each tail.

   (a)

   (i) State the hypotheses, in terms of the mean rate $\lambda$ per minute, and the distribution of $X$ under $\mathrm{H}_0$.
   (ii) Find the critical region for the test, and state the actual significance level.

   [6 marks]

   (b)

   The technician counts $15$ emails in a single $4$-minute period.
   (i) Complete the test.
   (ii) Explain why the Poisson model might not be suitable for the arrival of emails, and how this could affect the conclusion.

   [6 marks]

   Total for question 4: 12 marks

End of questions