Sums of Poisson variables and Poisson hypothesis testsAQA A-Level Further Maths: Subtopic test

1. 1
   Emails arrive at two servers independently of one another. The number of emails arriving at server A in a minute is $X\sim\mathrm{Po}(3)$ and the number arriving at server B in a minute is $Y\sim\mathrm{Po}(2)$.

   (a)

   Which of the following is the distribution of the total number of emails, $X+Y$, arriving at the two servers in a minute?

   [1 mark]

   • A$\mathrm{Po}(6)$
   • B$\mathrm{Po}(1)$
   • C$\mathrm{Po}(5)$
   • D$\mathrm{Po}(2.5)$

   (b)

   Find the probability that exactly 4 emails arrive at the two servers together in a minute.

   [1 mark]

   • A$0.1680$
   • B$0.0902$
   • C$0.7350$
   • D$0.1755$

   (c)

   Find the probability that at least 2 emails arrive at the two servers together in a minute.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of complaints received by a café in a week has historically followed $\mathrm{Po}(4)$. After staff training, the manager claims that the mean number of complaints per week, $\lambda$, has fallen. In the first week after the training the café receives 1 complaint. A hypothesis test is carried out at the 5% significance level.

   (a)

   Which of the following are the correct hypotheses for the test?

   [1 mark]

   • A$\mathrm{H}_0:\lambda=4,\ \mathrm{H}_1:\lambda<4$
   • B$\mathrm{H}_0:\lambda=4,\ \mathrm{H}_1:\lambda\ne4$
   • C$\mathrm{H}_0:\lambda=1,\ \mathrm{H}_1:\lambda<1$
   • D$\mathrm{H}_0:\lambda<4,\ \mathrm{H}_1:\lambda=4$

   (b)

   Assuming $\mathrm{H}_0$ is true, find the probability of 1 or fewer complaints in a week.

   [1 mark]

   • A$0.0183$
   • B$0.0916$
   • C$0.0733$
   • D$0.9084$

   (c)

   State the conclusion of the test, in context, using your answer to (b).

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In a textile factory, flaws in fabric occur at random. Machine A produces flaws at a mean rate of 2.1 per metre and machine B at a mean rate of 1.4 per metre. The numbers of flaws per metre are modelled by $X\sim\mathrm{Po}(2.1)$ for machine A and $Y\sim\mathrm{Po}(1.4)$ for machine B, with $X$ and $Y$ independent.

   (a)

   One metre of fabric is taken from each machine. Find the probability that the two pieces together have exactly 2 flaws.

   [3 marks]

   (b)

   Machine B is serviced. The supervisor believes the mean number of flaws per metre from machine B has increased. A randomly chosen metre of fabric from machine B has 5 flaws. Test the supervisor's belief at the 5% significance level.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The numbers of ambulance call-outs in a week to village A and to the neighbouring village B are modelled by $X\sim\mathrm{Po}(2.5)$ and $Y\sim\mathrm{Po}(1.5)$ respectively. The two numbers are independent, and call-outs in different weeks are independent.

   (a)

   Historically the mean for village A has been as modelled. After a new housing estate was built in village A, there were 7 call-outs to village A in one week. Carry out a test, at the 5% significance level, to see whether there is evidence that the mean number of call-outs per week to village A has changed.

   [6 marks]

   (b)

   (i) Find the probability that the total number of call-outs to the two villages in one week is at least 6.
   (ii) Find the probability that the total number of call-outs to the two villages over four weeks is at most 12.

   [6 marks]

   Total for question 4: 12 marks

End of questions