Further Statistics 1: Poisson distributionAQA A-Level Further Maths: Topic test

1. 1
   Meteors are seen at random, independently and at a constant average rate of $4$ per hour by a group of observers at a dark-sky site. The number of meteors seen in one hour is $X\sim\mathrm{Po}(4)$.

   (a)

   What is $P(X=3)$?

   [1 mark]

   • A$0.147$
   • B$0.195$
   • C$0.433$
   • D$0.0183$

   (b)

   What is $P(X\geq2)$?

   [1 mark]

   • A$0.0916$
   • B$0.762$
   • C$0.0733$
   • D$0.908$

   (c)

   Find the probability that exactly $6$ meteors are seen in a period of $90$ minutes.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student models the number of cars, $X$, passing a point on a quiet lane in one minute by a Poisson distribution with mean $2.4$.

   (a)

   What is the standard deviation of $X$?

   [1 mark]

   • A$1.55$
   • B$2.4$
   • C$5.76$
   • D$1.2$

   (b)

   Which of the following is a condition for a Poisson distribution to be a suitable model?

   [1 mark]

   • ACars pass at equal intervals of time
   • BThe number of cars in a minute has a fixed upper limit
   • CCars pass at random, independently of one another, at a constant average rate
   • DThe mean number of cars per minute is a whole number

   (c)

   State two reasons why this model might be unsuitable for the same lane at rush hour.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Defects occur at random, independently and at a constant average rate of $0.8$ per kilometre along a long optical fibre cable.

   (a)

   Find the probability that a $2.5$ km length of cable has at most $1$ defect.

   [3 marks]

   (b)

   Given that a $1$ km length has at least one defect, find the probability that it has at least two defects.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Kingfishers are seen at random, independently and at constant average rates at two stretches of a river: at a mean rate of $1.2$ per hour at the upper stretch and $0.8$ per hour at the lower stretch. Sightings at the two stretches are independent of each other.

   (a)

   Find the probability that the total number of kingfishers seen at the two stretches in a two-hour period is at least $5$.

   [6 marks]

   (b)

   Since a weir was built at the lower stretch, the observers think the rate at which kingfishers are seen there has fallen. In a four-hour period they see none at the lower stretch. Test this belief at the $5\%$ significance level.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   In a hockey league, the number of goals scored by the home team in a match is modelled by $X\sim\mathrm{Po}(1.5)$ and the number of goals scored by the away team by $Y\sim\mathrm{Po}(1.0)$, where $X$ and $Y$ are independent.

   (a)

   What is the distribution of $T=X+Y$, the total number of goals in a match?

   [1 mark]

   • A$\mathrm{Po}(1.25)$
   • B$\mathrm{Po}(1.5)$
   • C$\mathrm{Po}(2.5)$
   • D$\mathrm{Po}(0.5)$

   (b)

   What is $P(T\geq2)$?

   [1 mark]

   • A$0.713$
   • B$0.287$
   • C$0.257$
   • D$0.442$

   (c)

   Find the probability that neither team scores in a match.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A railway operator states that the number of late-running trains on a line each day follows a Poisson distribution with mean $3$. After a timetable change, a manager believes the mean has fallen. On the first day after the change only $1$ train runs late. The manager tests the belief at the $10\%$ significance level, using this single day's figure.

   (a)

   Which pair of hypotheses should the manager use?

   [1 mark]

   • A$\mathrm{H}_0:\lambda=3,\ \mathrm{H}_1:\lambda\neq3$
   • B$\mathrm{H}_0:\lambda=1,\ \mathrm{H}_1:\lambda<1$
   • C$\mathrm{H}_0:\lambda<3,\ \mathrm{H}_1:\lambda=3$
   • D$\mathrm{H}_0:\lambda=3,\ \mathrm{H}_1:\lambda<3$

   (b)

   What probability should the manager compare with the $10\%$ significance level?

   [1 mark]

   • A$0.149$
   • B$0.199$
   • C$0.950$
   • D$0.0498$

   (c)

   State the conclusion of the test, in context.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A website receives sign-ups at random, independently and at a constant average rate of $7$ per hour.

   (a)

   Find the probability that exactly $2$ sign-ups are received in a period of $30$ minutes.

   [3 marks]

   (b)

   Find the least whole number of minutes $n$ for which the probability of at least one sign-up in $n$ minutes exceeds $0.99$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   In an office, a printer jams at random, independently, at a constant mean rate of $0.5$ per day and a photocopier jams at random, independently of the printer, at a constant mean rate of $0.7$ per day. The office is open five days each week.

   (a)

   Find the probability that the total number of jams of the two machines in one week is at least $4$ and at most $8$.

   [6 marks]

   (b)

   After the printer is serviced, it jams $4$ times in the next $20$ days. Test, at the $5\%$ significance level, whether the printer's mean rate of jamming has decreased.

   [6 marks]

   Total for question 8: 12 marks

End of questions