AS June 2024 Q2
2. A manager keeps a record of accidents in a canteen.
Accidents occur randomly with an average of 2.7 per month. The manager decides to model the number of accidents with a Poisson distribution.
One day, two members of staff bump into each other in the canteen and each report the accident to the manager. The canteen manager is unsure whether to record this as one or two accidents.
Given that the manager still wants to model the number of accidents per month with a Poisson distribution,
- a property of the Poisson distribution that the manager should consider when deciding how to record this situation
- whether the manager should record this as one or two accidents
The manager introduces some new procedures to try and reduce the average number of accidents per month.
During the following 12 months the total number of accidents is 22
The manager claims that the accident rate has been reduced.
You should state your hypotheses clearly and the p-value used in your test. (4)
| Scheme | Marks | AO |
|---|---|---|
| Since accidents occur randomly/independently / at a constant/average rate | B1 | 2.4 |
| (1) |
Notes
B1 for a suitable reason picking up the underlined words from the context.
| Scheme | Marks | AO |
|---|---|---|
| (i) [\(A\) = no. of accidents in a month. \(A \sim \mathrm{Po}(2.7)\)] [\(\mathrm{P}(A \geqslant 3) = 1 - \mathrm{P}(A \leqslant 2) = 1 - 0.49362\ldots = 0.50637\ldots =\)] awrt 0.506 | B1 | 1.1b |
| (1) | ||
| (ii) [\(T\) = no. of accidents in a 3-month period.] \(T \sim \mathrm{Po}(3 \times 2.7 = [8.1])\) | M1 | 3.3 |
| [\(\mathrm{P}(T \leqslant 10)\)] \(= 0.805837\ldots\) = awrt 0.806 | A1 | 1.1b |
| (2) | ||
| (iii) [\(M\) = no. of months with no accidents.] \(M \sim \mathrm{B}(8, \mathrm{e}^{-2.7})\) | M1 | 3.3 |
| \(M \sim \mathrm{B}(8, 0.067(2)\ldots)\) | A1 | 1.1b |
| \(\mathrm{P}(M \geqslant 2) = 1 - \mathrm{P}(M \leqslant 1)\) | M1 | 3.4 |
| \(= 1 - 0.903542\ldots = 0.096457\ldots\) = awrt 0.0965 | A1 | 1.1b |
| (4) |
Notes
(i) B1 for awrt 0.506
(ii) M1 for selecting the \(\mathrm{Po}(8.1)\) model (sight of or implied by a correct answer)
A1 for awrt 0.806
(iii) 1st M1 for selecting a suitable binomial model e.g. \(\mathrm{B}(8, p)\) or \(\mathrm{B}(n, 0.067\ldots)\)
1st A1 for the correct model (\(p = \mathrm{e}^{-2.7}\) or 0.067 or better) seen or implied by a correct answer.
2nd M1 for using their binomial model to attempt \(\mathrm{P}(M \geqslant 2)\) or \(1 - \mathrm{P}(M \leqslant 1)\)
awrt 0.0964 is evidence for this M1 mark
2nd A1 for awrt 0.0965
| Scheme | Marks | AO |
|---|---|---|
| For a Poisson model accidents (events) must occur singly/independently so manager should record as one accident | B1 | 3.5b/2.4 |
| (1) |
Notes
B1 for stating the accidents (events) “occur singly” oe or accidents (events) are “independent”
AND should record as one accident
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0: \lambda = 2.7\) (or \(\mu = 32.4\)) \(\qquad \mathrm{H}_1: \lambda \lt 2.7\) (or \(\mu \lt 32.4\)) | B1 | 2.5 |
| [\(Y\) = no. of accidents in a year.] \(Y \sim \mathrm{Po}(32.4)\) | M1 | 3.3 |
| \(\mathrm{P}(Y \leqslant 22) = 0.03512\ldots\) | A1 | 3.4 |
| [Significant result so reject \(\mathrm{H}_0\)] there is evidence to support the manager’s claim / there is evidence that the number of accidents per month has decreased | A1 | 2.2b |
| (4) | ||
| (13 marks) |
Notes
B1 for both correct hypotheses in terms of \(\lambda\) or \(\mu\) (accept \(\mu = 2.7\) etc)
M1 for selecting the correct model (sight of or implied by the correct probability)
\(\mathrm{P}(Y = 22) =\) awrt 0.0129 is evidence for M1
1st A1 for awrt 0.035 (accept 0.04 if \(\mathrm{P}(Y \leqslant 22)\) and \(\mathrm{Po}(32.4)\) are explicitly seen)
2nd A1 dep on M1A1 indep of hyp’s for a correct conclusion in context
number of accidents reduced is A0 must be rate / per month / average number