| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2024 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Explain or apply conditions in context |
| Difficulty | Moderate -0.8 Part (a) is a standard 'state conditions' question requiring recall of Poisson assumptions (random, independent events). This is routine bookwork for FS1 with no problem-solving required, similar to stating when to use a binomial distribution. While it's a Further Maths topic, the question itself demands only basic recall of standard conditions. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Accidents occur randomly/independently at a constant/average rate | B1 (1) | For a suitable reason picking up the underlined words from the context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([A = \text{no. of accidents in a month},\ A \sim Po(2.7)]\) \(P(A \geqslant 3) = 1 - P(A \leqslant 2) = 1 - 0.49362 = 0.50637\) | B1 (1) | For awrt 0.506 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([T = \text{no. of accidents in 3-month period},\ T \sim Po(3 \times 2.7) = Po(8.1)]\) | M1 | For selecting the \(Po(8.1)\) model (sight of or implied by a correct answer) |
| \(P(T \leqslant 10) = 0.805837\ldots = \text{awrt}\ \mathbf{0.806}\) | A1 (2) | For awrt 0.806 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([M = \text{no. of months with no accidents},\ M \sim B(8,\ e^{-2.7})]\) | M1 | For selecting a suitable binomial model e.g. \(B(8,p)\) or \(B(n, 0.067\ldots)\) |
| \(M \sim B(8,\ 0.067(2)\ldots)\) | A1 | For the correct model (\(p = e^{-2.7}\) or 0.067 or better) seen or implied by a correct answer |
| \(P(M \geqslant 2) = 1 - P(M \leqslant 1)\) | M1 | For using their binomial model to attempt \(P(M \geqslant 2)\) or \(1 - P(M \leqslant 1)\); awrt 0.0964 is evidence for this M1 |
| \(= 1 - 0.903542\ldots = 0.096457\ldots = \text{awrt}\ \mathbf{0.0965}\) | A1 (4) | For awrt 0.0965 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| For a Poisson model, accidents (events) must occur singly/independently, so manager should record as one accident | B1 (1) | For stating accidents "occur singly" or accidents are "independent" AND should record as one accident |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \lambda = 2.7\ (\text{or}\ \mu = 32.4)\), \(H_1: \lambda < 2.7\ (\text{or}\ \mu < 32.4)\) | B1 | For both correct hypotheses in terms of \(\lambda\) or \(\mu\) (accept \(\mu = 2.7\) etc) |
| \([Y = \text{no. of accidents in a year},\ Y \sim Po(32.4)]\) \(P(Y \leqslant 22) = 0.03512\ldots\) | M1 | For selecting the correct model (sight of or implied by the correct probability); \(P(Y=22) = \text{awrt}\ 0.0129\) is evidence for M1 |
| Significant result so reject \(H_0\) | A1 | For awrt 0.035 (accept 0.04 if \(P(Y \leqslant 22)\) and \(Po(32.4)\) are explicitly seen) |
| There is evidence to support the manager's claim / there is evidence that the number of accidents per month has decreased | A1 (4) | dep on M1A1 indep of hyp's; number of accidents reduced is A0 — must be rate/per month/average number |
# Question 2:
## Part 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Accidents occur randomly/independently at a constant/average rate | B1 (1) | For a suitable reason picking up the underlined words from the context |
## Part 2(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[A = \text{no. of accidents in a month},\ A \sim Po(2.7)]$ $P(A \geqslant 3) = 1 - P(A \leqslant 2) = 1 - 0.49362 = 0.50637$ | B1 (1) | For awrt 0.506 |
## Part 2(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[T = \text{no. of accidents in 3-month period},\ T \sim Po(3 \times 2.7) = Po(8.1)]$ | M1 | For selecting the $Po(8.1)$ model (sight of or implied by a correct answer) |
| $P(T \leqslant 10) = 0.805837\ldots = \text{awrt}\ \mathbf{0.806}$ | A1 (2) | For awrt 0.806 |
## Part 2(b)(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[M = \text{no. of months with no accidents},\ M \sim B(8,\ e^{-2.7})]$ | M1 | For selecting a suitable binomial model e.g. $B(8,p)$ or $B(n, 0.067\ldots)$ |
| $M \sim B(8,\ 0.067(2)\ldots)$ | A1 | For the correct model ($p = e^{-2.7}$ or 0.067 or better) seen or implied by a correct answer |
| $P(M \geqslant 2) = 1 - P(M \leqslant 1)$ | M1 | For using their binomial model to attempt $P(M \geqslant 2)$ or $1 - P(M \leqslant 1)$; awrt 0.0964 is evidence for this M1 |
| $= 1 - 0.903542\ldots = 0.096457\ldots = \text{awrt}\ \mathbf{0.0965}$ | A1 (4) | For awrt 0.0965 |
## Part 2(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| For a Poisson model, accidents (events) must occur singly/independently, so manager should record as **one** accident | B1 (1) | For stating accidents "occur singly" or accidents are "independent" **AND** should record as one accident |
## Part 2(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \lambda = 2.7\ (\text{or}\ \mu = 32.4)$, $H_1: \lambda < 2.7\ (\text{or}\ \mu < 32.4)$ | B1 | For both correct hypotheses in terms of $\lambda$ or $\mu$ (accept $\mu = 2.7$ etc) |
| $[Y = \text{no. of accidents in a year},\ Y \sim Po(32.4)]$ $P(Y \leqslant 22) = 0.03512\ldots$ | M1 | For selecting the correct model (sight of or implied by the correct probability); $P(Y=22) = \text{awrt}\ 0.0129$ is evidence for M1 |
| Significant result so reject $H_0$ | A1 | For awrt 0.035 (accept 0.04 if $P(Y \leqslant 22)$ and $Po(32.4)$ are explicitly seen) |
| There is evidence to support the manager's claim / there is evidence that the number of **accidents per month** has decreased | A1 (4) | dep on M1A1 **indep of hyp's**; **number** of accidents reduced is A0 — must be **rate/per month/average number** |
---\begin{enumerate}
\item A manager keeps a record of accidents in a canteen.
\end{enumerate}
Accidents occur randomly with an average of 2.7 per month. The manager decides to model the number of accidents with a Poisson distribution.\\
(a) Give a reason why a Poisson distribution could be a suitable model in this situation.\\
(b) Assuming that a Poisson model is suitable, find the probability of\\
(i) at least 3 accidents in the next month,\\
(ii) no more than 10 accidents in a 3-month period,\\
(iii) at least 2 months with no accidents in an 8-month period.
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,\\
(c) state
\begin{itemize}
\item a property of the Poisson distribution that the manager should consider when deciding how to record this situation
\item whether the manager should record this as one or two accidents
\end{itemize}
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.\\
(d) Use a $5 \%$ level of significance to carry out a suitable test to assess the manager's claim.\\
You should state your hypotheses clearly and the $p$-value used in your test.
\hfill \mbox{\textit{Edexcel FS1 AS 2024 Q2 [13]}}