| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Sum of independent Poisson processes |
| Difficulty | Standard +0.3 This is a straightforward application of standard Poisson distribution techniques: direct probability calculations, time-scaling (65/24 for hourly rate), sum of independent Poissons, and normal approximation. All are routine procedures covered in Further Maths Statistics with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.293\) | B1 | AWRT 0.293 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X \sim \text{Po}\!\left(\frac{65}{24}\right)\), \(P(X = 2) = 0.244\) | M1 | Selects and uses Poisson model with \(\lambda = \frac{65}{24}\) to find \(P(\text{motor claims} = 2)\) or \(P(\text{motor claims} \leq 2)\) |
| \(0.244\) | A1 | AWRT 0.244 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\lambda = 65 + 23 = 88\), \(Y \sim \text{Po}(88)\) | M1 | Selects Poisson model and attempts square root of \(\lambda\) or calculates 88 |
| Standard deviation \(= \sqrt{88} = 9.38\) | A1 | AWRT 9.38 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(Y \sim \text{Po}(88)\) | M1 | Uses Poisson model with \(\lambda = 88\) or their \(\lambda\) from (c)(i) to correctly find \(P(Y \leq 90)\) AWRT 0.61, \(P(Y \geq 91)\) AWRT 0.39, \(P(Y \leq 89)\) AWRT 0.57 or \(P(Y \geq 90)\) AWRT 0.43 |
| \(P(Y > 90) = 1 - P(Y \leq 90) = 1 - 0.611\ldots = 0.389\) | A1 | AWRT 0.389 |
## Question 6(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.293$ | B1 | AWRT 0.293 |
---
## Question 6(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim \text{Po}\!\left(\frac{65}{24}\right)$, $P(X = 2) = 0.244$ | M1 | Selects and uses Poisson model with $\lambda = \frac{65}{24}$ to find $P(\text{motor claims} = 2)$ or $P(\text{motor claims} \leq 2)$ |
| $0.244$ | A1 | AWRT 0.244 |
---
## Question 6(c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\lambda = 65 + 23 = 88$, $Y \sim \text{Po}(88)$ | M1 | Selects Poisson model and attempts square root of $\lambda$ or calculates 88 |
| Standard deviation $= \sqrt{88} = 9.38$ | A1 | AWRT 9.38 |
---
## Question 6(c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $Y \sim \text{Po}(88)$ | M1 | Uses Poisson model with $\lambda = 88$ or their $\lambda$ from (c)(i) to correctly find $P(Y \leq 90)$ AWRT 0.61, $P(Y \geq 91)$ AWRT 0.39, $P(Y \leq 89)$ AWRT 0.57 or $P(Y \geq 90)$ AWRT 0.43 |
| $P(Y > 90) = 1 - P(Y \leq 90) = 1 - 0.611\ldots = 0.389$ | A1 | AWRT 0.389 |
---6 An insurance company models the number of motor claims received in 1 day using a Poisson distribution with mean 65
6
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the company receives at most 60 motor claims in 1 day.
Give your answer to three decimal places.
6
\item The company receives motor claims using a telephone line which is open 24 hours a day.
Find the probability that the company receives exactly 2 motor claims in 1 hour.
Give your answer to three decimal places.\\
6
\item The company models the number of property claims received in 1 day using a Poisson distribution with mean 23
Assume that the number of property claims received is independent of the number of motor claims received.
6 (c) (i) Find the standard deviation of the variable that represents the total number of motor claims and property claims received in 1 day.
Give your answer to three significant figures.\\
6 (c) (ii) Find the probability that the company receives a total of more than 90 motor claims and property claims in 1 day.
Give your answer to three significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2023 Q6 [8]}}