| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Explain or apply conditions in context |
| Difficulty | Standard +0.3 This question tests standard Poisson distribution calculations including scaling the rate parameter, a routine normal approximation with continuity correction, and conceptual understanding of Poisson assumptions. Part (a) involves straightforward probability calculations, part (b) requires recognizing that constant rate is a Poisson assumption and calculating separate rates—all standard S2 material with no novel problem-solving required. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\lambda = 3\) | B1 | For mean = 3 |
| \(1 - e^{-3}(1 + 3 + \frac{3^2}{2} + \frac{3^3}{3!})\) or \(1 - e^{-3}(1 + 3 + 4.5 + 4.5)\) or \(1 - (0.04979 + 0.14936 + 0.22404 + 0.22404)\) | M1 | Any \(\lambda\). Allow one end error |
| \(= 0.353\) (3 sf) | A1 | No working scores B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(N(54, 54)\) | M1 | soi |
| \(\frac{39.5 - 54}{\sqrt{54}} (= -1.973)\) | M1 | Allow with wrong or no continuity correction. For standardising with their mean and variance |
| \(1 - \phi(\text{'1.973'})\) | M1 | For area consistent with their working |
| \(= 0.0242\) (3 sf) | A1 | Special case: if no working seen, 0.0242 scores SC B3, 0.0284 scores SC B2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 'Mean not constant' or 'number of hits per minute not constant' or 'not a constant rate' | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2p + p = 2 \times 0.3 \Rightarrow p = 0.2\) [where \(p\) is the rate per minute for night time] | M1 | May be implied by answer |
| [During day-time]: \(\text{Po}(0.4)\). [During night-time]: \(\text{Po}(0.2)\) | A1 | Accept Po(24) [per daytime hour], Po(12) [per night time hour]. Accept Po(288) [per day time shift], Po(144) [per night time shift]. Note: Po(432), Po(216) scores M0A0 |
## Question 3(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = 3$ | B1 | For mean = 3 |
| $1 - e^{-3}(1 + 3 + \frac{3^2}{2} + \frac{3^3}{3!})$ or $1 - e^{-3}(1 + 3 + 4.5 + 4.5)$ or $1 - (0.04979 + 0.14936 + 0.22404 + 0.22404)$ | M1 | Any $\lambda$. Allow one end error |
| $= 0.353$ (3 sf) | A1 | No working scores B1 |
## Question 3(a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $N(54, 54)$ | M1 | soi |
| $\frac{39.5 - 54}{\sqrt{54}} (= -1.973)$ | M1 | Allow with wrong or no continuity correction. For standardising with their mean and variance |
| $1 - \phi(\text{'1.973'})$ | M1 | For area consistent with their working |
| $= 0.0242$ (3 sf) | A1 | Special case: if no working seen, 0.0242 scores **SC B3**, 0.0284 scores **SC B2** |
## Question 3(b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| 'Mean not constant' or 'number of hits per minute not constant' or 'not a constant rate' | B1 | |
## Question 3(b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2p + p = 2 \times 0.3 \Rightarrow p = 0.2$ [where $p$ is the rate per minute for night time] | M1 | May be implied by answer |
| [During day-time]: $\text{Po}(0.4)$. [During night-time]: $\text{Po}(0.2)$ | A1 | Accept Po(24) [per daytime hour], Po(12) [per night time hour]. Accept Po(288) [per day time shift], Po(144) [per night time shift]. Note: Po(432), Po(216) scores M0A0 |3 A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.
\begin{enumerate}[label=(\alph*)]
\item Assume that the owner is correct.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that there will be at least 4 hits during a 10-minute period.
\item Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period.\\
A friend agrees that the website receives, on average, 0.3 hits per minute. However, she notices that the number of hits during the day-time ( 9.00 am to 9.00 pm ) is usually about twice the number of hits during the night-time ( 9.00 pm to 9.00 am ).
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution.
\item Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2023 Q3 [10]}}