| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (exact Poisson part) |
| Difficulty | Standard +0.3 This is a straightforward Poisson distribution question requiring standard techniques: scaling the parameter for different time periods, using tables/calculator for probabilities, applying normal approximation with continuity correction, and recognizing that constant rate is a Poisson assumption. All parts are routine applications 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 probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a)(i) | λ = 3 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| or 1 – (0.04979 + 0.14936 + 0.22404 + 0.22404) | M1 | Any λ. Allow one end error. |
| = 0.353 (3 sf) | A1 | No working scores B1. |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a)(ii) | N(54, 54) | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 54 | M1 | Allow with wrong or no continuity correction. |
| Answer | Marks | Guidance |
|---|---|---|
| 1 – ɸ ('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, |
| Answer | Marks |
|---|---|
| 3(b)(i) | ‘Mean not constant’ or’ ‘number of hits per minute not constant’ or ‘not a |
| constant rate’ | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3(b)(ii) | 2p + p = 2 × 0.3 [p = 0.2] | |
| [where p is the rate per minute for night time] | M1 | May be implied by answer. |
| [During day-time]: Po(0.4). [During night-time]: Po(0.2) | A1 | Accept Po(24) [per daytime hour], Po(12) [per night |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3(a)(i) ---
3(a)(i) | λ = 3 | B1 | For mean = 3.
1 – e–3(1 + 3 + 32 + 33 ) or 1 – e–3(1 + 3 + 4.5 + 4.5)
2 3!
or 1 – (0.04979 + 0.14936 + 0.22404 + 0.22404) | M1 | Any λ. Allow one end error.
= 0.353 (3 sf) | A1 | No working scores B1.
3
--- 3(a)(ii) ---
3(a)(ii) | N(54, 54) | M1 | soi
39.5−54
(= –1.973)
54 | M1 | Allow with wrong or no continuity correction.
For standardising with their mean and variance.
1 – ɸ ('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.
4
--- 3(b)(i) ---
3(b)(i) | ‘Mean not constant’ or’ ‘number of hits per minute not constant’ or ‘not a
constant rate’ | B1
1
--- 3(b)(ii) ---
3(b)(ii) | 2p + p = 2 × 0.3 [p = 0.2]
[where p is the rate per minute for night time] | M1 | May be implied by answer.
[During day-time]: Po(0.4). [During night-time]: 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.
2
Question | Answer | Marks | GuidanceA 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. [3]
\item Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period. [4]
\end{enumerate}
\end{enumerate}
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 (9.00am to 9.00pm) is usually about twice the number of hits during the night-time (9.00pm to 9.00am).
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\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. [1]
\item Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2023 Q3 [10]}}