| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Standard +0.3 Part (i) is a straightforward Poisson probability calculation with λ=4.8. Part (ii) requires recognizing when to use normal approximation (large λ=144) and applying continuity correction, which is standard S2 material but slightly above routine recall due to the approximation step and time conversion. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((\lambda = 2 \times 2.4) = 4.8\) | M1 | Any \(\lambda\) |
| \(e^{-4.8}\left(1 + 4 + \frac{4.8^2}{2} + \frac{4.8^3}{3!}\right)\) | ||
| \(0.294\) (3 sf) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((\lambda = 60 \times 2.4) = 144\), \(N('144', '144')\) | M1 | N and \(\sigma^2 = \mu\) SOI |
| \(\frac{139.5 - '144'}{\sqrt{'144'}} (= -0.375)\) | M1 | Allow with no continuity correction |
| \(\phi('0.375')\) | M1 | Correct area consistent with their working |
| \(0.646\) (3 sf) | A1 | |
| 4 |
**Question 2:**
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(\lambda = 2 \times 2.4) = 4.8$ | M1 | Any $\lambda$ |
| $e^{-4.8}\left(1 + 4 + \frac{4.8^2}{2} + \frac{4.8^3}{3!}\right)$ | | |
| $0.294$ (3 sf) | A1 | |
| | **2** | |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(\lambda = 60 \times 2.4) = 144$, $N('144', '144')$ | M1 | N and $\sigma^2 = \mu$ SOI |
| $\frac{139.5 - '144'}{\sqrt{'144'}} (= -0.375)$ | M1 | Allow with no continuity correction |
| $\phi('0.375')$ | M1 | Correct area consistent with their working |
| $0.646$ (3 sf) | A1 | |
| | **4** | |
---2 Cars arrive at a filling station randomly and at a constant average rate of 2.4 cars per minute.\\
(i) Calculate the probability that fewer than 4 cars arrive in a 2 -minute period.\\
(ii) Use a suitable approximating distribution to calculate the probability that at least 140 cars arrive in a 1-hour period.\\
\hfill \mbox{\textit{CAIE S2 2019 Q2 [6]}}