| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.8 This is a straightforward application of a standard approximation technique. Part (i) requires routine calculation using the Poisson approximation with λ=np=2.25, and part (ii) asks for standard justification criteria (n large, p small, np moderate). Both parts are textbook exercises requiring recall and direct application rather than problem-solving or insight. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Po}(2.25)\) | B1 | Stated or implied |
| \(e^{-2.25}(1 + 2.25 + \frac{2.25^2}{2})\) | M1 | Allow any \(\lambda\), one end error |
| \(= 0.609\) (3 sf) | A1 | SC B1 Use of \(B(75, 0.03)\) leading to \(0.608\) |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mu = 2.25\), which is less than 5; \(n\) large | B1 | Allow \(np < 5\) and \(n\) large or \(p < 0.1\) and \(n > 50\), no contradictions |
| Total: 1 |
## Question 1:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Po}(2.25)$ | B1 | Stated or implied |
| $e^{-2.25}(1 + 2.25 + \frac{2.25^2}{2})$ | M1 | Allow any $\lambda$, one end error |
| $= 0.609$ (3 sf) | A1 | SC B1 Use of $B(75, 0.03)$ leading to $0.608$ |
| **Total: 3** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mu = 2.25$, which is less than 5; $n$ large | B1 | Allow $np < 5$ and $n$ large or $p < 0.1$ and $n > 50$, no contradictions |
| **Total: 1** | | |
---1 A random variable $X$ has the distribution $\mathrm { B } ( 75,0.03 )$.\\
(i) Use the Poisson approximation to the binomial distribution to calculate $\mathrm { P } ( X < 3 )$.\\
(ii) Justify the use of the Poisson approximation.\\
\hfill \mbox{\textit{CAIE S2 2018 Q1 [4]}}