| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| 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 the Poisson approximation to the binomial distribution. Part (a) requires calculating P(X < 3) using λ = np = 3.6, which is routine. Part (b) asks for standard justification (n large, p small, np moderate), which is textbook recall. The question involves no problem-solving or novel insight, just direct application of a standard approximation technique. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Poisson | B1 | SOI |
| Mean \(= 3.6\) | B1 | Can be awarded for \(N(3.6, \ldots)\) |
| \(e^{-3.6}(1 + 3.6 + \frac{3.6^2}{2})\) | M1 | Allow any \(\lambda\); allow one end error; expression must be seen |
| \(0.303\) (3 s.f.) | A1 | If M0 awarded allow SC B1 for \(0.303\); SC use of binomial: B1 for answer \(0.300\) (3 sf) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Binomial with] \(200 > 50\) | B1 | |
| \([200 \times 0.018 =]\ 3.6 < 5\) or \([p =]\ 0.018 < 0.1\) | B1 | If B0 B0 then SC \(n\) large, \(p\) small: B1 or \(n\) large \(np < 5\): B1 or \(n > 50\) and either \(np < 5\) or \(p < 0.1\): B1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Poisson | B1 | SOI |
| Mean $= 3.6$ | B1 | Can be awarded for $N(3.6, \ldots)$ |
| $e^{-3.6}(1 + 3.6 + \frac{3.6^2}{2})$ | M1 | Allow any $\lambda$; allow one end error; expression must be seen |
| $0.303$ (3 s.f.) | A1 | If M0 awarded allow SC B1 for $0.303$; SC use of binomial: B1 for answer $0.300$ (3 sf) |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Binomial with] $200 > 50$ | B1 | |
| $[200 \times 0.018 =]\ 3.6 < 5$ or $[p =]\ 0.018 < 0.1$ | B1 | If B0 B0 then SC $n$ large, $p$ small: B1 or $n$ large $np < 5$: B1 or $n > 50$ and either $np < 5$ or $p < 0.1$: B1 |
---3 It is known that $1.8 \%$ of children in a certain country have not been vaccinated against measles. A random sample of 200 children in this country is chosen.
\begin{enumerate}[label=(\alph*)]
\item Use a suitable approximating distribution to find the probability that there are fewer than 3 children in the sample who have not been vaccinated against measles.
\item Justify your approximating distribution.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q3 [6]}}