| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Poisson to the Normal distribution |
| Type | Combined Poisson approximation and exact calculation |
| Difficulty | Standard +0.3 This is a straightforward application of the normal approximation to the Poisson distribution with a large parameter (λ=145). Part (a) requires standardizing with continuity correction and using normal tables, while part (b) asks for a standard justification (λ>10 or similar). Both parts are routine bookwork for S2 level with no problem-solving required beyond applying a memorized procedure. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(N(145, 145)\) | B1 | Stated or implied |
| \(\pm\frac{150.5-145}{\sqrt{145}}\) \([= \pm0.457]\) | M1 | Condone incorrect or omitted continuity correction |
| \(\Phi(0.457)\) | M1 | For area consistent with their working |
| \(= 0.676\) (3sf) | A1 | SC: Unsupported answer of 0.676 scores B3. Unsupported answer of 0.646 or 0.661 scores B2. Unsupported answer of 0.6799 scores B1 |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(145 > 15\) | B1 | Explicit. \(\lambda > 15\) B0 if \(\lambda = 145\) not stated. Accept \(\geqslant\). Accept mean for \(\lambda\) |
| Total: 1 |
## Question 1:
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $N(145, 145)$ | B1 | Stated or implied |
| $\pm\frac{150.5-145}{\sqrt{145}}$ $[= \pm0.457]$ | M1 | Condone incorrect or omitted continuity correction |
| $\Phi(0.457)$ | M1 | For area consistent with their working |
| $= 0.676$ (3sf) | A1 | SC: Unsupported answer of 0.676 scores **B3**. Unsupported answer of 0.646 or 0.661 scores **B2**. Unsupported answer of 0.6799 scores **B1** |
| **Total: 4** | | |
**Part (b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $145 > 15$ | B1 | Explicit. $\lambda > 15$ B0 if $\lambda = 145$ not stated. Accept $\geqslant$. Accept mean for $\lambda$ |
| **Total: 1** | | |
---1 A random variable $X$ has the distribution $\mathrm { Po } ( 145 )$.
\begin{enumerate}[label=(\alph*)]
\item Use a suitable approximating distribution to calculate $\mathrm { P } ( X \leqslant 150 )$.
\item Justify the use of your approximating distribution in this case.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q1 [5]}}