| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Poisson to the Normal distribution |
| Type | Multiple approximations in one question |
| Difficulty | Moderate -0.3 This question tests standard approximation procedures (Poisson to Normal, Binomial to Poisson) that are routine applications of learned formulas. Part (a) requires continuity correction and z-score calculation, while part (b) only asks to write down an expression using the Poisson approximation formula. Both are textbook exercises with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x! |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(X \sim N(25, 25)\) | B1 | soi |
| \(\frac{30.5 - 25}{\sqrt{25}}\) \([= 1.1]\) | M1 | Standardising with their values. Allow with missing or incorrect continuity correction. |
| \(1 - \phi(\text{'1.1'})\) | M1 | For area consistent with their working. |
| \(= 0.136\) (3 sf) | A1 | |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(e^{-100p}\left(1 + 100p + \frac{(100p)^2}{2!}\right)\) | M1 | For \(P_o(100p)\) expression. Accept un-simplified terms (e.g. \(p^0/0!\) for M1). Allow one end error (e.g. for correct with extra term \(e^{-100p} \times \frac{(100p)^3}{3!}\) oe), or brackets omitted. |
| \(e^{-100p}\left(1 + 100p + \frac{(100p)^2}{2!}\right)\) or \(e^{-100p} + e^{-100p} \times 100p + e^{-100p} \times \frac{(100p)^2}{2!}\) or \(e^{-100p}(1 + 100p + 5000p^2)\) oe | A1 | Must have brackets. Allow with or without ! sign (but not \(0!\) or \(p^0\)). ISW once a fully correct answer seen. |
| 2 |
## Question 1:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \sim N(25, 25)$ | **B1** | soi |
| $\frac{30.5 - 25}{\sqrt{25}}$ $[= 1.1]$ | **M1** | Standardising with their values. Allow with missing or incorrect continuity correction. |
| $1 - \phi(\text{'1.1'})$ | **M1** | For area consistent with their working. |
| $= 0.136$ (3 sf) | **A1** | |
| | **4** | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{-100p}\left(1 + 100p + \frac{(100p)^2}{2!}\right)$ | **M1** | For $P_o(100p)$ expression. Accept un-simplified terms (e.g. $p^0/0!$ for M1). Allow one end error (e.g. for correct with extra term $e^{-100p} \times \frac{(100p)^3}{3!}$ oe), or brackets omitted. |
| $e^{-100p}\left(1 + 100p + \frac{(100p)^2}{2!}\right)$ or $e^{-100p} + e^{-100p} \times 100p + e^{-100p} \times \frac{(100p)^2}{2!}$ or $e^{-100p}(1 + 100p + 5000p^2)$ oe | **A1** | Must have brackets. Allow with or without ! sign (but not $0!$ or $p^0$). ISW once a fully correct answer seen. |
| | **2** | |
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\begin{enumerate}[label=(\alph*)]
\item A random variable $X$ has the distribution $\operatorname { Po } ( 25 )$.\\
Use the normal approximation to the Poisson distribution to find $\mathrm { P } ( X > 30 )$.
\item A random variable $Y$ has the distribution $\mathrm { B } ( 100 , p )$ where $p < 0.05$.
Use the Poisson approximation to the binomial distribution to write down an expression, in terms of $p$, for $\mathrm { P } ( Y < 3 )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2023 Q1 [6]}}