| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Poisson to the Normal distribution |
| Type | Multiple approximations in one question |
| Difficulty | Standard +0.3 This is a straightforward application of standard distribution approximations with continuity corrections. Part (i) uses Poisson→Normal (λ=42 is large enough), part (ii) uses Binomial→Poisson (n large, p small). Both require routine justification of conditions (λ>15 or np<5) and standard continuity correction application. No novel insight needed—pure textbook procedure slightly above average due to requiring two different approximations and formal justifications. |
| 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 |
| \(X \sim N(42, 42)\) | B1 | Stated or implied |
| \(\frac{39.5 - \text{"42"}}{\sqrt{\text{"42"}}}\ (= -0.386)\) | M1 | Allow with wrong or no cc |
| \(1 - \phi(\text{"}-0.386\text{"}) = \phi(\text{"0.386"})\) | M1 | Correct area consistent with their working |
| \(= 0.65(0)\) (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(42 > \) (e.g. 15) or mean is large | B1 | \(\lambda > 15\) or higher, \(\lambda =\) large; ignore subsequent work if not undermining what already written |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(Y \sim Po(1.2)\) | B1 | Stated or implied |
| \(1 - e^{-1.2}(1 + 1.2 + \frac{1.2^2}{2})\) | M1 | Allow any \(\lambda\); allow one end error |
| \(= 0.121\) (3 sf) | A1 | Using binomial: \(0.119\) SR B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(60 \times 0.02 = 1.2 < 5\) or mean is small | B1FT | Or large \(n\) small \(p\); FT Poisson only |
## Question 5(i)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \sim N(42, 42)$ | B1 | Stated or implied |
| $\frac{39.5 - \text{"42"}}{\sqrt{\text{"42"}}}\ (= -0.386)$ | M1 | Allow with wrong or no cc |
| $1 - \phi(\text{"}-0.386\text{"}) = \phi(\text{"0.386"})$ | M1 | Correct area consistent with their working |
| $= 0.65(0)$ (3 sf) | A1 | |
---
## Question 5(i)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $42 > $ (e.g. 15) or mean is large | B1 | $\lambda > 15$ or higher, $\lambda =$ large; ignore subsequent work if not undermining what already written |
---
## Question 5(ii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Y \sim Po(1.2)$ | B1 | Stated or implied |
| $1 - e^{-1.2}(1 + 1.2 + \frac{1.2^2}{2})$ | M1 | Allow any $\lambda$; allow one end error |
| $= 0.121$ (3 sf) | A1 | Using binomial: $0.119$ SR **B1** |
---
## Question 5(ii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $60 \times 0.02 = 1.2 < 5$ or mean is small | B1FT | Or large $n$ small $p$; FT Poisson only |5 (i) A random variable $X$ has the distribution $\operatorname { Po } ( 42 )$.
\begin{enumerate}[label=(\alph*)]
\item Use an appropriate approximating distribution to find $\mathrm { P } ( X \geqslant 40 )$.
\item Justify your use of the approximating distribution.\\
(ii) A random variable $Y$ has the distribution $\mathrm { B } ( 60,0.02 )$.\\
(a) Use an appropriate approximating distribution to find $\mathrm { P } ( Y > 2 )$.\\
(b) Justify your use of the approximating distribution.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2017 Q5 [9]}}