| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.3 This is a straightforward application of the Poisson approximation to the binomial distribution. Part (a) requires stating B(3600, 0.0012) and Po(4.32) with standard justification (large n, small p), then a routine Poisson probability calculation. Part (b) involves solving e^(-λ) > 0.1 for λ, then finding n. All steps are standard textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p)5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks |
|---|---|
| \(B(3600, 0.0012)\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(Po(4.32)\) | B2 | B1 for Po, B1 for \(\lambda = 4.32\) |
| \(n = 3600\) which is large, \(p = 0.12\) which is small and \(np = 4.32\) which is \(< 5\) | B1 |
| Answer | Marks |
|---|---|
| \(1 - e^{-4.32}\left(1 + 4.32 + \frac{4.32^2}{2}\right)\) | M1 |
| \(0.805\) (3 sf) | A1 |
| Answer | Marks |
|---|---|
| \(e^{-\lambda} > 0.1\) | M1 |
| \((-\lambda > \ln 0.1)\), \((\lambda < \ln 10)\), \(0.0012n < \ln 10\) | A1 |
| \((n < 1918.8)\), largest \(n\) is \(1918\) | A1 |
## Question 3:
**Part 3(a)(i):**
| $B(3600, 0.0012)$ | B1 | |
**Part 3(a)(ii):**
| $Po(4.32)$ | B2 | B1 for Po, B1 for $\lambda = 4.32$ |
| $n = 3600$ which is large, $p = 0.12$ which is small and $np = 4.32$ which is $< 5$ | B1 | |
**Part 3(a)(iii):**
| $1 - e^{-4.32}\left(1 + 4.32 + \frac{4.32^2}{2}\right)$ | M1 | |
| $0.805$ (3 sf) | A1 | |
**Part 3(b):**
| $e^{-\lambda} > 0.1$ | M1 | |
| $(-\lambda > \ln 0.1)$, $(\lambda < \ln 10)$, $0.0012n < \ln 10$ | A1 | |
| $(n < 1918.8)$, largest $n$ is $1918$ | A1 | |
---3 In the data-entry department of a certain firm, it is known that $0.12 \%$ of data items are entered incorrectly, and that these errors occur randomly and independently.
\begin{enumerate}[label=(\alph*)]
\item A random sample of 3600 data items is chosen. The number of these data items that are incorrectly entered is denoted by $X$.
\begin{enumerate}[label=(\roman*)]
\item State the distribution of $X$, including the values of any parameters.
\item State an appropriate approximating distribution for $X$, including the values of any parameters.
Justify your choice of approximating distribution.
\item Use your approximating distribution to find $\mathrm { P } ( X > 2 )$.
\end{enumerate}\item Another large random sample of $n$ data items is chosen. The probability that the sample contains no data items that are entered incorrectly is more than 0.1 .
Use an approximating distribution to find the largest possible value of $n$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q3 [9]}}