| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Calculate variance of linear transformation |
| Difficulty | Moderate -0.8 This is a straightforward application of standard results: part (a) uses the basic variance property Var(aX+b)=a²Var(X) with binomial variance np(1-p); part (b) applies the standard Poisson approximation to binomial when n is large and p is small (np=4), then calculates a probability using tables. All steps are routine recall and direct application of formulas with no problem-solving or insight required. |
| Spec | 5.02d 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 | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Var}(X) = 400 \times 0.01 \times 0.99 \ (= 3.96)\) | M1 | |
| \(\text{Var}(4X + 2) = 16 \times \text{Var}(X)\) | M1 | For \(16 \times \textit{their}\ \text{Var}(X)\) |
| \(63.36\) | A1 | Accept 63.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Po}(4)\) | B1 | |
| \(n = 400 > 50\) and either \(np = 4 < 5\) or \(p = 0.01 < 0.1\) | B1 | Must quote values 400 and 4 or clearly see \(n=400\) and \(np=4\) (or \(p=0.01\)) in working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(e^{-4}\left(\frac{4^2}{2!}+\frac{4^3}{3!}+\frac{4^4}{4!}+\frac{4^5}{5!}\right)\) | M1 | FT *their* '4'. Allow one end error. FT from (b)(i) Use of Normal allow M1 for attempt at standardising (with correct continuity correction) using *their* N(4,3.96) and attempt at probability. FT from (b)(i) Use of Binomial allow M1 for attempt at P(2,3,4,5) Binomial terms clearly seen and added |
| \(0.694\) (3 sf) | A1 | CWO. SCB1 only for unsupported answer of 0.694 |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Var}(X) = 400 \times 0.01 \times 0.99 \ (= 3.96)$ | M1 | |
| $\text{Var}(4X + 2) = 16 \times \text{Var}(X)$ | M1 | For $16 \times \textit{their}\ \text{Var}(X)$ |
| $63.36$ | A1 | Accept 63.4 |
## Question 2(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Po}(4)$ | B1 | |
| $n = 400 > 50$ and either $np = 4 < 5$ or $p = 0.01 < 0.1$ | B1 | Must quote values 400 and 4 or clearly see $n=400$ and $np=4$ (or $p=0.01$) in working |
## Question 2(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $e^{-4}\left(\frac{4^2}{2!}+\frac{4^3}{3!}+\frac{4^4}{4!}+\frac{4^5}{5!}\right)$ | M1 | FT *their* '4'. Allow one end error. FT from **(b)(i)** Use of Normal allow M1 for attempt at standardising (with correct continuity correction) using *their* N(4,3.96) and attempt at probability. FT from **(b)(i)** Use of Binomial allow M1 for attempt at P(2,3,4,5) Binomial terms clearly seen and added |
| $0.694$ (3 sf) | A1 | CWO. **SCB1** only for unsupported answer of 0.694 |2 The random variable $X$ has the distribution $\mathrm { B } ( 400,0.01 )$.
\begin{enumerate}[label=(\alph*)]
\item Find $\operatorname { Var } ( 4 X + 2 )$.
\item \begin{enumerate}[label=(\roman*)]
\item State an appropriate approximating distribution for $X$, giving the values of any parameters. Justify your choice of approximating distribution.
\item Use your approximating distribution to find $\mathrm { P } ( 2 \leqslant X \leqslant 5 )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q2 [7]}}