| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State conditions for Poisson approximation |
| Difficulty | Moderate -0.8 This question tests standard bookwork recall (stating p must be small for Poisson approximation) and routine application of the approximation with straightforward calculations. The hypothesis test parts are also standard S2 procedures requiring no novel insight, though part (c)(ii) requires recognizing when normal approximation conditions fail. |
| Spec | 5.02n Sum of Poisson variables: is Poisson5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(p\) is small | B1 | B1 correct condition allow '\(p\) is close to 0' allow '\(p < 0.1\)' or any value less than 0.1 (condone \(np < 10\) or \(np\) condone comments about \(np\)) |
| (1) | ||
| (b) Let \(N =\) number of candles not suitable for sale. \(N \sim B(125, 0.02)\) | M1 | 1st M1 recognising Binomial distribution (may be implied by Po(2.5)) |
| \(\approx C \sim \text{Po}(2.5)\) | A1 | 1st A1 correct distribution Po(2.5) |
| \(P(C \leqslant 6) = 0.9858\) | M1 A1 | 2nd M1 writing or using \(P(C \leqslant 6)\) from Poisson distribution. 2nd A1 awrt 0.986 from correct distribution used (calc: 0.9858126...). Answer only 0.9858 or better scores 4 out of 4, but answer of 0.986 must see Po(2.5) to award full marks. [NB: Use of binomial gives 0.98678...] |
| (4) | ||
| (c)(i) \(H_0: p = 0.05\) \(H_1: p < 0.05\) | B1 M1 A1 M1 | B1 correct hypotheses in terms of \(p\) or \(\pi\). 1st M1 writing or using \(B(30, 0.05)\) (may be implied by 1st A1). 1st A1 awrt 0.215 |
| \(D \sim B(30, 0.05)\) | 2nd M1 a correct ft statement consistent with their \(p\)-value and 0.05. No context needed but do not allow contradicting non-contextual comments. | |
| \(P(D = 0) = 0.2146\) | 2nd A1 correct conclusion in context which must be not rejecting \(H_0\). Must use underlined words (oe). No hypotheses then A0. Condone e.g. '5% of candle holders have minor defects' | |
| Do not reject \(H_0\) / not significant | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| (c)(ii) Impossible to reject \(H_0\) (since \(P(D = 0) > 0.05\)) | B1 | B1 correct reasoning which implies there is no critical region/\(H_0\) cannot be rejected. Sample size is too small on its own is B0. |
| (1) | ||
| (d) \(0.95^{50}\)=[0.0769...] ..... or \(X \sim B(50, 0.05), P(X = 0)\) (is still) \(> 0.05\) (so still not possible to reject \(H_0\)) hence Ashley's change does not make the test appropriate. | M1 A1 | M1 for \(0.95^{50}\) or for \(X \sim B(50, 0.05)\) and \(P(X = 0) > 0.05\). A1 test is (still) not appropriate with M1 scored |
| (2) | ||
| [13 marks] |
**(a)** $p$ is small | B1 | B1 correct condition allow '$p$ is close to 0' allow '$p < 0.1$' or any value less than 0.1 (condone $np < 10$ or $np$ condone comments about $np$)
| | (1)
**(b)** Let $N =$ number of candles not suitable for sale. $N \sim B(125, 0.02)$ | M1 | 1st M1 recognising Binomial distribution (may be implied by Po(2.5))
$\approx C \sim \text{Po}(2.5)$ | A1 | 1st A1 correct distribution Po(2.5)
$P(C \leqslant 6) = 0.9858$ | M1 A1 | 2nd M1 writing or using $P(C \leqslant 6)$ from Poisson distribution. 2nd A1 awrt 0.986 from correct distribution used (calc: 0.9858126...). Answer only 0.9858 or better scores 4 out of 4, but answer of 0.986 must see Po(2.5) to award full marks. [NB: Use of binomial gives 0.98678...]
| | (4)
**(c)(i)** $H_0: p = 0.05$ $H_1: p < 0.05$ | B1 M1 A1 M1 | B1 correct hypotheses in terms of $p$ or $\pi$. 1st M1 writing or using $B(30, 0.05)$ (may be implied by 1st A1). 1st A1 awrt 0.215
$D \sim B(30, 0.05)$ | | 2nd M1 a correct ft statement consistent with their $p$-value and 0.05. No context needed but do not allow contradicting non-contextual comments.
$P(D = 0) = 0.2146$ | | 2nd A1 correct conclusion in context which must be not rejecting $H_0$. Must use underlined words (oe). No hypotheses then A0. Condone e.g. '5% of candle holders have minor defects'
Do not reject $H_0$ / not significant | | (5)
The manufacturer's claim is not supported/There is not enough evidence to suggest that the proportion(oe) of candle holders with minor defects is less than 5%/ Charlie's claim is supported
**(c)(ii)** Impossible to reject $H_0$ (since $P(D = 0) > 0.05$) | B1 | B1 correct reasoning which implies there is no critical region/$H_0$ cannot be rejected. Sample size is too small on its own is B0.
| | (1)
**(d)** $0.95^{50}$=[0.0769...] ..... or $X \sim B(50, 0.05), P(X = 0)$ (is still) $> 0.05$ (so still not possible to reject $H_0$) hence Ashley's change does not make the test appropriate. | M1 A1 | M1 for $0.95^{50}$ or for $X \sim B(50, 0.05)$ and $P(X = 0) > 0.05$. A1 test is (still) not appropriate with M1 scored
| | (2)
| | [13 marks]\begin{enumerate}
\item (a) Given $n$ is large, state a condition for which the binomial distribution $\mathrm { B } ( n , p )$ can be reasonably approximated by a Poisson distribution.
\end{enumerate}
A manufacturer produces candles. Those candles that pass a quality inspection are suitable for sale.
It is known that 2\% of the candles produced by the manufacturer are not suitable for sale.
A random sample of 125 candles produced by the manufacturer is taken.\\
(b) Use a suitable approximation to find the probability that no more than 6 of the candles are not suitable for sale.
The manufacturer also produces candle holders.\\
Charlie believes that 5\% of candle holders produced by the factory have minor defects.\\
The manufacturer claims that the true proportion is less than $5 \%$\\
To test the manufacturer's claim, a random sample of 30 candle holders is taken and none of them are found to contain minor defects.\\
(c) (i) Carry out a test of the manufacturer's claim using a $5 \%$ level of significance. You should state your hypotheses clearly.\\
(ii) Give a reason why this is not an appropriate test.
Ashley suggests changing the sample size to 50\\
(d) Comment on whether or not this change would make the test appropriate. Give a reason for your answer.
\hfill \mbox{\textit{Edexcel S2 2023 Q4 [13]}}