| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State conditions for Poisson approximation |
| Difficulty | Easy -1.2 Part (a) is pure recall of standard conditions (n large, p small, np moderate). Parts (b) and (c) are routine hypothesis test application with straightforward Poisson approximation—standard S2 bookwork with minimal problem-solving required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(n\) large (allow \(n > 50\) or any number greater than 50) ["too" large is OK]; \(p\) small (allow \(p < 0.2\) or a probability less than 0.2) | B1 | (1) |
| \(H_0: p = 0.009\); \(H_1: p > 0.009\) | B1 | (1) |
| Po(4.5) | B1 | |
| Probability: \(P(X \geq 9) = 1 - P(X \leq 8) = 1 - 0.9597 = 0.0403\) | M1, A1 | |
| Critical Region (CR): \(P(X \leq 7) = 0.9134\); \(P(X \leq 8) = 0.9597\); CR: \(X \geq 9\) | M1d, A1cso | |
| Reject \(H_0\) or Significant. There is evidence that the farmer's claim is true. Or There is evidence that the proportion of eggs with a double yolk is \(> 0.009\) | M1d, A1cso | (5) |
| [7] |
| Answer | Marks |
|---|---|
| Item | Guidance |
| (b) B1 | both hypotheses correct. Must mention \(p\) (or \(\pi\)). Words only is B0 |
| (c) B1 | writing or using Po(4.5) (Check their probs using tables if Po(4.5) is not seen) |
| 1st M1: writing \(1 - P(X \leq 8)\). May be implied by sight of \(1 - 0.9597\) or for CR method: \(P(X \leq 7) = 0.9134\) or \(P(X \leq 8) = 0.9597\) | |
| 1st A1: for probability awrt 0.0403 or CR of \(X > 8\) or \(X \geq 9\) | |
| Allow awrt 0.9597 if accompanied by a correct comparison with 0.95 | |
| 2nd dM1 | correct statement that must agree with hypotheses. Dependent on B1. Contradictory non-contextual statements such as "not significant" so "reject \(H_0\)" score M0 |
| 2nd A1cso | correct contextual statement. Depends on all other marks in (c) being scored. Must mention "farmer" and "claim" or "eggs" and "double yolk" |
| NB | A correct calculation followed only by a correct contextual comment scores the final M1(implied) and A1 |
| If 2-tail hypotheses in (b) | Score B0 in (b). Could score B1 M1A1 and M1 for a correct non-contextual comment but A0 since they should not be rejecting \(H_0\) in this case (or they have scored A0 earlier so not cso) |
| Answer/Working | Marks | Guidance |
|---|---|---|
| $n$ large (allow $n > 50$ or any number greater than 50) ["too" large is OK]; $p$ small (allow $p < 0.2$ or a probability less than 0.2) | B1 | (1) |
| $H_0: p = 0.009$; $H_1: p > 0.009$ | B1 | (1) |
| Po(4.5) | B1 | |
| **Probability:** $P(X \geq 9) = 1 - P(X \leq 8) = 1 - 0.9597 = 0.0403$ | M1, A1 | |
| **Critical Region (CR):** $P(X \leq 7) = 0.9134$; $P(X \leq 8) = 0.9597$; CR: $X \geq 9$ | M1d, A1cso | |
| Reject $H_0$ or Significant. There is evidence that the farmer's claim is true. Or There is evidence that the proportion of eggs with a double yolk is $> 0.009$ | M1d, A1cso | (5) |
| | [7] | |
**Notes:**
| Item | Guidance |
|---|---|
| (b) B1 | both hypotheses correct. Must mention $p$ (or $\pi$). Words only is B0 |
| (c) B1 | writing or using Po(4.5) (Check their probs using tables if Po(4.5) is not seen) |
| | 1st M1: writing $1 - P(X \leq 8)$. May be implied by sight of $1 - 0.9597$ or for CR method: $P(X \leq 7) = 0.9134$ or $P(X \leq 8) = 0.9597$ |
| | 1st A1: for probability awrt 0.0403 or CR of $X > 8$ or $X \geq 9$ |
| | Allow awrt 0.9597 if accompanied by a correct comparison with 0.95 |
| | 2nd dM1 | correct statement that must agree with hypotheses. Dependent on B1. Contradictory non-contextual statements such as "not significant" so "reject $H_0$" score M0 |
| | 2nd A1cso | correct contextual statement. Depends on all other marks in (c) being scored. Must mention "farmer" and "claim" or "eggs" and "double yolk" |
| | NB | A correct calculation followed only by a correct contextual comment scores the final M1(implied) and A1 |
| | **If 2-tail hypotheses in (b)** | Score B0 in (b). Could score B1 M1A1 and M1 for a correct non-contextual comment but A0 since they should not be rejecting $H_0$ in this case (or they have scored A0 earlier so not cso) |
---\begin{enumerate}
\item (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.
\end{enumerate}
A farmer supplies a bakery with eggs. The manager of the bakery claims that the proportion of eggs having a double yolk is 0.009 The farmer claims that the proportion of his eggs having a double yolk is more than 0.009\\
(b) State suitable hypotheses for testing these claims.
In a batch of 500 eggs the baker records 9 eggs with a double yolk.\\
(c) Using a suitable approximation, test at the $5 \%$ level of significance whether or not this supports the farmer's claim.\\
\hfill \mbox{\textit{Edexcel S2 2014 Q1 [7]}}