| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Standard +0.3 Part (a) is a direct application of Poisson probability formula with λ=1.25, requiring straightforward calculation of P(X<3). Part (b) is a standard hypothesis test for Poisson mean using given data (λ=5 over 4 weeks), requiring clear hypotheses, test statistic calculation, and comparison with critical values—all routine S2 procedures with no novel problem-solving required. Slightly above average due to the two-part structure and hypothesis testing component, but still a textbook exercise. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(X \sim P_0(1.25)\) | implied | B1 |
| \(P(X < 3) = P(0) + P(1) + P(2)\) or \(P(X \leq 2)\) | M1 | |
| \(= e^{-1.25}\left(1 + 1.25 + \frac{(1.25)^2}{2!}\right)\) | A1 | |
| \(= 0.868467 \ldots\) | awrt 0.868 or 0.8685 | A1 |
| (b) \(H_0: \lambda = 1.25\); \(H_1: \lambda \neq 1.25\) (or \(H_0: \lambda = 5\); \(H_1: \lambda \neq 5\)) | \(\lambda\) or \(\mu\) | B1; B1 |
| Let \(Y\) represent the number of breakdowns in 4 weeks | may be implied | B1 |
| Under \(H_0\), \(Y \sim P_0(5)\) | M1 | |
| \(P(Y \geq 11) = 1 - P(Y \leq 10)\) or \(P(X \geq 11) = 0.0137\) or \(P(X \geq 10) = 0.0318\) | One needed for M | M1 |
| \(= 0.0137\) | CR \(X \geq 11\) | A1 |
| \(0.0137 < 0.025, 0.0274 < 0.05, 0.9863 > 0.975, 0.9726 > 0.95\) or \(11 \geq 11\) | any allow % \(\sqrt{}\) from \(H_1\) context From their p | M1 |
| Evidence that the rate of breakdowns has changed/decreased | B1√ |
**(a)** $X \sim P_0(1.25)$ | implied | B1
$P(X < 3) = P(0) + P(1) + P(2)$ or $P(X \leq 2)$ | M1 |
$= e^{-1.25}\left(1 + 1.25 + \frac{(1.25)^2}{2!}\right)$ | A1 |
$= 0.868467 \ldots$ | awrt 0.868 or 0.8685 | A1 |
**(b)** $H_0: \lambda = 1.25$; $H_1: \lambda \neq 1.25$ (or $H_0: \lambda = 5$; $H_1: \lambda \neq 5$) | $\lambda$ or $\mu$ | B1; B1
Let $Y$ represent the number of breakdowns in 4 weeks | may be implied | B1
Under $H_0$, $Y \sim P_0(5)$ | M1 |
$P(Y \geq 11) = 1 - P(Y \leq 10)$ or $P(X \geq 11) = 0.0137$ or $P(X \geq 10) = 0.0318$ | One needed for M | M1
$= 0.0137$ | CR $X \geq 11$ | A1
$0.0137 < 0.025, 0.0274 < 0.05, 0.9863 > 0.975, 0.9726 > 0.95$ or $11 \geq 11$ | any allow % $\sqrt{}$ from $H_1$ context From their p | M1
Evidence that the rate of breakdowns has changed/decreased | B1√ |Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week. [4]
\end{enumerate}
Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Test, at the 5\% level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2006 Q4 [11]}}