| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2001 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Hypothesis test on Poisson rate |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with standard bookwork: parts (a)-(b) are direct probability calculations, (c) requires scaling λ to 3 weeks (λ=6) and computing P(X≤5), and (d) is a routine hypothesis test. All steps follow textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \underline{0.0902}\) or \(\underline{0.090}\) or \(\underline{0.09}\) | H1, A1 | (2) |
| (b) \(P(X > 5) = 1 - P(X \leq 5) = 1 - 0.9834 = \underline{0.0166}\) | M1, A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(Y \leq 5) = \underline{0.4457}\) | B1, M1, A1 | (3) |
| (d) \(H_0: \lambda = 2\) (or \(\mu = 8\)); \(H_1: \lambda < 2\) (or \(\mu < 8\)) | B1; B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(R \leq 3) = 0.0424\) [C.R. \(\leq 3\) or prob \(< 5\%\)] significant | M1, A1 | |
| There is evidence that the rate of requests has decreased | A1✓ | (5) — (12) |
## Question 5:
$X$ = no. of requests for bulbs in a week, $X \sim Po(2)$
**(a)** $P(X = 4) = \frac{e^{-2} \cdot 2^4}{4!}$ or $[P(X \leq 4) - P(X \leq 3)]$, $= 0.9473 - 0.8571$
$= \underline{0.0902}$ or $\underline{0.090}$ or $\underline{0.09}$ | H1, A1 | **(2)**
**(b)** $P(X > 5) = 1 - P(X \leq 5) = 1 - 0.9834 = \underline{0.0166}$ | M1, A1 | **(2)**
**(c)** $Y$ = no. of requests in 3 weeks, $Y \sim Po(6)$
$P(Y \leq 5) = \underline{0.4457}$ | B1, M1, A1 | **(3)**
**(d)** $H_0: \lambda = 2$ (or $\mu = 8$); $H_1: \lambda < 2$ (or $\mu < 8$) | B1; B1 |
$R$ = no. of requests in 4 weeks, $R \sim Po(8)$
$P(R \leq 3) = 0.0424$ [C.R. $\leq 3$ or prob $< 5\%$] significant | M1, A1 |
There is evidence that the rate of requests has decreased | A1✓ | **(5)** — **(12)**
---5. The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week.
Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is
\begin{enumerate}[label=(\alph*)]
\item exactly 4,
\item more than 5 .
Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.
\item Find the probability that the department can meet all requests for replacement light bulbs before the end of term.
The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.
\item Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2001 Q5 [12]}}