| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.3 This is a straightforward S2 hypothesis test question with standard structure: calculate mean/variance from grouped data, justify Poisson model, then perform a one-tailed test with clearly defined parameters. The scaling (15 minutes → 5 minutes) is routine, and all steps follow textbook procedures with no novel insight required. Slightly easier than average due to clear guidance and standard technique application. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b 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 probabilities5.02l Poisson conditions: for modelling |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| Number of intervals | 7 | 13 | 25 | 19 | 15 | 10 | 7 | 4 |
| Answer | Marks |
|---|---|
| (a) Mean \(= \frac{300}{100} = 3\) | M1 A1 |
| Variance \(= \frac{1222}{100} - 3^2 = 3.22\) | M1 A1 |
| (b) Mean \(\approx\) Variance, and positive skewness | B1 B1 |
| (c) \(H_0: \lambda = 3\) and \(H_1: \lambda > 3\) | B1 B1 |
| Under \(H_0\), no. of lorries in 15 minutes \(\sim \text{Po}(9)\) | M1 A1 |
| \(P(X \geq 18) = 1 - 0.995 = 0.005 < 1\%\) so reject \(H_0\) at 1% level, i.e. accept that mean has increased | M1 A1 A1 |
(a) Mean $= \frac{300}{100} = 3$ | M1 A1 |
Variance $= \frac{1222}{100} - 3^2 = 3.22$ | M1 A1 |
(b) Mean $\approx$ Variance, and positive skewness | B1 B1 |
(c) $H_0: \lambda = 3$ and $H_1: \lambda > 3$ | B1 B1 |
Under $H_0$, no. of lorries in 15 minutes $\sim \text{Po}(9)$ | M1 A1 |
$P(X \geq 18) = 1 - 0.995 = 0.005 < 1\%$ so reject $H_0$ at 1% level, i.e. accept that mean has increased | M1 A1 A1 |
**Total: 13 marks**
---5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results:
\begin{center}
\begin{tabular}{ | l | | c | c | c | c | c | c | c | c | }
\hline
\begin{tabular}{ c }
Number of lorries in \\
five-minute interval, $X$ \\
\end{tabular} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
Number of intervals & 7 & 13 & 25 & 19 & 15 & 10 & 7 & 4 \\
\hline
\end{tabular}
\end{center}
Q. 5 continued on next page ...
\section*{STATISTICS 2 (A) TEST PAPER 9 Page 2}
\begin{enumerate}
\item continued ...\\
(a) Show that the mean of $X$ is 3 , and find the variance of $X$.\\
(b) Give two reasons for thinking that $X$ can be modelled by a Poisson distribution. (2 marks)
\end{enumerate}
After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.\\
(c) Test whether the group's claim is valid. Work at the $5 \%$ significance level, and state your hypotheses clearly.\\
\hfill \mbox{\textit{Edexcel S2 Q5 [13]}}