| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Comment on test validity or assumptions |
| Difficulty | Standard +0.8 This is a chi-squared goodness-of-fit test for a Poisson distribution requiring multiple steps: estimating the parameter λ from data, calculating expected frequencies, combining cells appropriately, computing the test statistic, determining degrees of freedom (n-2 due to parameter estimation), and interpreting the result. While methodical, it requires careful execution of several statistical procedures and understanding of when to pool categories, making it moderately challenging but still a standard S3 question type. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Number of breakdowns | 0 | 1 | 2 | \(>2\) |
| Frequency | 38 | 32 | 10 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): Poisson distribution is a suitable model; \(H_1\): Poisson distribution is not a suitable model | both | B1 |
| From these data \(\lambda = \frac{52}{80} = 0.65\) | M1 A1 | |
| Expected frequencies: 41.76, 27.15, 8.82, 2.27 | \(80 \times P(X = x)\) | M1 A2/1/0 |
| Amalgamation | M1; B1 \(\checkmark\); B1 \(\checkmark\) | |
| \(\alpha = 0.05\), \(\nu = 3 - 1 - 1 = 1\); critical value = 3.841 | ||
| \(\sum \frac{(O - E)^2}{E} = 1.312\) | M1 A1 | |
| Since 1.312 is not in the critical region there is insufficient evidence to reject \(H_0\) and we can conclude that the Poisson model is a suitable one. | M1 A1 \(\checkmark\) | |
| Total: 13 marks |
## Part (a)
| $H_0$: Poisson distribution is a suitable model; $H_1$: Poisson distribution is not a suitable model | both | B1 |
| From these data $\lambda = \frac{52}{80} = 0.65$ | M1 A1 |
| Expected frequencies: 41.76, 27.15, 8.82, 2.27 | $80 \times P(X = x)$ | M1 A2/1/0 |
| Amalgamation | M1; B1 $\checkmark$; B1 $\checkmark$ |
| $\alpha = 0.05$, $\nu = 3 - 1 - 1 = 1$; critical value = 3.841 | |
| $\sum \frac{(O - E)^2}{E} = 1.312$ | M1 A1 |
| Since 1.312 is not in the critical region there is insufficient evidence to reject $H_0$ and we can conclude that the Poisson model is a suitable one. | M1 A1 $\checkmark$ |
| **Total: 13 marks** |
---Breakdowns on a certain stretch of motorway were recorded each day for 80 consecutive days. The results are summarised in the table below.
\begin{tabular}{|c|c|c|c|c|}
\hline
Number of breakdowns & 0 & 1 & 2 & $>2$ \\
\hline
Frequency & 38 & 32 & 10 & 0 \\
\hline
\end{tabular}
It is suggested that the number of breakdowns per day can be modelled by a Poisson distribution.
Using a 5% level of significance, test whether or not the Poisson distribution is a suitable model for these data. State your hypotheses clearly. [13]
\hfill \mbox{\textit{Edexcel S3 Q4 [13]}}