| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Find critical region |
| Difficulty | Standard +0.3 This is a straightforward application of a one-tailed Poisson hypothesis test with standard procedures: finding a critical region using tables, comparing the observed value, and calculating P(Type I error) = significance level. While it requires understanding of hypothesis testing concepts, the execution is mechanical with no novel problem-solving required. Slightly easier than average due to the routine nature of the question. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c 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 |
|---|---|---|
| (i) \(H_0: \lambda = 4.8\), \(H_1: \lambda < 4.8\) | B1, M1 | For both \(H_0\) and \(H_1\). For evaluating \(P(0)\) and \(P(1)\) and \(P(2)\). |
| Under \(H_0\): \(P(0) = e^{-4.8} = 0.00823\), \(P(1) = 0.0395\), \(P(2) = 0.0948\) | ||
| Critical region is \(X = 0\) or \(1\) | M1, A1 (5 marks) | For stating/showing that \(P(0) + P(1) + P(2) > 10\%\) for critical region. For correct conclusion. |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(P(\text{Type I error}) = P(0) + P(1) = 0.0477\) | M1, A1 (2 marks) | For identifying correct outcome. For correct answer. |
(i) $H_0: \lambda = 4.8$, $H_1: \lambda < 4.8$ | B1, M1 | For both $H_0$ and $H_1$. For evaluating $P(0)$ and $P(1)$ and $P(2)$.
Under $H_0$: $P(0) = e^{-4.8} = 0.00823$, $P(1) = 0.0395$, $P(2) = 0.0948$ | |
Critical region is $X = 0$ or $1$ | M1, A1 (5 marks) | For stating/showing that $P(0) + P(1) + P(2) > 10\%$ for critical region. For correct conclusion.
SR If M0, M0 allow M1 for stating / showing $P(0) + P(1) < 10\%$
(ii) $P(\text{Type I error}) = P(0) + P(1) = 0.0477$ | M1, A1 (2 marks) | For identifying correct outcome. For correct answer.
---The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.
\begin{enumerate}[label=(\roman*)]
\item A hypothesis test at the 10% level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test. [5]
\item Find the probability of a Type I error. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2002 Q4 [7]}}