| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Find power function or power value |
| Difficulty | Challenging +1.2 This is a straightforward application of hypothesis testing definitions to a Poisson distribution. Part (a) requires knowing that power = P(reject H₀ | λ) = P(X ≤ 3 | λ), part (b) is simply evaluating at λ=7 (the significance level), and part (c) uses the relationship that Type II error = 1 - power. All parts involve direct application of standard definitions with routine Poisson probability calculations, requiring no novel insight beyond textbook knowledge of hypothesis testing terminology. |
| Spec | 5.02i Poisson distribution: random events model5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks |
|---|---|
| Power \(= P(X \leq 3 / \lambda) = e^{-\lambda} + e^{-\lambda} \lambda + \frac{e^{-\lambda}2^2}{2} + \frac{e^{-\lambda}2^3}{6}\) | M1 |
| \(= \frac{e^{-\lambda}}{6}(6 + 6\lambda + 3\lambda^2 + \lambda^3)\) | A1 A1 |
Total: 3 marks
| Answer | Marks |
|---|---|
| CR is \(X \leq 3\) | M1 |
| Size \(= P[X \leq 3 / \lambda = 7] = 0.0818\) | A1 |
Total: 2 marks
| Answer | Marks |
|---|---|
| \(P(\text{Type II error}) = 1 - \text{power} = 1 - \frac{e^{-4}}{6}(6 + 6 \times 4 + 3 \times 4^2 + 4^3)\) | M1 |
| \(= 0.5665...\) | A1 |
Total: 2 marks
**Part a:**
Power $= P(X \leq 3 / \lambda) = e^{-\lambda} + e^{-\lambda} \lambda + \frac{e^{-\lambda}2^2}{2} + \frac{e^{-\lambda}2^3}{6}$ | M1 |
$= \frac{e^{-\lambda}}{6}(6 + 6\lambda + 3\lambda^2 + \lambda^3)$ | A1 A1 |
Total: 3 marks
**Part b:**
CR is $X \leq 3$ | M1 |
Size $= P[X \leq 3 / \lambda = 7] = 0.0818$ | A1 |
Total: 2 marks
**Part c:**
$P(\text{Type II error}) = 1 - \text{power} = 1 - \frac{e^{-4}}{6}(6 + 6 \times 4 + 3 \times 4^2 + 4^3)$ | M1 |
$= 0.5665...$ | A1 |
Total: 2 marks
---5. The number of tornadoes per year to hit a particular town follows a Poisson distribution with mean $\lambda$. A weatherman claims that due to climate changes the mean number of tornadoes per year has decreased. He records the number of tornadoes $x$ to hit the town last year.
To test the hypotheses $\mathrm { H } _ { 0 } : \lambda = 7$ and $\mathrm { H } _ { 1 } : \lambda < 7$, a critical region of $x \leq 3$ is used.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms $\lambda$ the power function of this test.
\item Find the size of this test.
\item Find the probability of a Type II error when $\lambda = 4$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2007 Q5 [7]}}