| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Find Type II error probability |
| Difficulty | Challenging +1.2 This is a Further Maths Statistics question requiring understanding of hypothesis testing with Poisson distributions, including calculating Type I and Type II error probabilities. While it involves multiple concepts (defining errors, finding critical regions, calculating probabilities under different parameter values), the calculations are relatively straightforward once the setup is understood. The Poisson probability calculations are mechanical, and the conceptual framework is standard for S4 level, making it moderately above average difficulty but not requiring exceptional insight. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (i) Rejecting \(H_0\) when \(H_0\) is true | B1 | |
| (ii) Accepting \(H_0\) when \(H_0\) is false | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(X \sim \text{Po}(6)\), \(H_0: \lambda = 6\), \(H_1: \lambda < 6\) | B1 | |
| Critical region: \(P(X \leqslant c) \leqslant 0.05\) | M1 | |
| \(P(X \leqslant 2) = 0.0620\), \(P(X \leqslant 1) = 0.0174\) | A1 | |
| Critical region is \(X \leqslant 1\), P(Type I error) \(= 0.0174\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(X \sim \text{Po}(4)\) | M1 | |
| \(P(\text{Type II error}) = P(X \geqslant 2 \mid \lambda = 4) = 1 - P(X \leqslant 1)\) | M1 | |
| \(= 1 - 0.0916 = 0.9084\) | A1 |
# Question 2:
**(a)**
| Answer | Mark | Guidance |
|--------|------|----------|
| (i) Rejecting $H_0$ when $H_0$ is true | B1 | |
| (ii) Accepting $H_0$ when $H_0$ is false | B1 | |
**(b)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $X \sim \text{Po}(6)$, $H_0: \lambda = 6$, $H_1: \lambda < 6$ | B1 | |
| Critical region: $P(X \leqslant c) \leqslant 0.05$ | M1 | |
| $P(X \leqslant 2) = 0.0620$, $P(X \leqslant 1) = 0.0174$ | A1 | |
| Critical region is $X \leqslant 1$, P(Type I error) $= 0.0174$ | A1 | |
**(c)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $X \sim \text{Po}(4)$ | M1 | |
| $P(\text{Type II error}) = P(X \geqslant 2 \mid \lambda = 4) = 1 - P(X \leqslant 1)$ | M1 | |
| $= 1 - 0.0916 = 0.9084$ | A1 | |
---2. (a) Define
\begin{enumerate}[label=(\roman*)]
\item a Type I error,
\item a Type II error.
Rolls of material, manufactured by a machine, contain defects at a mean rate of 6 per roll.
The machine is modified. A single roll is selected at random and a test is carried out to see whether or not the mean number of defects per roll has decreased. The significance level is chosen to be as close as possible to $5 \%$.\\
(b) Calculate the probability of a Type I error for this test.\\
(c) Given that the true mean number of defects per roll of material made by the machine is now 4, calculate the probability of a Type II error.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2014 Q2 [7]}}