| 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 | Sequential or two-stage test design |
| Difficulty | Challenging +1.8 This S4 two-stage hypothesis test requires understanding of sequential testing procedures (uncommon in standard A-level), careful probability calculations with Poisson distributions across multiple scenarios, and correct interpretation of size/power. The multi-stage decision tree and conditional probability structure elevate this significantly above routine hypothesis testing questions, though the calculations themselves are methodical rather than requiring deep insight. |
| Spec | 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02m Poisson: mean = variance = lambda |
**Total: 7 marks**
The cloth produced by a certain manufacturer has defects that occur randomly at a constant rate of $\lambda$ per square metre. If $\lambda$ is thought to be greater than $1.5$ then action has to be taken.
Using $H_0: \lambda = 1.5$ and $H_1: \lambda > 1.5$ a quality control officer takes a $4 \text{ m}^2$ sample of cloth and rejects $H_0$ if there are $11$ or more defects. If there are $8$ or fewer defects she accepts $H_0$.
If there are $9$ or $10$ defects a second sample of $4 \text{ m}^2$ is taken and $H_0$ is rejected if there are $11$ or more defects in this second sample, otherwise it is accepted.
**(a) Find the size of this test.**
(4 marks)
**(b) Find the power of this test when $\lambda = 2$**
(3 marks)
---2. The cloth produced by a certain manufacturer has defects that occur randomly at a constant rate of $\lambda$ per square metre. If $\lambda$ is thought to be greater than 1.5 then action has to be taken.
Using $\mathrm { H } _ { 0 } : \lambda = 1.5$ and $\mathrm { H } _ { 1 } : \lambda > 1.5$ a quality control officer takes a $4 \mathrm {~m} ^ { 2 }$ sample of cloth and rejects $\mathrm { H } _ { 0 }$ if there are 11 or more defects. If there are 8 or fewer defects she accepts $\mathrm { H } _ { 0 }$. If there are 9 or 10 defects a second sample of $4 \mathrm {~m} ^ { 2 }$ is taken and $\mathrm { H } _ { 0 }$ is rejected if there are 11 or more defects in this second sample, otherwise it is accepted.
\begin{enumerate}[label=(\alph*)]
\item Find the size of this test.
\item Find the power of this test when $\lambda = 2$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2014 Q2 [7]}}