| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson with binomial combination |
| Difficulty | Standard +0.8 This S2 question combines Poisson probability calculations with hypothesis testing, requiring rate scaling (2 per 5m² to various areas), binomial-Poisson compound probability in part (a), and finding two-tailed critical regions. While methodical, it demands careful parameter adjustment, understanding of significance levels, and multiple computational steps—more demanding than standard single-concept S2 questions but not requiring novel insight. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a 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 probabilities5.02l Poisson conditions: for modelling |
\begin{enumerate}
\item In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per $5 \mathrm {~m} ^ { 2 }$
\end{enumerate}
The quality control manager randomly selects 12 pieces of cloth each of area $15 \mathrm {~m} ^ { 2 }$.\\
(a) Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects.
The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of $25 \mathrm {~m} ^ { 2 }$ of cloth from the next batch produced to test if there has been any change in the rate of defects.\\
(b) (i) Write down suitable hypotheses for this test.\\
(ii) Describe a suitable test statistic that the manager should use.\\
(iii) Explain what is meant by the critical region for this test.\\
(c) Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%\\
(d) Find the actual significance level for this test.
\hfill \mbox{\textit{Edexcel S2 2017 Q5 [14]}}