Edexcel S2 — Question 2 5 marks

Exam BoardEdexcel
ModuleS2 (Statistics 2)
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPoisson distribution
TypePoisson hypothesis test
DifficultyStandard +0.3 This is a straightforward hypothesis testing question requiring students to find a critical region for a one-tailed Poisson test. While it involves cumulative probability calculations and understanding of significance levels, it's a standard textbook procedure with no conceptual surprises—slightly easier than average since it's purely mechanical application of the critical region method.
Spec5.05a Sample mean distribution: central limit theorem

It is thought that a random variable \(X\) has a Poisson distribution whose mean, \(\lambda\), is equal to 8. Find the critical region to test the hypothesis \(H_0 : \lambda = 8\) against the hypothesis \(H_1 : \lambda < 8\), working at the 1\% significance level. [5 marks]

AnswerMarks Guidance
\(X \sim \text{Po}(\lambda)\) Under \(H_0\), \(P(X \leq 2) > 1\%\), \(P(X \leq 1) < 1\%\)M1 A1 A1
\(X = 0\) or \(X = 1\) will lead to rejection of \(H_0\) at 1% levelM1 A1 Total: 5 marks
$X \sim \text{Po}(\lambda)$ Under $H_0$, $P(X \leq 2) > 1\%$, $P(X \leq 1) < 1\%$ | M1 A1 A1 |

$X = 0$ or $X = 1$ will lead to rejection of $H_0$ at 1% level | M1 A1 | **Total: 5 marks**
It is thought that a random variable $X$ has a Poisson distribution whose mean, $\lambda$, is equal to 8.
Find the critical region to test the hypothesis $H_0 : \lambda = 8$ against the hypothesis $H_1 : \lambda < 8$,
working at the 1\% significance level. [5 marks]

\hfill \mbox{\textit{Edexcel S2  Q2 [5]}}