| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson hypothesis testing with standard procedures: defining critical regions, finding significance levels from tables, and conducting a one-tailed test. The multi-part structure and definitional questions add length but not conceptual difficulty—all steps are routine for S2 students with no novel problem-solving required. |
| Spec | 2.05a 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 probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The range of values/region/area/set of values of the test statistic that would lead you to reject \(H_0\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The probability of incorrectly rejecting \(H_0\), or probability of rejecting \(H_0\) when \(H_0\) is true | B1 | (2 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X \sim \text{Po}(8)\) | M1 | May be implied by correct critical region |
| \(P(X \leq 4) = 0.0996\), \(P(X \leq 3) = 0.0424\) | ||
| Critical region \([0,3]\) | A1 | Allow \(0 \leq X \leq 3\) or \(CR \leq 3\) or \(X \leq 3\); not \(P(X\leq 3)\) alone |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| awrt \(0.0424\) | B1 | (3 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \lambda = 8\) (or \(\mu = 8\)) | B1 | Both hypotheses correct; must use \(\lambda\) or \(\mu\) |
| \(H_1: \lambda > 8\) (or \(\mu > 8\)) | ||
| \(P(X \geq 13) = 1 - P(X \leq 12)\) or \(P(X \leq 13) = 0.9658\) or \(P(X \geq 14) = 0.0342\) | M1 | |
| \(= 0.0638\); \(CR\ X \geq 14\) | A1 | |
| Insufficient evidence to reject \(H_0\)/not significant/not in critical region | M1 dep | |
| There is insufficient evidence of an increase/change in the rate/number of sales per month, or the estate agents claim is incorrect | A1 | (5 marks total, 10 overall) |
# Question 7:
## Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The range of values/region/area/set of values of the test statistic that would lead you to reject $H_0$ | B1 | |
## Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The probability of incorrectly rejecting $H_0$, or probability of rejecting $H_0$ when $H_0$ is true | B1 | (2 marks total) |
## Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim \text{Po}(8)$ | M1 | May be implied by correct critical region |
| $P(X \leq 4) = 0.0996$, $P(X \leq 3) = 0.0424$ | | |
| Critical region $[0,3]$ | A1 | Allow $0 \leq X \leq 3$ or $CR \leq 3$ or $X \leq 3$; not $P(X\leq 3)$ alone |
## Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| awrt $0.0424$ | B1 | (3 marks total) |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \lambda = 8$ (or $\mu = 8$) | B1 | Both hypotheses correct; must use $\lambda$ or $\mu$ |
| $H_1: \lambda > 8$ (or $\mu > 8$) | | |
| $P(X \geq 13) = 1 - P(X \leq 12)$ or $P(X \leq 13) = 0.9658$ or $P(X \geq 14) = 0.0342$ | M1 | |
| $= 0.0638$; $CR\ X \geq 14$ | A1 | |
| Insufficient evidence to reject $H_0$/not significant/not in critical region | M1 dep | |
| There is insufficient evidence of an increase/change in the **rate/number** of sales per month, **or** the estate **agents** claim is incorrect | A1 | (5 marks total, 10 overall) |7. (a) Explain briefly what you understand by
\begin{enumerate}[label=(\roman*)]
\item a critical region of a test statistic,
\item the level of significance of a hypothesis test.\\
(b) An estate agent has been selling houses at a rate of 8 per month. She believes that the rate of sales will decrease in the next month.
\item Using a $5 \%$ level of significance, find the critical region for a one tailed test of the hypothesis that the rate of sales will decrease from 8 per month.
\item Write down the actual significance level of the test in part (b)(i).
The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.\\
(c) Test the estate agent's claim at the $5 \%$ level of significance. State your hypotheses clearly.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2012 Q7 [10]}}