| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Test with normal approximation |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question requiring routine application of Poisson test procedures and normal approximation. While it has multiple parts (14 marks total), each component follows textbook methods: one-tailed Poisson test with critical value, stating standard Poisson assumptions, and applying normal approximation for larger parameter. No novel insight or complex problem-solving required, just systematic application of learned techniques, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial2.04f Find normal probabilities: Z transformation2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \lambda = 7\) | \(H_1: \lambda > 7\) | B1 |
| \(X =\) number of visits. \(X \sim \text{Po}(7)\) | B1 | |
| \(P(X \geq 10) = 1 - P(X \leq 9) = 0.1695\) or \(1 - P(X \leq 10) = 0.0985\); \(1 - P(X \leq 9) = 0.1695\) CR \(X \geq 11\) | M1, A1 | |
| \(0.1695 > 0.10\), Not significant or it is not in the critical region or do not reject \(H_0\); The rate of visits on a Saturday is not greater/is unchanged | M1, A1 no ft | (7 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(X = 11\) | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| (The visits occur) randomly/independently or singly or constant rate | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| [\(H_0: \lambda = 7\) \(H_1: \lambda > 7\) (or \(H_0: \lambda = 14\) \(H_1: \lambda > 14\))] | B1;B1 | |
| \(X \sim N_r(14,14)\) | ||
| \(P(X \geq 20) = P(z \geq \frac{19.5-14}{\sqrt{14}})\) | +/- 0.5, stand | M1, M1 |
| \(= P(z \geq 1.47) = 0.0708\) or \(z = 1.2816\) | A1dep both M; A1dep 2nd M | (6 marks) |
| \(0.0708 < 0.10\) therefore significant. The rate of visits is greater on a Saturday | A1dep 2nd M |
## Part (a)(i)
$H_0: \lambda = 7$ | $H_1: \lambda > 7$ | B1 |
$X =$ number of visits. $X \sim \text{Po}(7)$ | B1 |
$P(X \geq 10) = 1 - P(X \leq 9) = 0.1695$ or $1 - P(X \leq 10) = 0.0985$; $1 - P(X \leq 9) = 0.1695$ CR $X \geq 11$ | M1, A1 |
$0.1695 > 0.10$, Not significant or it is not in the critical region or do not reject $H_0$; The rate of visits on a Saturday is not greater/is unchanged | M1, A1 no ft | (7 marks)
## Part (a)(ii)
$X = 11$ | B1 | (1 mark)
## Part (b)
(The visits occur) randomly/independently or singly or constant rate | B1 | (1 mark)
## Part (c)
[$H_0: \lambda = 7$ $H_1: \lambda > 7$ (or $H_0: \lambda = 14$ $H_1: \lambda > 14$)] | B1;B1 |
$X \sim N_r(14,14)$ |
$P(X \geq 20) = P(z \geq \frac{19.5-14}{\sqrt{14}})$ | +/- 0.5, stand | M1, M1 |
$= P(z \geq 1.47) = 0.0708$ or $z = 1.2816$ | A1dep both M; A1dep 2nd M | (6 marks)
$0.0708 < 0.10$ therefore significant. The rate of visits is greater on a Saturday | A1dep 2nd M |
---A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Test, at the 10\% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
\item State the minimum number of visits required to obtain a significant result.
\end{enumerate}
[7]
\item State an assumption that has been made about the visits to the server. [1]
\end{enumerate}
In a random two minute period on a Saturday the web server is visited 20 times.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Using a suitable approximation, test at the 10\% level of significance, whether or not the rate of visits is greater on a Saturday. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2009 Q6 [14]}}