| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2003 |
| Session | January |
| Marks | 20 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Critique inappropriate sampling methods |
| Difficulty | Moderate -0.8 This is a straightforward S2 question testing standard concepts: parts (a)-(d) are basic definitions requiring recall only, part (e) is direct binomial probability calculation, part (f) is a routine one-tailed hypothesis test, and part (g) is standard normal approximation to binomial. The critical thinking element about sampling bias (using only timely payers) is mentioned but not deeply explored. All techniques are textbook exercises with no novel problem-solving required. |
| Spec | 2.01a Population and sample: terminology2.01b Informal inferences: from samples2.01c Sampling techniques: simple random, opportunity, etc2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| All subscribers to the magazine | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| A list of all members that had paid their subscriptions | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Members who have paid | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Advantage: total accuracy | B1 | |
| Disadvantage: time consuming to obtain data and analyse it | B1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(X\) represent the number agreeing to change the name; \(X \sim B(25, 0.4)\) | B1 | |
| \(P(X = 10) = P(X \leq 10) - P(X \leq 9) = 0.1612\) | M1 A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.40,\quad H_1: p < 0.40\) | B1, B1 | |
| \(P(X \leq 6) = 0.0736 > 0.05 \Rightarrow\) not significant | M1 A1 | |
| No reason to reject \(H_0\); conclude \(\%\) is less than the editor believes | A1 | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(X\) represent the number agreeing to change the name; \(X \sim B(200, 0.4)\) | ||
| \(P(71 \leq X < 83) \approx P(70.5 \leq Y < 82.5)\) where \(Y \sim N(80, 48)\) | B1 B1 | |
| \(\approx P\!\left(\frac{70.5 - 80}{\sqrt{48}} \leq X < \frac{82.5 - 80}{\sqrt{48}}\right)\) | M1 M1 | |
| \(\approx P(-1.37 \leq X < 0.36)\) | A1 A1 | |
| \(= 0.5533\) | A1 | (7 marks) |
# Question 6:
## Part (a)
| All subscribers to the magazine | B1 | (1 mark) |
## Part (b)
| A list of all members that had paid their subscriptions | B1 | (1 mark) |
## Part (c)
| Members who have paid | B1 | (1 mark) |
## Part (d)
| Advantage: total accuracy | B1 | |
| Disadvantage: time consuming to obtain data and analyse it | B1 | (2 marks) |
## Part (e)
| Let $X$ represent the number agreeing to change the name; $X \sim B(25, 0.4)$ | B1 | |
| $P(X = 10) = P(X \leq 10) - P(X \leq 9) = 0.1612$ | M1 A1 | (3 marks) |
## Part (f)
| $H_0: p = 0.40,\quad H_1: p < 0.40$ | B1, B1 | |
| $P(X \leq 6) = 0.0736 > 0.05 \Rightarrow$ not significant | M1 A1 | |
| No reason to reject $H_0$; conclude $\%$ is less than the editor believes | A1 | (5 marks) |
## Part (g)
| Let $X$ represent the number agreeing to change the name; $X \sim B(200, 0.4)$ | | |
| $P(71 \leq X < 83) \approx P(70.5 \leq Y < 82.5)$ where $Y \sim N(80, 48)$ | B1 B1 | |
| $\approx P\!\left(\frac{70.5 - 80}{\sqrt{48}} \leq X < \frac{82.5 - 80}{\sqrt{48}}\right)$ | M1 M1 | |
| $\approx P(-1.37 \leq X < 0.36)$ | A1 A1 | |
| $= 0.5533$ | A1 | (7 marks) |6. A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that $40 \%$ of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.
\begin{enumerate}[label=(\alph*)]
\item Define the population associated with the magazine.
\item Suggest a suitable sampling frame for the survey.
\item Identify the sampling units.
\item Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey.
As a pilot study the editor took a random sample of 25 subscribers.
\item Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name.
In fact only 6 subscribers agreed to the name being changed.
\item Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not the percentage agreeing to the change is less that the editor believes.
The full survey is to be carried out using 200 randomly chosen subscribers.
\item Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed.
\section*{END}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2003 Q6 [20]}}