| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Moderate -0.8 This is a straightforward S2 question testing basic sampling concepts and a standard normal approximation to binomial hypothesis test. Parts (a) and (b) require simple recall of sampling theory. Parts (c) and (d) involve routine application of the normal approximation with clearly stated parameters (n=120, p=0.05), finding critical values from tables, and stating significance level—all standard textbook procedures with no problem-solving insight required. |
| Spec | 2.01a Population and sample: terminology2.01c Sampling techniques: simple random, opportunity, etc2.04d Normal approximation to binomial2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks |
|---|---|
| e.g. quicker; may not be able to get all pupils to respond | B2 |
| Answer | Marks |
|---|---|
| school roll | B1 |
| Answer | Marks |
|---|---|
| let \(X\) = no. of students who play tennis \(\therefore X \sim B(120, \frac{1}{30})\) | M1 |
| \(H_0: p = \frac{1}{30}\) | B1 |
| \(H_1: p \ne \frac{1}{30}\) | |
| Using Po approx. \(X \approx \text{Po}(6)\) | M1 |
| \(P(X \le 2) = 0.0620\); \(P(X \le 10) = 0.9574\) | M1 A1 |
| \(\therefore\) C.R. is \(X \le 2\) or \(X \ge 11\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.0620 + 0.0426 = 0.1046\) | A1 | (10) |
**Part (a)**
e.g. quicker; may not be able to get all pupils to respond | B2 |
**Part (b)**
school roll | B1 |
**Part (c)**
let $X$ = no. of students who play tennis $\therefore X \sim B(120, \frac{1}{30})$ | M1 |
$H_0: p = \frac{1}{30}$ | B1 |
$H_1: p \ne \frac{1}{30}$ |
Using Po approx. $X \approx \text{Po}(6)$ | M1 |
$P(X \le 2) = 0.0620$; $P(X \le 10) = 0.9574$ | M1 A1 |
$\therefore$ C.R. is $X \le 2$ or $X \ge 11$ | A1 |
**Part (d)**
$0.0620 + 0.0426 = 0.1046$ | A1 | (10)
---A teacher wants to investigate the sports played by students at her school in their free time. She decides to ask a random sample of 120 pupils to complete a short questionnaire.
\begin{enumerate}[label=(\alph*)]
\item Give two reasons why the teacher might choose to use a sample survey rather than a census. [2 marks]
\item Suggest a suitable sampling frame that she could use. [1 mark]
\end{enumerate}
The teacher believes that 1 in 20 of the students play tennis in their free time. She uses the data collected from her sample to test if the proportion is different from this.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Using a suitable approximation and stating the hypotheses that she should use, find the critical region for this test. The probability for each tail of the region should be as close as possible to 5\%. [6 marks]
\item State the significance level of this test. [1 mark]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [10]}}