| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Explain sampling frames and units |
| Difficulty | Moderate -0.8 Part (a) is basic terminology recall requiring no calculation. Part (b) is a standard normal approximation to binomial hypothesis test with symmetric two-tailed critical region—routine S2 material following a textbook template. Part (c) is immediate from part (b). The question requires minimal problem-solving beyond applying memorized procedures. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks |
|---|---|
| Frame – list of all learners she has taught | B1 |
| Units – individual learners | B1 |
| Let \(X =\) no. of learners failing first 2 attempts \(\therefore X \sim B(120, \frac{1}{30})\) | M1 |
| \(H_0 : p = \frac{1}{30}\) \(H_1 : p \neq \frac{1}{30}\) | B1 |
| Po approx. \(X \sim Po(6)\) | M1 |
| \(P(X \leq 1) = 0.0174\), \(P(X \leq 11) = 0.9799\) | M1 A1 |
| \(\therefore\) C.R. is \(X \leq 1\) or \(X \geq 12\) | A1 |
| \(0.0174 + 0.0201 = 0.0375\) | A1 |
Frame – list of all learners she has taught | B1 |
Units – individual learners | B1 |
Let $X =$ no. of learners failing first 2 attempts $\therefore X \sim B(120, \frac{1}{30})$ | M1 |
$H_0 : p = \frac{1}{30}$ $H_1 : p \neq \frac{1}{30}$ | B1 |
Po approx. $X \sim Po(6)$ | M1 |
$P(X \leq 1) = 0.0174$, $P(X \leq 11) = 0.9799$ | M1 A1 |
$\therefore$ C.R. is $X \leq 1$ or $X \geq 12$ | A1 |
$0.0174 + 0.0201 = 0.0375$ | A1 |
---2. A driving instructor keeps records of all the learners she has taught. In order to analyse her success rate she wishes to take a random sample of 120 of these learners.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable sampling frame and identify the sampling units.
She believes that only 1 in 20 of the people she teaches fail to pass their test in their first two attempts. She decides to use her sample to test whether or not the proportion is different from this.
\item Using a suitable approximation and stating clearly the hypotheses she should use, find the largest critical region for this test such that the probability in each "tail" is less than $2.5 \%$.
\item State the significance level of this test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q2 [9]}}