| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Expected value and most likely value |
| Difficulty | Moderate -0.8 This is a straightforward application of binomial distribution basics: parts (a) and (c) require simple expectation calculations (np), part (b) is a standard one-tailed hypothesis test with clear setup, and part (d) involves binomial probability calculation. All steps are routine S2 techniques with no conceptual challenges or novel problem-solving required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks |
|---|---|
| (a) \(30 \times \frac{1}{4} = 7.5\) | M1 A1 |
| (b) \(X \sim B(30, p)\) with \(H_0: p = 0.25\) and \(H_1: p > 0.25\) | B1 B1 |
| Under \(H_0\), \(P(X \geq 15) = 1 - 0.9973 = 0.0027 < 5\%\), so reject \(H_0\) | M1 A1 A1 |
| (c) \(30 \times 0.9 = 27\) | B1 |
| (d) \(P(\text{More than 25 right}) = P(X < 5) \text{ in } B(30, 0.1) = 0.825\) | M1 A1 |
(a) $30 \times \frac{1}{4} = 7.5$ | M1 A1 |
(b) $X \sim B(30, p)$ with $H_0: p = 0.25$ and $H_1: p > 0.25$ | B1 B1 |
Under $H_0$, $P(X \geq 15) = 1 - 0.9973 = 0.0027 < 5\%$, so reject $H_0$ | M1 A1 A1 |
(c) $30 \times 0.9 = 27$ | B1 |
(d) $P(\text{More than 25 right}) = P(X < 5) \text{ in } B(30, 0.1) = 0.825$ | M1 A1 |
**Total: 10 marks**
---3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.
\begin{enumerate}[label=(\alph*)]
\item Phil ticks answers at random. Find how many of the 30 he should expect to get right.
\item If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a $5 \%$ significance level.
Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
\item Determine the expected number of answers that Sarah will get right.
\item Find the probability that Sarah gets more than 25 correct answers out of 30.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [10]}}