| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Moderate -0.3 This is a straightforward binomial hypothesis test with standard procedures: setting up H₀ and H₁, calculating probabilities under B(20, 0.5), and comparing to significance levels. Part (b) requires finding critical values which is slightly more involved than part (a), but both are routine S2 textbook exercises requiring recall of method rather than problem-solving insight. |
| Spec | 2.05a 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) \(X \sim B(20, p)\); \(H_0: p = \frac{1}{2}\), \(H_1: p \ne \frac{1}{2}\) | B1 B1 |
| Assuming \(H_0\), \(P(X \ge 16 \text{ or } X \le 4) = 0.0059 \times 2 = 0.0118\) | M1 M1 A1 |
| \(> 1\%\), so do not reject \(H_0\) at \(1\%\) level | A1 |
| (b) For significance at \(0.1\%\) level, would need \(X \le 2\) or \(X \ge 18\) | B1 B1 |
(a) $X \sim B(20, p)$; $H_0: p = \frac{1}{2}$, $H_1: p \ne \frac{1}{2}$ | B1 B1 |
Assuming $H_0$, $P(X \ge 16 \text{ or } X \le 4) = 0.0059 \times 2 = 0.0118$ | M1 M1 A1 |
$> 1\%$, so do not reject $H_0$ at $1\%$ level | A1 |
(b) For significance at $0.1\%$ level, would need $X \le 2$ or $X \ge 18$ | B1 B1 |
**Total: 8 marks**3. A coin is tossed 20 times, giving 16 heads.
\begin{enumerate}[label=(\alph*)]
\item Test at the $1 \%$ significance level whether the coin is fair, stating your hypotheses clearly.
\item Find the critical region for the same test at the $0.1 \%$ significance level.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [8]}}