| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question requiring binomial table lookups and understanding of critical regions. Part (a) involves splitting 5% between two tails, part (b) is a one-tailed test. Both require careful cumulative probability calculations but follow routine procedures taught in S2 with no novel problem-solving required. Slightly easier than average due to being a textbook application. |
| Spec | 2.04b Binomial distribution: as model B(n,p)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) From tables, extreme 2.5% tails are given by \(X \leq 3\) and \(X \geq 13\), so this is the critical region | M1 A1 A1 A1 |
| (b) The bottom 5% tail is still given by \(X \leq 3\); region is \(\{0, 1, 2, 3\}\) | M1 M1 A1 |
(a) From tables, extreme 2.5% tails are given by $X \leq 3$ and $X \geq 13$, so this is the critical region | M1 A1 A1 A1 |
(b) The bottom 5% tail is still given by $X \leq 3$; region is $\{0, 1, 2, 3\}$ | M1 M1 A1 |
**Total: 7 marks**
---3. The random variable $X$ is modelled by a binomial distribution $\mathrm { B } ( n , p )$, with $n = 20$ and $p$ unknown. It is suspected that $p = 0 \cdot 4$.
\begin{enumerate}[label=(\alph*)]
\item Find the critical region for the test of $\mathrm { H } _ { 0 } : p = 0.4$ against $\mathrm { H } _ { 1 } : p \neq 0.4$, at the $5 \%$ significance level.
\item Find the critical region if, instead, the alternative hypothesis is $\mathrm { H } _ { 1 } : p < 0.4$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [7]}}