| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type II error probability |
| Difficulty | Challenging +1.2 This question requires understanding of hypothesis testing mechanics (finding critical region for given α, then calculating P(Type II error) under alternative distribution) and conceptual knowledge of how Type II errors behave. While it involves multiple steps and binomial probability calculations, these are standard S2 procedures without requiring novel insight or complex problem-solving—slightly above average due to the multi-part nature and need to work with two different probability distributions. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(B(14, 0.25)\): Critical region \(\geq 7\) | M1 | Use \(B(14, 0.25)\) and find \(r\) for an upper tail. *All marks need upper tail*. e.g. CV 5 or 6 or 7, or .1117, .0383, .0103, 0.8883, 0.9617, .9897 |
| \(CR \geq 7\) or \(AR \leq 6\) stated or clearly implied | A1 | Not just "CV = 7" |
| \(B(14, 0.4)\): \(P(\leq 6) = 0.6925\) | M1 | Find \(P\)(in AR when \(p = 0.4\)) *[indept of M1]*. *Not* \(P(\geq r)\), e.g. final answer 0.3075 |
| \(= 0.6925\) | A1 | Answer 0.692 or 0.693 or a.r.t. 0.6925 or 0.6924 only, *not* isw \([0.692452]\). NB: expect CV 8 or 9 and answer 0.9825 or 0.9417: M0M0 |
| Total: 4 |
# Question 8:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B(14, 0.25)$: Critical region $\geq 7$ | M1 | Use $B(14, 0.25)$ and find $r$ for an upper tail. *All marks need upper tail*. e.g. CV 5 or 6 or 7, or .1117, .0383, .0103, 0.8883, 0.9617, .9897 |
| $CR \geq 7$ or $AR \leq 6$ stated or clearly implied | A1 | Not just "CV = 7" |
| $B(14, 0.4)$: $P(\leq 6) = 0.6925$ | M1 | Find $P$(in AR when $p = 0.4$) *[indept of M1]*. *Not* $P(\geq r)$, e.g. final answer 0.3075 |
| $= 0.6925$ | A1 | Answer 0.692 or 0.693 or a.r.t. 0.6925 or 0.6924 only, *not* isw $[0.692452]$. NB: expect CV 8 or 9 and answer 0.9825 or 0.9417: M0M0 |
| | **Total: 4** | |8 The random variable $R$ has the distribution $\mathrm { B } ( 14 , p )$. A test is carried out at the $\alpha \%$ significance level of the null hypothesis $\mathrm { H } _ { 0 } : p = 0.25$, against $\mathrm { H } _ { 1 } : p > 0.25$.\\
(i) Given that $\alpha$ is as close to 5 as possible, find the probability of a Type II error when the true value of $p$ is 0.4 .\\
(ii) State what happens to the probability of a Type II error as
\begin{enumerate}[label=(\alph*)]
\item $p$ increases from 0.4,
\item $\alpha$ increases, giving a reason.
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2013 Q8 [6]}}