| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed test critical region |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial hypothesis testing with standard procedures: calculating binomial probabilities using given parameters, stating null/alternative hypotheses for a one-tailed test, and finding a critical region at a specified significance level. While it requires multiple steps and understanding of hypothesis testing framework, all techniques are routine for AS-level statistics with no novel problem-solving or conceptual challenges beyond textbook exercises. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim B(30, 0.92)\), \(P(X = 28) = 0.2696\) | B1, B1 [2] | AO3.3, AO1.1 (BC) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 27) = 1 - 0.4346\) oe \(= 0.5654\) | M1, A1 [2] | AO1.1; OR for sum of at least two correct probabilities from \(0.2696 + {}_{30}C_{29} \times 0.92^{29} \times 0.08^1 + 0.92^{30}\) (BC) |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(p\) = probability that a train arrives on time | B1 | AO2.5; For definition of \(p\) |
| \(H_0: p = 0.92\) | B1 | AO1.1; For \(H_0\) and \(H_1\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq 13) = 0.0116\ [> 1\%]\) | M1 | AO1.1; For probability \(P(X \leq\) any whole number value 1 to 18); Allow FT from \(H_1\): \(p < 0.92\) OR \(H_1\): \(p \neq 0.92\) |
| \(P(X \leq 12) = 0.0021\ [< 1\%]\) | M1 | AO1.1; Both \(P(X \leq 13)\) and \(P(X \leq 12)\) |
| The critical region is \(X \leq 12\) | A1 [5] | AO2.2a; For correct critical region stated |
## Question 10(a)(i):
$X \sim B(30, 0.92)$, $P(X = 28) = 0.2696$ | B1, B1 [2] | AO3.3, AO1.1 (BC)
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## Question 10(a)(ii):
$P(X > 27) = 1 - 0.4346$ oe $= 0.5654$ | M1, A1 [2] | AO1.1; OR for sum of at least two correct probabilities from $0.2696 + {}_{30}C_{29} \times 0.92^{29} \times 0.08^1 + 0.92^{30}$ (BC)
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## Question 10(b):
Let $p$ = probability that a train arrives on time | B1 | AO2.5; For definition of $p$
$H_0: p = 0.92$ | B1 | AO1.1; For $H_0$ and $H_1$
$H_1: p < 0.92$
Let $X \sim B(18, 0.92)$
$P(X \leq 13) = 0.0116\ [> 1\%]$ | M1 | AO1.1; For probability $P(X \leq$ any whole number value 1 to 18); Allow FT from $H_1$: $p < 0.92$ OR $H_1$: $p \neq 0.92$
$P(X \leq 12) = 0.0021\ [< 1\%]$ | M1 | AO1.1; Both $P(X \leq 13)$ and $P(X \leq 12)$
The critical region is $X \leq 12$ | A1 [5] | AO2.2a; For correct critical region stated
---10 A company operates trains. The company claims that $92 \%$ of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.
\begin{enumerate}[label=(\alph*)]
\item Assuming that $92 \%$ of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
\begin{enumerate}[label=(\roman*)]
\item exactly 28 trains arrive on time,
\item more than 27 trains arrive on time.
A journalist believes that the percentage of trains operated by this company which arrive on time is lower than $92 \%$.
\end{enumerate}\item To investigate the journalist's belief a hypothesis test will be carried out at the $1 \%$ significance level. A random sample of 18 trains is selected. For this hypothesis test,
\begin{itemize}
\item state the hypotheses,
\item find the critical region.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 Q10 [9]}}