| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Multiple binomial probability calculations |
| Difficulty | Standard +0.3 This is a straightforward multi-part binomial hypothesis testing question covering standard S1 content. Part (i) involves routine binomial probability calculations and expectation. Parts (ii) and (iii) are textbook two-tailed hypothesis tests with the critical probability value provided in (iii), requiring only standard procedure application without novel insight or complex reasoning. |
| 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 |
|---|---|
| Marks | Guidance |
| B1 | For either reject \(H_0\) or significant, dep on correct comparison |
| E1* | Dep on good attempt at correct hypotheses in part (ii). If they have \(H_1: p > 0.35\), allow SC1 if all correct including comparison with 5%. |
| [2] |
## Part (iii)
**Answer:** $0.0022 < 2.5\%$
So reject $H_0$. Significant.
Conclude that there is enough evidence to indicate that the probability is different.
| **Marks** | **Guidance** |
|-----------|--------------|
| B1 | For either reject $H_0$ or significant, dep on correct comparison |
| E1* | Dep on good attempt at correct hypotheses in part (ii). If they have $H_1: p > 0.35$, allow SC1 if all correct including comparison with 5%. |
| **[2]** | |
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# APPENDIX
## NOTE RE OVER-SPECIFICATION OF ANSWERS
If answers are grossly over-specified, deduct the final answer mark in every case. Probabilities should also be rounded to a sensible degree of accuracy. In general final non-probability answers should not be given to more than 4 significant figures. Allow probabilities given to 5 sig fig.
## Additional notes re Q7 part ii
**Comparison with 97.5% method:**
If 97.5% seen anywhere then B1 for $P(X < 9)$, B1 for 0.8782, M1* for comparison with 97.5% dep on second B1, A1* for not significant or E1*
**Smallest critical region method:**
Smallest critical region that 10 could fall into is $\{10,11,12,13,14,15,16,17,18,19,20\}$ gets **B1 and has size 0.1218 gets B1**. This is $> 2.5\%$ gets M1*, A1*, E1* as per scheme
NB **These marks only awarded if 10 used, not other values.**
**Use of $k$ method with no probabilities quoted:**
This gets zero marks.
**Use of $k$ method with one probability quoted:**
Mark as per scheme.
**Line diagram method and Bar chart method:**
No marks unless correct probabilities shown on diagram, then mark as per scheme.
**Upper tailed test done with** $H_1: p>0.35$
Hyp gets max B1B1B0E0
If compare with 5% give SC2 for $P(X \geq 10) = 1 - 0.8782 = 0.1218 > 5\%$ and SC1 for final conclusion (must be 'larger than' not 'different from')
If compare with 2.5% – no further marks B0B0M0A0E0
**Lower tailed test done with** $H_1: p<0.35$
No marks out of last 5.7 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.
\begin{enumerate}[label=(\roman*)]
\item 10 customers are selected at random.\\
(A) Find the probability that exactly 5 of them are accessing the internet.\\
(B) Find the probability that at least 5 of them are accessing the internet.\\
(C) Find the expected number of these customers who are accessing the internet.
Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
\item Carry out a hypothesis test at the $5 \%$ significance level to investigate whether the probability for this coffee shop is different from 0.35 . Give a reason for your choice of alternative hypothesis.
\item To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if $X$ has the binomial distribution with parameters $n = 200$ and $p = 0.35$, then $\mathrm { P } ( X \geqslant 90 ) = 0.0022$. Using the same hypotheses and significance level which you used in part (ii), complete this test.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S1 2013 Q7 [18]}}