| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Comment on claim using CI |
| Difficulty | Standard +0.3 This is a straightforward confidence interval interpretation question followed by a standard two-sample proportion test. Part (i) requires only checking whether 0.43 lies within given intervals (basic interpretation), while part (ii) is a routine application of the two-sample z-test for proportions with clear parameters. The question is slightly above average difficulty due to the two-part structure and requiring knowledge of the two-sample test, but involves no novel insight or complex reasoning. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) No — \(0.43\) belongs to relevant interval | B1 | Must be with reason |
| (b) Yes — \(0.43\) is outside relevant interval | B1 | |
| B1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p_R = p_T\), \(H_1: p_R \neq p_T\) | B1 | Proportions |
| Estimate of \(p = 74/165\) | B1 | |
| Variance estimate of difference \(= \left(\frac{74}{165}\right)\left(\frac{91}{165}\right)\left(\frac{1}{80} + \frac{1}{85}\right)\) | B1 | May be implied by later work |
| \(z = (28/80 - 46/85)/\sigma_{est}\) | M1 A1 | Standardising; completely correct expression |
| \(= -2.468\) | A1 | \(+\) or \(-\), \(2.47\) |
| Compare correctly with CV: \(-2.468 < -2.326\), or \(2.468 > 2.326\) | M1 | |
| Reject \(H_0\) and accept that the proportions differ on the island | A1 | 8 |
# Question 6(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| (a) No — $0.43$ belongs to relevant interval | B1 | Must be with reason |
| (b) Yes — $0.43$ is outside relevant interval | B1 | |
| | B1 | **3** | |
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# Question 6(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p_R = p_T$, $H_1: p_R \neq p_T$ | B1 | Proportions |
| Estimate of $p = 74/165$ | B1 | |
| Variance estimate of difference $= \left(\frac{74}{165}\right)\left(\frac{91}{165}\right)\left(\frac{1}{80} + \frac{1}{85}\right)$ | B1 | May be implied by later work |
| $z = (28/80 - 46/85)/\sigma_{est}$ | M1 A1 | Standardising; completely correct expression |
| $= -2.468$ | A1 | $+$ or $-$, $2.47$ |
| Compare correctly with CV: $-2.468 < -2.326$, or $2.468 > 2.326$ | M1 | |
| Reject $H_0$ and accept that the proportions differ on the island | A1 | **8** | Conclusion in context |
---6 An anthropologist was studying the inhabitants of two islands, Raloa and Tangi. Part of the study involved the incidence of blood group type A. The blood of 80 randomly chosen inhabitants of Raloa and 85 randomly chosen inhabitants of Tangi was tested. The number of inhabitants with type A blood was 28 for the Raloa sample and 46 for the Tangi sample. The anthropologist calculated $90 \%$ confidence intervals for the population proportions of inhabitants with type A blood. They were $( 0.262,0.438 )$ for Raloa and $( 0.452,0.630 )$ for Tangi, where each figure is correct to 3 decimal places. It is known that $43 \%$ of the world's population have type A blood.\\
(i) State, giving your reasons, whether there is evidence for the following assertions about the proportions of people with type A blood.
\begin{enumerate}[label=(\alph*)]
\item The proportion in Raloa is different from the world proportion.
\item The proportion in Tangi is different from the world proportion.\\
(ii) Carry out a suitable test, at the $2 \%$ significance level, of whether the proportions of people with type A blood differ on the two islands.
\end{enumerate}
\hfill \mbox{\textit{OCR S3 2006 Q6 [11]}}