| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2012 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Standard +0.3 This is a straightforward application of standard chi-squared tests with clear data and routine calculations. Part (i) is a basic goodness-of-fit test, part (ii) is a standard contingency table test of independence, and part (iii) requires simple interpretation of results. All procedures are textbook exercises requiring no novel insight, though the multi-part structure and contingency table setup place it slightly above average difficulty. |
| Spec | 5.06a Chi-squared: contingency tables5.06b Fit prescribed distribution: chi-squared test |
| Treatment | \(A\) | \(B\) | \(C\) |
| Number in group | 31 | 25 | 34 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: p_1 = p_2 = p_3 = \frac{1}{3}\), \(H_1\): Not all equal (or equivalent in words) | B1 | \(p = \frac{1}{3}\) only is insufficient |
| E-values all 30 | B1 | |
| \(\chi^2 = \frac{(1^2 + 5^2 + 4^2)}{30}\) | M1 | |
| \(= 1.4\) | A1 | Accept 1.39, 1.399 etc |
| (Critical value \(= 5.991\)) Compare \(\chi^2\) with CV and do not reject \(H_0\). Insufficient evidence that groups not randomly chosen | M1, A1ft [6] | With valid comparison, or accept that groups randomly chosen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (\(H_0\): Outcome independent of treatment) Contingency table: Improved: A=14, B=18, C=25; Not Improved: A=17, B=7, C=9 | B1 | For correct contingency table. If incorrect contingency table attempted e.g. 31 25 34 / 14 18 25, M1s available. Max 3/8 (TS=1.61) |
| E-values: 19.6, 15.8, 21.5 / 11.4, 9.2, 12.5 | M1, A1 | At least 2 correct FT table; All correct CAO |
| \(\chi^2 = 5.6^2(19.6^{-1} + 11.4^{-1}) + 2.2^2(15.8^{-1} + 9.2^{-1}) + 3.5^2(21.5^{-1} + 12.5^{-1})\) | M1, A1 | At least 2 correct ft; All correct |
| \(= 6.7\) | A1 | |
| \(> 5.991\) and reject \(H_0\) | M1 | |
| Sufficient evidence at the 5% SL that the outcome depends on treatment | A1ft [8] | Ft TS if attempt at correct table made |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Treatment A shows fewer improved than expected, Treatments B and C show more | M1 | Or consider proportion(s) improved: 0.45, 0.72, 0.74 (M1) |
| So abandon Treatment A | A1 [2] | SC B1 if A chosen with no proportions of successful treatments given |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: p_1 = p_2 = p_3 = \frac{1}{3}$, $H_1$: Not all equal (or equivalent in words) | B1 | $p = \frac{1}{3}$ only is insufficient |
| E-values all 30 | B1 | |
| $\chi^2 = \frac{(1^2 + 5^2 + 4^2)}{30}$ | M1 | |
| $= 1.4$ | A1 | Accept 1.39, 1.399 etc |
| (Critical value $= 5.991$) Compare $\chi^2$ with CV and do not reject $H_0$. Insufficient evidence that groups not randomly chosen | M1, A1ft **[6]** | With valid comparison, or accept that groups randomly chosen |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| ($H_0$: Outcome independent of treatment) Contingency table: Improved: A=14, B=18, C=25; Not Improved: A=17, B=7, C=9 | B1 | For correct contingency table. If incorrect contingency table attempted e.g. 31 25 34 / 14 18 25, M1s available. Max 3/8 (TS=1.61) |
| E-values: 19.6, 15.8, 21.5 / 11.4, 9.2, 12.5 | M1, A1 | At least 2 correct FT table; All correct CAO |
| $\chi^2 = 5.6^2(19.6^{-1} + 11.4^{-1}) + 2.2^2(15.8^{-1} + 9.2^{-1}) + 3.5^2(21.5^{-1} + 12.5^{-1})$ | M1, A1 | At least 2 correct ft; All correct |
| $= 6.7$ | A1 | |
| $> 5.991$ and reject $H_0$ | M1 | |
| Sufficient evidence at the 5% SL that the outcome depends on treatment | A1ft **[8]** | Ft TS if attempt at correct table made |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Treatment A shows fewer improved than expected, Treatments B and C show more | M1 | Or consider proportion(s) improved: 0.45, 0.72, 0.74 (M1) |
| So abandon Treatment A | A1 **[2]** | SC B1 if A chosen with no proportions of successful treatments given |7 A study was carried out into whether patients suffering from a certain respiratory disorder would benefit from particular treatments. Each of 90 patients who agreed to take part was given one of three treatments $A$, $B$ or $C$ as shown in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
Treatment & $A$ & $B$ & $C$ \\
\hline
Number in group & 31 & 25 & 34 \\
\hline
\end{tabular}
\end{center}
(i) It is claimed that each patient was equally likely to have been given any of the treatments. Test at the $5 \%$ significance level whether the numbers given each treatment are consistent with this claim.\\
(ii) After 3 months the numbers of patients showing improvement for treatments $A , B$ and $C$ were 14, 18 and 25 respectively. By setting up a $2 \times 3$ contingency table, test whether the outcome is dependent on the treatment. Use a $5 \%$ significance level.\\
(iii) If one of the treatments is abandoned, explain briefly which it should be.
\section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
\hfill \mbox{\textit{OCR S3 2012 Q7 [16]}}