| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2011 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Yates’ continuity correction |
| Type | Perform chi-squared test with Yates' |
| Difficulty | Standard +0.3 This is a straightforward application of chi-squared test with Yates' correction to a 2×2 contingency table. Part (i) requires simple recall of when to apply Yates' correction, and part (ii) involves standard calculation of expected frequencies, applying the correction formula, and comparing to critical values—all routine procedures for S3 students with no novel problem-solving required. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Vaccines | |
| \cline { 2 - 3 } | \(A\) | \(B\) |
| Developed typhoid | 19 | 4 |
| Did not develop typhoid | 310 | 367 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| In a \(2\times 2\) contingency table | B1 1 | Or equivalent. Accept df=1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): Vaccine type and outcome are independent | B1M*dep | Accept omission of \(H_1\) |
| \(H_1\): They are not independent | ||
| E-values: 10.81, 12.19, 318.19, 358.81 | M1, A1 | 1 correct E value; Accept 1 dp |
| \(\chi^2 = 7.69^{-2}(10.81^{-1}+12.19^{-1}+318.19^{-1}+358.81^{-1})\) | M1 | 1 correct \(\chi^2\) value ft E values |
| \(= 10.67\) | M1, A1 | Using Yates' correctly; Accept 10.7 |
| \(CV = 6.635\) | B1 | |
| \(\mathbf{10.67 > CV}\) | M1 | |
| Reject \(H_0\), sufficient evidence at 1% that outcome depends on vaccine used | A1\(\sqrt{}\) dep*M | \(\sqrt{10.67}\) |
| Result significant at level less than \(\frac{1}{2}\%\), evidence is very strong | A1\(\sqrt{}\) 10 [11] | Sensible comment, \(\sqrt{10.67}\) |
## Question 7:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| In a $2\times 2$ contingency table | B1 **1** | Or equivalent. Accept df=1 |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: Vaccine type and outcome are independent | B1M*dep | Accept omission of $H_1$ |
| $H_1$: They are not independent | | |
| E-values: 10.81, 12.19, 318.19, 358.81 | M1, A1 | 1 correct E value; Accept 1 dp |
| $\chi^2 = 7.69^{-2}(10.81^{-1}+12.19^{-1}+318.19^{-1}+358.81^{-1})$ | M1 | 1 correct $\chi^2$ value ft E values |
| $= 10.67$ | M1, A1 | Using Yates' correctly; Accept 10.7 |
| $CV = 6.635$ | B1 | |
| $\mathbf{10.67 > CV}$ | M1 | |
| Reject $H_0$, sufficient evidence at 1% that outcome depends on vaccine used | A1$\sqrt{}$ dep*M | $\sqrt{10.67}$ |
| Result significant at level less than $\frac{1}{2}\%$, evidence is very strong | A1$\sqrt{}$ **10** [11] | Sensible comment, $\sqrt{10.67}$ |
---7 (i) When should Yates' correction be applied when carrying out a $\chi ^ { 2 }$ test?
Two vaccines against typhoid fever, $A$ and $B$, were tested on a total of 700 people in Nepal during a particular year. The vaccines were allocated randomly and whether or not typhoid had developed was noted during the following year. The results are shown in the table.
\begin{center}
\begin{tabular}{ | l | c | r | }
\hline
\multirow{2}{*}{} & \multicolumn{2}{|c|}{Vaccines} \\
\cline { 2 - 3 }
& $A$ & \multicolumn{1}{c|}{$B$} \\
\hline
Developed typhoid & 19 & 4 \\
\hline
Did not develop typhoid & 310 & 367 \\
\hline
\end{tabular}
\end{center}
(ii) Carry out a suitable $\chi ^ { 2 }$ test at the $1 \%$ significance level to determine whether the outcome depends on the vaccine used. Comment on the result.
\hfill \mbox{\textit{OCR S3 2011 Q7 [11]}}