| Exam Board | OCR MEI |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2008 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 3×3 contingency table |
| Difficulty | Standard +0.3 This is a standard one-way ANOVA question with straightforward calculations using provided summary statistics. Part (i) requires recall of the ANOVA model, part (ii) is a routine application with given sums, and part (iii) asks for recognition of a randomized block design. All components are textbook exercises requiring no novel insight, though the multi-part structure and Further Maths context place it slightly above average difficulty. |
| Spec | 5.06c Fit other distributions: discrete and continuous5.06d Goodness of fit: chi-squared test |
| Contractor A | Contractor B | Contractor C | Contractor D |
| 41 | 54 | 56 | 41 |
| 49 | 45 | 45 | 36 |
| 50 | 50 | 54 | 46 |
| 44 | 50 | 50 | 38 |
| 56 | 47 | 49 | 35 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x_{ij} = \mu + \alpha_i + e_{ij}\) | 1 | |
| \(\mu\) = population grand mean for whole experiment | 1 | |
| \(\alpha_i\) = population mean by which \(i\)th treatment differs from \(\mu\) | 1 | |
| \(e_{ij}\) are experimental errors \(\sim \text{ind } N(0, \sigma^2)\) | 1, 3 | Allow "uncorrelated"; 1 for ind N; 1 for 0; 1 for \(\sigma^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Totals are 240, 246, 254, 264, 196 each from sample of size 5. Grand total 936 | ||
| "Correction factor" \(CF = \dfrac{936^2}{20} = 43804.8\) | ||
| Total SS \(= 44544 - CF = 739.2\) | ||
| Between contractors SS \(= \dfrac{240^2}{5} + \ldots + \dfrac{196^2}{5} - CF = 44209.6 - CF = 404.8\) | M1, M1 | For correct methods for any two, if each calculated SS is correct |
| Residual SS (by subtraction) \(= 739.2 - 404.8 = 334.4\) | A1 | |
| ANOVA table: Between Contractors: SS=404.8, df=3, MS=134.93, MS ratio=6.456 | M1, M1, 1, A1 | |
| Residual: SS=334.4, df=16, MS=20.9 | ||
| Total: SS=739.2, df=19 | 1 | CAO |
| Refer to \(F_{3,16}\) | 1 | NO FT IF WRONG |
| Upper 5% point is 3.24 | 1 | NO FT IF WRONG |
| Significant | 1 | |
| Seems performances of contractors are not all the same | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Randomised blocks | B1 | |
| Description | E1, E1 | Take the subject areas as "blocks", ensure each contractor is used at least once in each block |
# Question 4:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x_{ij} = \mu + \alpha_i + e_{ij}$ | 1 | |
| $\mu$ = population grand mean for whole experiment | 1 | |
| $\alpha_i$ = population mean by which $i$th treatment differs from $\mu$ | 1 | |
| $e_{ij}$ are experimental errors $\sim \text{ind } N(0, \sigma^2)$ | 1, 3 | Allow "uncorrelated"; 1 for ind N; 1 for 0; 1 for $\sigma^2$ |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Totals are 240, 246, 254, 264, 196 each from sample of size 5. Grand total 936 | | |
| "Correction factor" $CF = \dfrac{936^2}{20} = 43804.8$ | | |
| Total SS $= 44544 - CF = 739.2$ | | |
| Between contractors SS $= \dfrac{240^2}{5} + \ldots + \dfrac{196^2}{5} - CF = 44209.6 - CF = 404.8$ | M1, M1 | For correct methods for any two, if each calculated SS is correct |
| Residual SS (by subtraction) $= 739.2 - 404.8 = 334.4$ | A1 | |
| ANOVA table: Between Contractors: SS=404.8, df=3, MS=134.93, MS ratio=6.456 | M1, M1, 1, A1 | |
| Residual: SS=334.4, df=16, MS=20.9 | | |
| Total: SS=739.2, df=19 | 1 | CAO |
| Refer to $F_{3,16}$ | 1 | NO FT IF WRONG |
| Upper 5% point is 3.24 | 1 | NO FT IF WRONG |
| Significant | 1 | |
| Seems performances of contractors are not all the same | 1 | |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Randomised blocks | B1 | |
| Description | E1, E1 | Take the subject areas as "blocks", ensure each contractor is used at least once in each block |4 (i) State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.\\
(ii) An examinations authority is considering using an external contractor for the typesetting and printing of its examination papers. Four contractors are being investigated. A random sample of 20 examination papers over the entire range covered by the authority is selected and 5 are allocated at random to each contractor for preparation. The authority carefully checks the printed papers for errors and assigns a score to each to indicate the overall quality (higher scores represent better quality). The scores are as follows.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Contractor A & Contractor B & Contractor C & Contractor D \\
\hline
41 & 54 & 56 & 41 \\
49 & 45 & 45 & 36 \\
50 & 50 & 54 & 46 \\
44 & 50 & 50 & 38 \\
56 & 47 & 49 & 35 \\
\hline
\end{tabular}
\end{center}
[The sum of these data items is 936 and the sum of their squares is 44544 .]\\
Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a $5 \%$ significance level. Report briefly on your conclusions.\\
(iii) The authority thinks that there might be differences in the ways the contractors cope with the preparation of examination papers in different subject areas. For this purpose, the subject areas are broadly divided into mathematics, sciences, languages, humanities, and others. The authority wishes to design a further investigation, ensuring that each of these subject areas is covered by each contractor. Name the experimental design that should be used and describe briefly the layout of the investigation.
\hfill \mbox{\textit{OCR MEI S4 2008 Q4 [24]}}