OCR MEI S4 2009 June — Question 4 24 marks

Exam BoardOCR MEI
ModuleS4 (Statistics 4)
Year2009
SessionJune
Marks24
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicNon-parametric tests
TypeExperimental design types
DifficultyStandard +0.3 This is a straightforward Further Maths Statistics question testing standard knowledge of experimental design and ANOVA. Part (i) requires describing a Latin square (textbook recall), part (ii) asks for standard distributional assumptions (e_ij ~ N(0,σ²)), and part (iii) involves routine ANOVA table completion with simple arithmetic (subtraction for SS, counting df, division for MS, F-test). All components are standard bookwork or mechanical calculations with no novel problem-solving required, making it slightly easier than average.
Spec2.01c Sampling techniques: simple random, opportunity, etc2.01d Select/critique sampling: in context
4
  1. Describe, with the aid of a specific example, an experimental situation for which a Latin square design is appropriate, indicating carefully the features which show that a completely randomised or randomised blocks design would be inappropriate.
  2. The model for the one-way analysis of variance may be written, in a customary notation, as $$x _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$ State the distributional assumptions underlying \(e _ { i j }\) in this model. What is the interpretation of the term \(\alpha _ { i }\) ?
  3. An experiment for comparing 5 treatments is carried out, with a total of 20 observations. A partial one-way analysis of variance table for the analysis of the results is as follows.
    Source of variationSums of squaresDegrees of freedomMean squaresMean square ratio
    Between treatments
    Residual68.76
    Total161.06
    Copy and complete the table, and carry out the appropriate test using a \(1 \%\) significance level.
Question 4:
Part (i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Description must be in context. If no context given, mark according to scheme and give half-marks, rounded down
Clear description of "rows"E1, E1
And "columns"E1, E1
As extraneous factors to be taken account of in the design, with "treatments" to be comparedE1, E1
Need same numbers of eachE1
Clear contrast with situations for completely randomised design and randomised trendsE1, E1
Part (ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(e_{ij} \sim \text{ind } N(0, \sigma^2)\)1, 1, 1 Allow uncorrelated; For 0; For \(\sigma^2\)
\(\alpha_i\) is population mean effect by which \(i\)th treatment differs from overall mean1, 1
Part (iii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
ANOVA table: Between Treatments: SS = 92.30, df = 4, MS = 23.075, MS ratio = 5.0341
Residual: SS = 68.76, df = 15, MS = 4.5841
Total: SS = 161.06, df = 191
Refer to \(F_{4,15}\)1 No FT if wrong
Upper 1% point is 4.89; Significant, seems treatments are not all the same1, 1 No FT if wrong 
# Question 4:

## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Description must be in context. If no context given, mark according to scheme and give half-marks, rounded down | | |
| Clear description of "rows" | E1, E1 | |
| And "columns" | E1, E1 | |
| As extraneous factors to be taken account of in the design, with "treatments" to be compared | E1, E1 | |
| Need same numbers of each | E1 | |
| Clear contrast with situations for completely randomised design and randomised trends | E1, E1 | | **[9 marks]**

## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $e_{ij} \sim \text{ind } N(0, \sigma^2)$ | 1, 1, 1 | Allow uncorrelated; For 0; For $\sigma^2$ |
| $\alpha_i$ is population mean effect by which $i$th treatment differs from overall mean | 1, 1 | | **[5 marks]**

## Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| ANOVA table: Between Treatments: SS = 92.30, df = 4, MS = 23.075, MS ratio = 5.034 | 1 | |
| Residual: SS = 68.76, df = 15, MS = 4.584 | 1 | |
| Total: SS = 161.06, df = 19 | 1 | |
| Refer to $F_{4,15}$ | 1 | No FT if wrong |
| Upper 1% point is 4.89; Significant, seems treatments are not all the same | 1, 1 | No FT if wrong | **[10 marks]**
4 (i) Describe, with the aid of a specific example, an experimental situation for which a Latin square design is appropriate, indicating carefully the features which show that a completely randomised or randomised blocks design would be inappropriate.\\
(ii) The model for the one-way analysis of variance may be written, in a customary notation, as

$$x _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$

State the distributional assumptions underlying $e _ { i j }$ in this model. What is the interpretation of the term $\alpha _ { i }$ ?\\
(iii) An experiment for comparing 5 treatments is carried out, with a total of 20 observations. A partial one-way analysis of variance table for the analysis of the results is as follows.

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Source of variation & Sums of squares & Degrees of freedom & Mean squares & Mean square ratio \\
\hline
Between treatments &  &  &  &  \\
\hline
Residual & 68.76 &  &  &  \\
\hline
Total & 161.06 &  &  &  \\
\hline
\end{tabular}
\end{center}

Copy and complete the table, and carry out the appropriate test using a $1 \%$ significance level.

\hfill \mbox{\textit{OCR MEI S4 2009 Q4 [24]}}
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