Question 4 - A-Level Maths

OCR MEI S4 — Question 4 12 marks

Exam Board	OCR MEI
Module	S4 (Statistics 4)
Marks	12
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Chi-squared test of independence
Type	Standard 3×3 contingency table
Difficulty	Standard +0.8 This is a multi-part question covering experimental design (Latin squares), one-way ANOVA model specification, and hypothesis testing with known variances. While it requires understanding of multiple statistical concepts and careful interpretation, the individual components are standard Further Maths Statistics topics with straightforward application of learned procedures rather than requiring novel insight or complex derivations.
Spec	2.01c Sampling techniques: simple random, opportunity, etc 2.01d Select/critique sampling: in context

4 An experiment is carried out to compare five industrial paints, A, B, C, D, E, that are intended to be used to protect exterior surfaces in polluted urban environments. Five different types of surface (I, II, III, IV, V) are to be used in the experiment, and five specimens of each type of surface are available. Five different external locations ( \(1,2,3,4,5\) ) are used in the experiment. The paints are applied to the specimens of the surfaces which are then left in the locations for a period of six months. At the end of this period, a "score" is given to indicate how effective the paint has been in protecting the surface.

Name a suitable experimental design for this trial and give an example of an experimental layout. Initial analysis of the data indicates that any differences between the types of surface are negligible, as also are any differences between the locations. It is therefore decided to analyse the data by one-way analysis of variance.
State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
The data for analysis are as follows. Higher scores indicate better performance. The underlying distributions of strengths are assumed to be Normal for both suppliers, with variances 2.45 for supplier A and 1.40 for supplier B.
Test at the \(5 \%\) level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
Provide a two-sided 90\% confidence interval for the true mean difference.
Show that the test procedure used in part (i), with samples of sizes 7 and 5 and a \(5 \%\) significance level, leads to acceptance of the null hypothesis of equal means if \(- 1.556 < \bar { x } - \bar { y } < 1.556\), where \(\bar { x }\) and \(\bar { y }\) are the observed sample means from suppliers A and B . Hence find the probability of a Type II error for this test procedure if in fact the true mean strength from supplier A is 2.0 units more than that from supplier B.
A manager suggests that the Wilcoxon rank sum test should be used instead, comparing the median strengths for the samples of sizes 7 and 5 . Give one reason why this suggestion might be sensible and two why it might not.

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4 An experiment is carried out to compare five industrial paints, A, B, C, D, E, that are intended to be used to protect exterior surfaces in polluted urban environments. Five different types of surface (I, II, III, IV, V) are to be used in the experiment, and five specimens of each type of surface are available. Five different external locations ( $1,2,3,4,5$ ) are used in the experiment.

The paints are applied to the specimens of the surfaces which are then left in the locations for a period of six months. At the end of this period, a "score" is given to indicate how effective the paint has been in protecting the surface.\\
(i) Name a suitable experimental design for this trial and give an example of an experimental layout.

Initial analysis of the data indicates that any differences between the types of surface are negligible, as also are any differences between the locations. It is therefore decided to analyse the data by one-way analysis of variance.\\
(ii) State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.\\
(iii) The data for analysis are as follows. Higher scores indicate better performance.

The underlying distributions of strengths are assumed to be Normal for both suppliers, with variances 2.45 for supplier A and 1.40 for supplier B.\\
(i) Test at the $5 \%$ level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.\\
(ii) Provide a two-sided 90\% confidence interval for the true mean difference.\\
(iii) Show that the test procedure used in part (i), with samples of sizes 7 and 5 and a $5 \%$ significance level, leads to acceptance of the null hypothesis of equal means if $- 1.556 < \bar { x } - \bar { y } < 1.556$, where $\bar { x }$ and $\bar { y }$ are the observed sample means from suppliers A and B . Hence find the probability of a Type II error for this test procedure if in fact the true mean strength from supplier A is 2.0 units more than that from supplier B.\\
(iv) A manager suggests that the Wilcoxon rank sum test should be used instead, comparing the median strengths for the samples of sizes 7 and 5 . Give one reason why this suggestion might be sensible and two why it might not.

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