Standard 3×3 contingency table

3 There are three bus companies in a city. The council is investigating whether the buses reliably arrive at their destination on time. The results from random samples of buses from each company are summarised in the following table. Test, at the \(5 \%\) significance level, whether the reliability of buses is independent of bus company.

2 In the colleges in three regions of a particular country, students are given individual targets to achieve. Their performance is measured against their individual target and graded as 'above target', 'on target' or 'below target'. For a random sample of students from each of the three regions, the observed frequencies are summarised in the following table. Test, at the 10\% significance level, whether performance is independent of region.

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.

1. 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.
2. State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
3. The data for analysis are as follows. Higher scores indicate better performance.

   Paint A	Paint B	Paint C	Paint D	Paint E
   64	66	59	65	64
   58	68	56	78	52
   73	76	69	69	56
   60	70	60	72	61
   67	71	63	71	58

   [The sum of these data items is 1626 and the sum of their squares is 106838 .]
   Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a 5\% significance level. Report briefly on your conclusions.
   [0pt] [12]

4 An agricultural company conducts a trial of five fertilisers (A, B, C, D, E) in an experimental field at its research station. The fertilisers are applied to plots of the field according to a completely randomised design. The yields of the crop from the plots, measured in a standard unit, are analysed by the one-way analysis of variance, from which it appears that there are no real differences among the effects of the fertilisers. A statistician notes that the residual mean square in the analysis of variance is considerably larger than had been anticipated from knowledge of the general behaviour of the crop, and therefore suspects that there is some inadequacy in the design of the trial.

1. Explain briefly why the statistician should be suspicious of the design.
2. Explain briefly why an inflated residual leads to difficulty in interpreting the results of the analysis of variance, in particular that the null hypothesis is more likely to be accepted erroneously. Further investigation indicates that the soil at the west side of the experimental field is naturally more fertile than that at the east side, with a consistent 'fertility gradient' from west to east.
3. What experimental design can accommodate this feature? Provide a simple diagram of the experimental field indicating a suitable layout. The company decides to conduct a new trial in its glasshouse, where experimental conditions can be controlled so that a completely randomised design is appropriate. The yields are as follows.

   Fertiliser A	Fertiliser B	Fertiliser C	Fertiliser D	Fertiliser E
   23.6	26.0	18.8	29.0	17.7
   18.2	35.3	16.7	37.2	16.5
   32.4	30.5	23.0	32.6	12.8
   20.8	31.4	28.3	31.4	20.4

   [The sum of these data items is 502.6 and the sum of their squares is 13610.22 .]
4. Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a \(5 \%\) significance level. Report briefly on your conclusions.
5. State the assumptions about the distribution of the experimental error that underlie your analysis in part (iv).

4

1. State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
2. 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.

   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

   [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.
3. 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.

4 At an agricultural research station, a trial is made of four varieties (A, B, C, D) of a certain crop in an experimental field. The varieties are grown on plots in the field and their yields are measured in a standard unit.

1. It is at first thought that there may be a consistent trend in the natural fertility of the soil in the field from the west side to the east, though no other trends are known. Name an experimental design that should be used in these circumstances and give an example of an experimental layout. Initial analysis suggests that any natural fertility trend may in fact be ignored, so the data from the trial are analysed by one-way analysis of variance.
2. The usual model for one-way analysis of variance of the yields \(y _ { i j }\) may be written as $$y _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$ where the \(e _ { i j }\) represent the experimental errors. Interpret the other terms in the model. State the usual distributional assumptions for the \(e _ { i j }\).
3. The data for the yields are as follows, each variety having been used on 5 plots.

   Variety
   A	B	C	D
   12.3	14.2	14.1	13.6
   11.9	13.1	13.2	12.8
   12.8	13.1	14.6	13.3
   12.2	12.5	13.7	14.3
   13.5	12.7	13.4	13.8

   $$\left[ \Sigma \Sigma y _ { i j } = 265.1 , \quad \Sigma \Sigma y _ { i j } ^ { 2 } = 3524.31 . \right]$$ Construct the usual one-way analysis of variance table and carry out the usual test, at the 5\% significance level. Report briefly on your conclusions. {www.ocr.org.uk} after the live examination series.
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4 A gardening research organisation is running a trial to examine the growth and the size of flowers of various plants.

1. In the trial, seeds of three types of plant are sown. The growth of each plant is classified as good, average or poor. The results are shown in the table.

   \multirow{2}{*}{}		Growth			\multirow[t]{2}{*}{Row totals}
   		Good	Average	Poor
   \multirow{3}{*}{Type of plant}	Coriander	12	28	15	55
   	Aster	7	18	23	48
   	Fennel	14	22	11	47
   Column totals		33	68	49	150

   Carry out a test at the \(5 \%\) significance level to examine whether there is any association between growth and type of plant. State carefully your null and alternative hypotheses. Include a table of the contributions of each cell to the test statistic.
2. It is known that the diameter of marigold flowers is Normally distributed with mean 47 mm and standard deviation 8.5 mm . A certain fertiliser is expected to cause flowers to have a larger mean diameter, but without affecting the standard deviation. A large number of marigolds are grown using this fertiliser. The diameters of a random sample of 50 of the flowers are measured and the mean diameter is found to be 49.2 mm . Carry out a hypothesis test at the \(1 \%\) significance level to check whether flowers grown with this fertiliser appear to be larger on average. Use hypotheses \(\mathrm { H } _ { 0 } : \mu = 47 , \mathrm { H } _ { 1 } : \mu > 47\), where \(\mu \mathrm { mm }\) represents the mean diameter of all marigold flowers grown with this fertiliser.

4 A researcher is investigating the sizes of pebbles at various locations in a river. Three sites in the river are chosen and each pebble sampled at each site is classified as large, medium or small. The results are as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between pebble size and site. Your working should include a table of the contributions of each cell to the test statistic.
2. By referring to each site, comment briefly on how the size of the pebbles compares with what would be expected if there were no association. You should support your answers by referring to your table of contributions.

2 A random sample of 200 American voters were asked about which political party they supported and their attitude to a proposed new form of taxation. The voters' responses are summarised in the table. Attitude Carry out a \(\chi ^ { 2 }\) test, at the \(1 \%\) level of significance, to investigate whether there is an association between party supported and attitude to the proposed form of taxation.

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.

1. 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.
2. State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
3. 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.
4. Test at the \(5 \%\) level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
5. Provide a two-sided 90\% confidence interval for the true mean difference.
6. 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.
7. 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.

8 A manufacturer produces three types of car: hatchbacks, saloons and estates. Each type of car is available in one of three colours: silver, blue and red. The manufacturer wants to know whether the popularity of the colour of the car is related to the type of car. A random sample of 300 cars chosen by customers gives the information summarised in the following table. Test at the \(10 \%\) significance level whether the colour of car chosen by customers is independent of the type of car.

6. A market researcher recorded the number of adverts for vehicles in each of three categories on ITV, Channel 4 and Channel 5 over a period of time. The results are shown in the table below.

1. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the proportion of adverts for each type of vehicle being dependent on the channel.
2. Suggest a reason for your result in part (a).

1. A bag contains a large number of balls, all of the same size and weight. The balls are coloured Red, Blue or Yellow.

Jasmine asks each child in a group of 150 children to close their eyes, select a ball from the bag and show it to her. The child then replaces the ball and repeats the process a second time. If both balls are the same colour the child receives a prize.
The results are given in the table below. Jasmine carries out a test, at the \(5 \%\) level of significance, to see whether or not the colour of the 2nd ball is independent of the colour of the 1st ball.

1. Calculate the expected frequencies for the cases where both balls are the same colour. The test statistic Jasmine obtained was 12.712 to three decimal places.
2. Use this value to complete the test, stating the critical value and conclusion clearly. With reference to your calculations in part (a) and the nature of the experiment, (c) give a plausible reason why Jasmine may have obtained her conclusion in part (b).

3 At a cinema there are three film sessions each Saturday, "early", "middle" and "late". The numbers of the audience, in different age groups, at the three showings on a randomly chosen Saturday are given in Table 1. \begin{table}[h] \end{table}