5.06d Goodness of fit: chi-squared test

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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6 A machine is used to toss a coin repeatedly. Rosa believes that the outcome of each toss made by the machine is not independent of the previous toss. Rosa gets the machine to toss a coin 6 times and record the number of heads, \(X\), obtained. After recording the number of heads obtained, Rosa resets the machine and gets it to toss the coin 6 more times. Rosa again records the number of heads obtained and she repeats this procedure until she has recorded 88 independent values of \(X\).

1. The sample mean and sample variance of \(X\) are 3.35 and 3.392 respectively. Explain what these results suggest about the validity of a binomial model \(\mathrm { B } ( 6 , p )\) for the data. Rosa uses a computer spreadsheet to work out the probabilities for a more sophisticated model in which the outcome of each toss is dependent on the outcome of the previous toss. Her model suggests that the probabilities \(\mathrm { P } ( X = x )\), for \(x = 0,1,2,3,4,5,6\), are approximately in the ratio \(5 : 6 : 7 : 8 : 7 : 6 : 5\). She carries out a \(\chi ^ { 2 }\) test to investigate whether this model is a good fit for the data. The following table shows the full results of the experiments, together with some of the calculations needed for the test.

   \(x\)	0	1	2	3	4	5	6	Total
   Observed frequency	7	10	16	15	15	11	14	88
   Expected frequency
   Contribution to \(\chi ^ { 2 }\) statistic	0.9	0.3333	0.2857	0.0625	0.0714

2. In the Printed Answer Booklet, complete the table.
3. Carry out the test, using a 10\% significance level.
4. Rosa says that the results definitely show that one of the two proposed models is correct. Comment on this statement.

At a bird feeding station, birds are captured and ringed. If a bird is recaptured, the ring enables it to be identified. The table below shows the number of recaptures, \(x\), during a period of a month, for each bird of a particular species in a random sample of \(40\) birds.

Number of recaptures, \(x\)	0	1	2	3	4	5	6	7	8	9	10
Frequency	2	5	5	9	10	4	3	1	0	1	0

1. The sample mean of \(x\) is \(3.4\). Calculate the sample variance of \(x\). [2]
2. Briefly comment on whether the results of part (i) support a suggestion that a Poisson model might be a good fit to the data. [1]

The screenshot below shows part of a spreadsheet for a \(\chi^2\) test to assess the goodness of fit of a Poisson model. The sample mean of \(3.4\) has been used as an estimate of the Poisson parameter. Some values in the spreadsheet have been deliberately omitted. \includegraphics{figure_2}

1. State the null and alternative hypotheses for the test. [1]
2. Calculate the missing values in cells
3. Complete the test at the \(10\%\) significance level. [5]
4. The screenshot below shows part of a spreadsheet for a \(\chi^2\) test for a different species of bird. Find the value of the Poisson parameter used. \includegraphics{figure_3} [3]