Chi-squared test of independence

4

1. A researcher believes that there may be some association between a student's sex and choice of certain subjects at A-level. A random sample of 250 A -level students is selected. The table below shows, for each sex, how many study either or both of the two subjects, Mathematics and English.

   	Mathematics only	English only	Both	Neither	Row totals
   Male	38	19	6	32	95
   Female	42	55	9	49	155
   Column totals	80	74	15	81	250

   Carry out a test at the \(5 \%\) significance level to examine whether there is any association between a student's sex and choice of subjects. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [12]
2. Over a long period it has been determined that the mean score of students in a particular English module is 67.4 and the standard deviation is 8.9. A new teaching method is introduced with the aim of improving the results. A random sample of 12 students taught by the new method is selected. Their mean score is found to be 68.3. Carry out a test at the \(10 \%\) level to investigate whether the new method appears to have been successful. State carefully your null and alternative hypotheses. You should assume that the scores are Normally distributed and that the standard deviation is unchanged.

4 A survey of a random sample of 250 people is carried out. Their musical preferences are categorized as pop, classical or jazz. Their ages are categorized as under 25, 25 to 50, or over 50. The results are as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between musical preference and age group. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
2. Discuss briefly how musical preferences vary between the age groups, as shown by the contributions to the test statistic.

4 The sexes and ages of a random sample of 300 runners taking part in marathons are classified as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between age group and sex. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
2. Does your analysis support the suggestion that women are less likely than men to enter marathons as they get older? Justify your answer. For marathons in general, on average \(3 \%\) of runners are 'Female, 50 and over'. The random variable \(X\) represents the number of 'Female, 50 and over' runners in a random sample of size 300.
3. Use a suitable approximating distribution to find \(\mathrm { P } ( X \geqslant 12 )\).

4 A student is investigating whether there is any association between the species of shellfish that occur on a rocky shore and where they are located. A random sample of 160 shellfish is selected and the numbers of shellfish in each category are summarised in the table below.

1. Write down null and alternative hypotheses for a test to examine whether there is any association between species and location. The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.

   Contribution		Location
   \cline { 3 - 5 }	Exposed	Sheltered	Pool
   \multirow{3}{*}{Species}	Limpet	0.0009	0.2585	0.4450
   \cline { 2 - 5 }	Mussel	10.3472	1.2756	4.8773
   \cline { 2 - 5 }	Other	8.0719	0.1402	7.4298

   The sum of these contributions is 32.85 .
2. Calculate the expected frequency for mussels in pools. Verify the corresponding contribution 4.8773 to the test statistic.
3. Carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly.
4. For each species, comment briefly on how its distribution compares with what would be expected if there were no association.
5. If 3 of the 160 shellfish are selected at random, one from each of the 3 types of location, find the probability that all 3 of them are limpets.

7 It is thought that a person's eye colour is related to the reaction of the person's skin to ultra-violet light. As part of a study, a random sample of 140 people were treated with a standard dose of ultra-violet light. The degree of reaction was classified as None, Mild or Strong. The results are given in Table 1. The corresponding expected frequencies for a \(\chi ^ { 2 }\) test of association between eye colour and reaction are shown in Table 2. \begin{table}[h] \end{table} \begin{table}[h] \end{table}

1. (a) State suitable hypotheses for the test.
   (b) Show how the expected frequency of 18.11 in Table 2 is obtained.
   (c) Show that the three cells in the top row together contribute 4.53 to the calculated value of \(\chi ^ { 2 }\), correct to 2 decimal places.
   (d) You are given that the total calculated value of \(\chi ^ { 2 }\) is 12.78 , correct to 2 decimal places. Give the smallest value of \(\alpha\) obtained from the tables for which the null hypothesis would be rejected at the \(\alpha \%\) significance level.
2. Test, at the \(5 \%\) significance level, whether the proportions of people in the whole population with blue eyes, brown eyes and other colours are in the ratios \(2 : 2 : 1\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

6 The Research and Development department of a paint manufacturer has produced paint of three different shades of grey, \(G _ { 1 } , G _ { 2 }\) and \(G _ { 3 }\). In order to find the reaction of the public to these shades, each of a random sample of 120 people was asked to state which shade they preferred. The results, classified by gender, are shown in Table 1. \begin{table}[h] \end{table} Table 2 shows the corresponding expected values, correct to 2 decimal places, for a test of independence. \begin{table}[h] \end{table}

1. Show how the value 17.88 for Male, \(G _ { 2 }\) was obtained.
2. Test, at the \(5 \%\) significance level, whether gender and preferred shade are independent.
3. Determine the smallest significance level obtained from tables or calculator for which there is evidence that not all shades are equally preferred by people in general, irrespective of gender.

4 A study in 1981 investigated the effect of water fluoridation on children's dental health. In a town with fluoridation, 61 out of a random sample of 107 children showed signs of increased tooth decay after six months. In a town without fluoridation the corresponding number was 106 out of a random sample of 143 children. The population proportions of children with increased tooth decay are denoted by \(p _ { 1 }\) and \(p _ { 2 }\) for the towns with fluoridation and without fluoridation respectively. A test is carried out of the null hypothesis \(p _ { 1 } = p _ { 2 }\) against the alternative hypothesis \(p _ { 1 } < p _ { 2 }\). Find the smallest significance level at which the null hypothesis is rejected.

5 Gloria is a market trader who sells jeans. She trades on Mondays, Wednesdays and Fridays. Wishing to investigate whether the volume of trade depends on the day of the week, Gloria analysed a random sample of 150 days' sales and classified them by day and volume (low, medium and high). The results are given in the table below. Gloria asked a statistician to perform a suitable test of independence and, as part of this test, expected frequencies were calculated. These are shown in the table below.

1. Show how the value 23.04 for medium volume on Wednesday has been obtained.
2. State, giving a reason, if it is necessary to combine any rows or columns in order to carry out the test. The value of the test statistic is found to be 21.15, correct to 2 decimal places.
3. Stating suitable hypotheses for the test, give its conclusion using a \(1 \%\) significance level. Gloria wishes to hold a sale and asks the statistician to advise her on which day to hold it in order to sell as much as possible.
4. State the day that the statistician should advise and give a reason for the choice.

4 The students in a large university department take a trial examination some time before the proper examination. A random sample of 60 students took both examinations during a particular course. 42 students passed the trial examination, 36 passed the proper examination and 13 failed both examinations.

1. Copy and complete the following contingency table.

   				Proper
   \cline { 2 - 4 } \multicolumn{1}{l}{}	Pass	Fail	Total
   \cline { 2 - 5 }	Pass			42
   \cline { 2 - 5 } Trial	Fail		13
   \cline { 2 - 5 }	Total	36		60

2. Carry out a test of independence at the \(\frac { 1 } { 2 } \%\) level of significance.

6 Certain types of food are now sold in metric units. A random sample of 1000 shoppers was asked whether they were in favour of the change to metric units or not. The results, classified according to age, were as shown in the table.

1. Use a \(\chi ^ { 2 }\) test to show that there is very strong evidence that shoppers' views about changing to metric units are not independent of their ages.
2. The data may also be regarded as consisting of two random samples of shoppers; one sample consists of 470 shoppers aged under 35 , of whom 187 were in favour of change, and the second sample consists of 530 shoppers aged 35 or over, of whom 161 were in favour of change. Determine whether a test for equality of population proportions supports the conclusion in part (i).

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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