Chi-squared test of independence

7 Mohammed is conducting a medical trial to study the effect of two drugs, \(A\) and \(B\), on the amount of time it takes to recover from a particular illness. Drug \(A\) is used by one group of 60 patients and drug \(B\) is used by a second group of 60 patients. The results are summarised in the table:

10 {}

1 Young children are learning to read using two different reading schemes, \(A\) and \(B\). The standards achieved are measured against the national average standard achieved and classified as above average, average or below average. For two randomly chosen groups of young children, the numbers in each category are shown in the table. Test at the \(5 \%\) significance level whether standard achieved is independent of the reading scheme used.

Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below.

	Black	Non-black
Men	69	71
Women	30	55

Test at the 5\% significance level whether Carl's belief is justified. [8]

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 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.

7 Wade and Odelia are investigating whether there is an association between the region where a person lives and the brand of washing powder they use. They decide to conduct a \(\chi ^ { 2 }\)-test for association and survey a random sample of 200 people. The expected frequencies for the test have been calculated and are shown in the contingency table below.

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 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.

6 During August, 102 candidates took their driving test at centre \(A\) and 60 passed. During the same month, 110 candidates took their driving test at centre \(B\) and 80 passed. 6

1. Test whether the driving test result is independent of the driving test centre using the \(5 \%\) level of significance. 6
2. Rebecca claims that if the result of the test in part (a) is to reject the null hypothesis then it is easier to pass a driving test at centre \(B\) than centre \(A\). State, with a reason, whether or not you agree with Rebecca's claim.

2 A large multinational company recruits employees from all four countries in the UK. For a sample of 250 recruits, the percentages of males and females from each of the countries are shown in Table 1. \begin{table}[h] \end{table}

1. Add the frequencies to the contingency table, Table 2, below.
2. Carry out a \(\chi ^ { 2 }\)-test at the \(10 \%\) significance level to investigate whether there is an association between country and gender of recruits.
3. By comparing observed and expected values, make one comment about the distribution of female recruits.
   [0pt] [1 mark] \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 2}

   	England	Scotland	Wales	Northern Ireland	Total
   Male					145
   Female					105
   Total					250

   \end{table}

5 Two companies, \(P\) and \(Q\), produce a certain type of paint brush. An independent examiner rates the quality of the brushes produced as poor, satisfactory or good. He takes a random sample of brushes from each company. The examiner's ratings are summarised in the table.

1. Test, at the \(5 \%\) significance level, whether quality of brushes is independent of company.
2. Compare the quality of the brushes produced by the two companies.

3 A sample survey, conducted to determine the attitudes of residents to a proposed reorganisation of local schools, gave the following results. Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether there is an association between the ages of residents and their attitudes to the proposed reorganisation of local schools.

2 A test for association is to be carried out. The tables below show the observed frequencies and the expected frequencies that are to be used for the test. It is necessary to merge some rows or columns before the test can be carried out.
Find the entry in the tables that provides evidence for this.
Circle your answer.
[0pt] [1 mark]
Observed A-Z
Observed B-Z
Expected A-X
Expected B-Y

6 A scientist is investigating whether the ability to remember depends on age. A random sample of 150 students in different age groups is chosen. Each student is shown a set of 20 objects for thirty seconds and then asked to list as many as they can remember. The students are graded \(A\) or \(B\) according to how many objects they remembered correctly: grade \(A\) for 16 or more correct and grade \(B\) for fewer than 16 correct. The results are shown in the table.

1. Carry out a \(\chi ^ { 2 }\)-test at the \(2.5 \%\) significance level to test whether grade is independent of age of student.
   The scientist decides instead to use three grades: grade \(A\) for 16 or more correct, grade \(B\) for 10 to 15 correct and grade \(C\) for fewer than 10 correct. The results are shown in the following table.

   \multirow{2}{*}{}	Age of students
   	11-12 years	13-14 years	15-16 years
   Grade \(A\)	25	16	19
   Grade \(B\)	12	27	11
   Grade \(C\)	16	18	6

   With this second set of data, the test statistic is calculated as 10.91.
2. Complete the \(\chi ^ { 2 }\)-test at the \(2.5 \%\) significance level for this second set of data.
3. State, with a reason, whether you would prefer to use the result from part (a) or part (b) to investigate whether the ability to remember depends on age.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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).

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.

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).

A \(\chi^2\) test is carried out in a school to test for association between the class a student belongs to and the number of times they are late to school in a week. The contingency table below gives the expected values for the test. Find a possible value for the degrees of freedom for the test. Circle your answer. [1 mark] 6 \quad 8 \quad 12 \quad 15

In a test of association of two factors, \(A\) and \(B\), a \(2 \times 2\) contingency table yielded \(5.63\) for the value of \(\chi^2\) with Yates' correction.

1. State the null hypothesis and alternative hypothesis for the test. [1]
2. State how Yates' correction is applied, and whether it increases or decreases the value of \(\chi^2\). [2]
3. Carry out the test at the \(2\frac{1}{2}\%\) significance level. [3]

5. A random sample of 500 adults completed a questionnaire on how often they took part in some form of exercise. They gave a response of 'never', 'sometimes' or 'regularly'. Of those asked, \(52 \%\) were females of whom \(10 \%\) never exercised and \(35 \%\) exercised regularly. Of the males, \(12.5 \%\) never exercised and \(55 \%\) sometimes exercised. Test, at the \(5 \%\) level of significance, whether or not there is any association between gender and the amount of exercise. State your hypotheses clearly.

5 Students at a science department of a university are offered the opportunity to study an optional language module, either German or Mandarin, during their second year of study. From a sample of 50 students who opted to study a language module, 31 were female. Of those who opted to study Mandarin, 8 were female and 12 were male. Test, using the \(5 \%\) level of significance, whether choice of language is independent of gender. The sample of students may be regarded as random.
[0pt] [8 marks] Turn over for the next question

Random samples of employees are taken from two companies, \(A\) and \(B\). Each employee is asked which of three types of coffee (Cappuccino, Latte, Ground) they prefer. The results are shown in the following table.

	Cappuccino	Latte	Ground
Company \(A\)	60	52	32
Company \(B\)	35	40	31

Test, at the 5\% significance level, whether coffee preferences of employees are independent of their company. [7] Larger random samples, consisting of \(N\) times as many employees from each company, are taken. In each company, the proportions of employees preferring the three types of coffee remain unchanged. Find the least possible value of \(N\) that would lead to the conclusion, at the 1\% significance level, that coffee preferences of employees are not independent of their company. [4]

6 A survey is carried out in an attempt to determine whether the salary achieved by the age of 30 is associated with having had a university education. The results of this survey are given in the table.

1. Use a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to determine whether the salary achieved by the age of 30 is associated with having had a university education.
2. What do you understand by a Type I error in this context?

5. Charlie carried out a survey on the main type of investment people have. The contingency table below shows the results of a survey of a random sample of people.

1. Find an expression, in terms of \(a , b , c\) and \(d\), for the difference between the observed and the expected value \(( O - E )\) for the group whose main type of investment is Bonds and are aged \(45 - 75\) Express your answer as a single fraction in its simplest form. Given that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.62\) for this information,
2. test, at the \(5 \%\) level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined.
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