Chi-squared test of independence

10 Three new flu vaccines, \(A , B\) and \(C\), were tested on 500 volunteers. The vaccines were assigned randomly to the volunteers and 178 received \(A , 149\) received \(B\) and 173 received \(C\). During the following year, 30 of the volunteers given \(A\) caught flu, 29 of the volunteers given \(B\) caught flu, and 16 of the volunteers given \(C\) caught flu. Carry out a suitable test for independence at the 5\% significance level. Without using a statistical test, decide which of the vaccines appears to be most effective.

6 A random sample of residents in a town took part in a survey. They were asked whether they would prefer the local council to spend money on improving the local bus service or on improving the quality of road surfaces. The responses are shown in the following table, classified according to the area of the town in which the residents live. Using a \(5 \%\) significance level, test whether there is an association between the area lived in and preference for improving the local bus service or improving the quality of road surfaces.

10 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. Test, at the 5\% significance level, whether coffee preferences of employees are independent of their company. 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.

8 Residents of three towns \(A , B\) and \(C\) were asked to grade the reliability of their digital television signal as good, satisfactory or poor. A random sample of responses from each town is taken and the numbers in each category are given in the following table. Test, at the 2.5\% significance level, whether grade of reliability is independent of town. Identify which town makes the greatest contribution to the test statistic and relate your answer to the context of the question.

A researcher is investigating the relationship between the political allegiance of university students and their childhood environment. He chooses a random sample of 100 students and finds that 60 have political allegiance to the Alliance party. He also classifies their childhood environment as rural or urban, and finds that 45 had a rural childhood. The researcher carries out a test, at the \(10 \%\) significance level, on this data and finds that political allegiance is independent of childhood environment. Given that \(A\) is the number of students in the sample who both support the Alliance party and have a rural childhood, find the greatest and least possible values of \(A\). A second random sample of size \(100 N\), where \(N\) is an integer, is taken from the university student population. It is found that the proportions supporting the Alliance party from urban and rural childhoods are the same as in the first sample. Given that the value of \(A\) in the first sample was 29, find the greatest possible value of \(N\) that would lead to the same conclusion (that political allegiance is independent of childhood environment) from a test, at the \(10 \%\) significance level, on this second set of data.

6 The reliability of the broadband connection received from two suppliers, \(A\) and \(B\), is classified as good, fair or poor by a random sample of householders. The information collected is summarised in the following table. Test, at the 5\% significance level, whether reliability is independent of supplier.

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.

8 Two salesmen, \(A\) and \(B\), work at a company that arranges different types of holidays: self-catering, hotel and cruise. The table shows, for a random sample of 150 holidays, the number of each type arranged by each salesman. Test at the 10\% significance level whether the type of holiday arranged is independent of the salesman.

8 The owner of three driving schools, \(A , B\) and \(C\), wished to assess whether there was an association between passing the driving test and the school attended. He selected a random sample of learner drivers from each of his schools and recorded the numbers of passes and failures at each school. The results that he obtained are shown in the table below. Using a \(\chi ^ { 2 }\)-test and a \(5 \%\) level of significance, test whether there is an association between passing or failing the driving test and the driving school attended.

10 Customers were asked which of three brands of coffee, \(A , B\) and \(C\), they prefer. For a random sample of 80 male customers and 60 female customers, the numbers preferring each brand are shown in the following table. Test, at the \(5 \%\) significance level, whether there is a difference between coffee preferences of male and female customers. A larger random sample is now taken. It consists of \(80 n\) male customers and \(60 n\) female customers, where \(n\) is a positive integer. It is found that the proportions choosing each brand are identical to those in the smaller sample. Find the least value of \(n\) that would lead to a different conclusion for the 5\% significance level hypothesis test.

8 Members of a Statistics club are voting to elect a new president of the club. Members must choose to vote either by post or by text or by email. The method of voting chosen by a random sample of 60 male members and 40 female members is given in the following table. Test, at the \(1 \%\) significance level, whether there is an association between method of voting and gender.

7 Two employees, \(A\) and \(B\), both produce the same toy for a company. The company records the total number of errors made per day by each employee during a 40-day period. The results are summarised in the following table. Employee The company claims that there is an association between employee and number of errors made per day. 7

1. Test the company's claim, using the \(5 \%\) level of significance.
   7
2. By considering observed and expected frequencies, interpret in context the association between employee and number of errors made per day. \includegraphics[max width=\textwidth, alt={}, center]{9be40ed6-6df8-426a-8afd-fefc17287de6-12_2492_1723_217_150}
   \includegraphics[max width=\textwidth, alt={}]{9be40ed6-6df8-426a-8afd-fefc17287de6-16_2496_1721_214_148}

7 A dairy industry researcher, Robyn, decided to investigate the milk yield, classified as low, medium or high, obtained from four different breeds of cow, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . The milk yield of a sample of 105 cows was monitored and the results are summarised in contingency Table 1. The sample of cows may be regarded as random.
Robyn decides to carry out a \(\chi ^ { 2 }\)-test for association between milk yield and breed using the information given in Table 1. 7

1. Contingency Table 2 gives some of the expected frequencies for this test.
   Complete Table 2 with the missing expected values.

   \multirow[t]{2}{*}{Table 2}		Yield
   		Low	Medium	High
   \multirow{4}{*}{Breed}	A			6
   	B	5.14	9.14	5.71
   	C
   	D	8.23	14.63	9.14

   7
2. 1. For Robyn's test, the test statistic \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 19.4\) correct to three significant figures.
      Use this information to carry out Robyn's test, using the \(1 \%\) level of significance.
      7
3. (ii) By considering the observed frequencies given in Table 1 with the expected frequencies in Table 2, interpret, in context, the association, if any, between milk yield and breed.

4 A school pupil keeps a note of whether her journeys to school and from school are delayed. The results for a random sample of journeys are shown in the table. Test at the 10\% significance level whether there is association between delays and the direction of the journey.

2 For a random sample of 160 employees of a large company, the principal method of transport for getting to work, arranged according to grade of employee, is shown in the table. A test is carried out at the \(5 \%\) significance level of whether there is association between grade of employee and method of transport.

1. State appropriate hypotheses for the test. The contributions to the test statistic are shown in the following table, correct to 3 decimal places.

   Grade	Walk or cycle	Private motorised transport	Public transport
   A	1.157	0.289	1.929
   B	1.878	0.225	0.327
   C	2.006	1.800	0.083

2. Show how the value 0.225 is obtained.
3. Complete the test, stating the conclusion.
4. Which combination of grade of employee and method of transport most strongly suggests association? Justify your answer.

5 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} The cinema manager carries out a test of whether there is any association between age group and session attended.

1. Show that it is necessary to combine cells in order to carry out the test. It is decided to combine the second and third rows of the table. Some of the expected frequencies for the table with rows combined, and the corresponding contributions to the \(\chi ^ { 2 }\) test statistic, are shown in the following incomplete tables. \begin{table}[h]

   \multirow{2}{*}{Expected frequencies}		Session
   		Early	Middle	Late
   \multirow{2}{*}{Age group}	< 25	29.4	23.1
   	\(\geqslant 25\)	26.6	20.9

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table} \begin{table}[h]

   \multirow{2}{*}{Contribution to \(\chi ^ { 2 }\)}		Session
   		Early	Middle	Late
   \multirow{2}{*}{Age group}	< 25	0.9918	0.4160
   	\(\geqslant 25\)	1.0962	0.4598

   \captionsetup{labelformat=empty} \caption{Table 3}
   \end{table}
2. In the Printed Answer Booklet, complete both tables.
3. Carry out the test at the \(5 \%\) significance level.
4. Use the figures in your completed Table 3 to comment on the numbers of the audience in different age groups.

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

6 Yusha investigates the proportion of left-handed people living in two cities, \(A\) and \(B\). He obtains data from random samples from the two cities. His results are shown in the table, in which \(L\) denotes "left-handed".

1. Test at the 10\% significance level whether there is association between being left-handed and living in a particular city. A person is chosen at random from one of the cities \(A\) and \(B\).
   Let \(A\) denote "the person lives in city \(A\) ".
2. State the relationship between \(\mathrm { P } ( L )\) and \(P ( L \mid A )\) according to the model implied by the null hypothesis of your test.
3. Use the data in the table to suggest a value for \(P ( L \mid A )\) given by an improved model.

4. A manager at a large estate agency believes that the type of property affects the time taken to sell it. A random sample of 125 properties sold is shown in the table. Test, at the \(5 \%\) level of significance, whether there is evidence for an association between the type of property and the time taken to sell it. You should state your hypotheses, expected frequencies, test statistic and the critical value used for this test.

1. Chen is treating vines to prevent fungus appearing. One month after the treatment, Chen monitors the vines to see if fungus is present.

The contingency table shows information about the type of treatment for a sample of 150 vines and whether or not fungus is present. Test, at the \(5 \%\) level of significance, whether or not there is any association between the type of treatment and the presence of fungus.
Show your working clearly, stating your hypotheses, expected frequencies, test statistic and critical value.

1. A random sample of 200 people were asked which hot drink they preferred from tea, coffee and hot chocolate. The results are given below.