Interpret association after test

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.

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 In a traffic survey a random sample of 400 cars passing a particular location during the rush hour is selected. The type of car and the sex of the driver are classified as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between type of car and sex of driver. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
2. For each type of car, comment briefly on how the number of drivers of each sex compares with what would be expected if there were no association. }{www.ocr.org.uk}) after the live examination series.
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4 In a survey a random sample of 63 runners is selected. The category of runner and the type of running are classified as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between category of runner and the type of running. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
2. For each category of runner, comment briefly on how the type of running compares with what would be expected if there were no association.

4 An art gallery is holding an exhibition. A random sample of 150 visitors to the exhibition is selected. The visitors are asked which of four artists they prefer. Their preferences, classified according to whether the visitor is female or male, are given in the table.

1. Carry out a test at the \(10 \%\) significance level to examine whether there is any association between artist preferred and sex of visitor. Your working should include a table showing the contributions of each cell to the test statistic.
2. For each artist, comment briefly on how the preferences of each sex compare with what would be expected if there were no association.

4 A researcher at a large company thinks that there may be some relationship between the numbers of working days lost due to illness per year and the ages of the workers in the company. The researcher selects a random sample of 190 workers. The ages of the workers and numbers of days lost for a period of 1 year are summarised below.

1. Carry out a test at the \(1 \%\) significance level to investigate whether the researcher's belief appears to be true. Your working should include a table showing the contributions of each cell to the test statistic.
2. For the 'Over 50' age group, comment briefly on how the working days lost compare with what would be expected if there were no association.
3. A student decides to reclassify the 'working days lost' into two groups, ' 0 to 4 ' and ' 5 or more', but leave the age groups as before. The test statistic with this classification is 7.08 . Carry out the test at the \(1 \%\) level with this new classification, using the same hypotheses as for the original test.
4. Comment on the results of the two tests. \section*{END OF QUESTION PAPER}

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.

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}

2 Year 12 students at Newstatus School choose to participate in one of four sports during the Spring term. The students' choices are summarised in the table.

1. Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether the choice of sport is independent of gender.
2. Interpret your result in part (a) as it relates to students choosing hockey.

7 A statistics unit is required to determine whether or not there is an association between students' performances in mathematics at Key Stage 3 and at GCE. A survey of the results of 500 students showed the following information:

1. Use a \(\chi ^ { 2 }\) test at the \(10 \%\) level of significance to determine whether there is an association between students' performances in mathematics at Key Stage 3 and at GCE.
2. Comment on the number of students who gained a grade A at GCE having gained a level 7 at Key Stage 3.

4 Julie, a driving instructor, believes that the first-time performances of her students in their driving tests are associated with their ages. Julie's records of her students' first-time performances in their driving tests are shown in the table.

1. Use a \(\chi ^ { 2 }\) test at the \(1 \%\) level of significance to investigate Julie's belief.
2. Interpret your result in part (a) as it relates to the 17-18 age group.

1 It is thought that the incidence of asthma in children is associated with the volume of traffic in the area where they live. Two surveys of children were conducted: one in an area where the volume of traffic was heavy and the other in an area where the volume of traffic was light. For each area, the table shows the number of children in the survey who had asthma and the number who did not have asthma.

1. Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether the incidence of asthma in children is associated with the volume of traffic in the area where they live.
2. Comment on the number of children in the survey who had asthma, given that they lived in an area where the volume of traffic was heavy.

6 Fiona, a lecturer in a school of engineering, believes that there is an association between the class of degree obtained by her students and the grades that they had achieved in A-level Mathematics. In order to investigate her belief, she collected the relevant data on the performances of a random sample of 200 recent graduates who had achieved grades A or B in A-level Mathematics. These data are tabulated below.

1. Conduct a \(\chi ^ { 2 }\) test, at the \(1 \%\) level of significance, to determine whether Fiona's belief is justified.
2. Make two comments on the degree performance of those students in this sample who achieved a grade B in A-level Mathematics.

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}

6 The captain of a sports team analyses the team's results according to the weather conditions, classified as "sunny" and "not sunny". The frequencies are shown in the following table.

1. Test at the \(5 \%\) significance level whether the team's performances are associated with weather conditions.
2. (a) Identify the cell that gives the largest contribution to the test statistic.
   (b) Interpret your answer to part (ii)(a).

7 A theatre has morning, afternoon and evening shows. On one particular day, the theatre asks all of its customers to state whether they enjoyed or did not enjoy the show. The results are summarised in the table. The theatre claims that there is no association between the show that a customer attends and whether they enjoyed the show. 7

1. Investigate the theatre's claim, using a \(2.5 \%\) level of significance.
   7
2. By considering observed and expected frequencies, interpret in context the association between the show that a customer attends and whether they enjoyed the show.

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.