Chi-squared test of independence

1. A doctor believes that the diet of her patients and their health are not independent.

She takes a random sample of 200 patients and records whether they are in good health or poor health and whether they have a good diet or a poor diet. The results are summarised in the table below. Stating your hypotheses clearly, test the doctor's belief using a \(5 \%\) level of significance. Show your working for your test statistic and state your critical value clearly.

1. A local village radio station, LSB, decides to survey adults in its broadcasting area about the programmes it produces. \(L S B\) broadcasts to 4 villages \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .
   The number of households in each of the villages is given below.

LSB decides to take a stratified sample of 200 households.

1. Explain how to select the households for this stratified sample.
   (3) One of the questions in the survey related to the age group of each member of the household and whether they listen to \(L S B\). The data received are shown below.

   \multirow{2}{*}{}	Age group
   	18-49	50-69	Older than 69
   Listen to LSB	130	162	65
   Do not listen to LSB	78	98	62

   The data are to be used to determine whether or not there is an association between the age group and whether they listen to \(L S B\).
2. Calculate the expected frequencies for the age group 50-69 that
   1. listen to \(L S B\)
   2. do not listen to \(L S B\) (2) Given that for the other 4 classes \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 4.657\) to 3 decimal places,
3. test at the \(5 \%\) level of significance, whether or not there is evidence of an association between age and listening to \(L S B\). Show your working clearly, stating the degrees of freedom and the critical value.

1. A Head of Department at a large university believes that gender is independent of the grade obtained by students on a Business Foundation course. A random sample was taken of 200 male students and 160 female students who had studied the course.

The results are summarised below. Stating your hypotheses clearly, test the Head of Department's belief using a \(5 \%\) level of significance. Show your working clearly.

5. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below. Test, at the \(5 \%\) level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly. \includegraphics[max width=\textwidth, alt={}, center]{304e58fa-eb82-4e2d-83f4-848f3eb461c8-15_2576_1774_141_159}

4. People over the age of 65 are offered an annual flu injection. A health official took a random sample from a list of patients who were over 65 . She recorded their gender and whether or not the offer of an annual flu injection was accepted or rejected. The results are summarised below. Using a \(5 \%\) significance level, test whether or not there is an association between gender and acceptance or rejection of an annual flu injection. State your hypotheses clearly.

5. The manager of a leisure centre collected data on the usage of the facilities in the centre by its members. A random sample from her records is summarised below. Making your method clear, test whether or not there is any evidence of an association between gender and use of the club facilities. State your hypotheses clearly and use a \(5 \%\) level of significance.
(11)

4. A new drug to treat the common cold was used with a randomly selected group of 100 volunteers. Each was given the drug and their health was monitored to see if they caught a cold. A randomly selected control group of 100 volunteers was treated with a dummy pill. The results are shown in the table below. Using a \(5 \%\) significance level, test whether or not the chance of catching a cold is affected by taking the new drug. State your hypotheses clearly.

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.

1. (a) State two reasons why stratified sampling might be chosen as a method of sampling when carrying out a statistical survey.
   (b) State one advantage and one disadvantage of quota sampling.
2. A sample of size 5 is taken from a population that is normally distributed with mean 10 and standard deviation 3 . Find the probability that the sample mean lies between 7 and 10 .
   (Total 6 marks)
3. A researcher carried out a survey of three treatments for a fruit tree disease. The contingency table below shows the results of a survey of a random sample of 60 diseased trees.

Test, at the \(5 \%\) level of significance, whether or not there is any association between the treatment of the trees and their survival. State your hypotheses and conclusion clearly.
(Total 11 marks)

6. A research worker studying colour preference and the age of a random sample of 50 children obtained the results shown below. Using a \(5 \%\) significance level, carry out a test to decide whether or not there is an association between age and colour preference. State your hypotheses clearly.

1. The Director of Studies at a large college believed that students' grades in Mathematics were independent of their grades in English. She examined the results of a random group of candidates who had studied both subjects and she recorded the number of candidates in each of the 6 categories shown.

1. Stating your hypotheses clearly, test the Director's belief using a \(10 \%\) level of significance. You must show each step of your working. The Head of English suggested that the Director was losing accuracy by combining the English grades C to U in one row. He suggested that the Director should split the English grades into two rows, grades C or D and grades E or U as for Mathematics.
2. State why this might lead to problems in performing the test.

2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results Showing your working clearly, test, at the \(1 \%\) level of significance, whether or not there is an association between gender and the type of course taken. State your hypotheses clearly.

1. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below.

Test, at the \(5 \%\) level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly.

3. A factory manufactures batches of an electronic component. Each component is manufactured in one of three shifts. A component may have one of two types of defect, \(D _ { 1 }\) or \(D _ { 2 }\), at the end of the manufacturing process. A production manager believes that the type of defect is dependent upon the shift that manufactured the component. He examines 200 randomly selected defective components and classifies them by defect type and shift. The results are shown in the table below. Stating your hypotheses, test, at the \(10 \%\) level of significance, whether or not there is evidence to support the manager's belief. Show your working clearly.

1. Two breeds of chicken are surveyed to measure their egg yield. The results are shown in the table below.

Showing each stage of your working clearly, test, at the \(5 \%\) significance level, whether or not there is an association between egg yield and breed of chicken. State your hypotheses clearly.

1. John thinks that a person's eye colour is related to their hair colour. He takes a random sample of 600 people and records their eye and hair colours. The results are shown in Table 1.

\begin{table}[h] \end{table} John carries out a \(\chi ^ { 2 }\) test in order to test whether eye colour and hair colour are related. He calculates the expected frequencies shown in Table 2. \begin{table}[h] \end{table}

1. Show how the value 47 in Table 2 has been calculated.
2. Write down the number of degrees of freedom John should use in this \(\chi ^ { 2 }\) test. Given that the value of the \(\chi ^ { 2 }\) statistic is 20.6 , to 3 significant figures,
3. find the smallest value of \(\alpha\) for which the null hypothesis will be rejected at the \(\alpha \%\) level of significance.
4. Use the data from Table 1 to test at the \(5 \%\) level of significance whether or not the proportions of people in the population with black, brown, red and blonde hair are in the ratio 2:6:1:3 State your hypotheses clearly.

1. A doctor takes a random sample of 100 patients and measures their intake of saturated fats in their food and the level of cholesterol in their blood. The results are summarised in the table below.

Using a \(5 \%\) level of significance, test whether or not there is an association between cholesterol level and intake of saturated fats. State your hypotheses and show your working clearly.

1. A survey asked a random sample of 200 people their age and the main use of their mobile phone.

The results are shown in Table 1 below. \begin{table}[h] \end{table} The data are to be used to test whether or not age and main use of their mobile phone are independent. Table 2 shows the expected frequencies for each group, assuming people's age and main use of their mobile phone are independent. \begin{table}[h] \end{table}

1. For users under 20 choosing the Internet as the main use of their mobile phone,
   1. verify that the expected frequency is 18.5
   2. show that the contribution to the \(\chi ^ { 2 }\) test statistic is 3.91 to 3 significant figures.
2. Given that the \(\chi ^ { 2 }\) test statistic for the data is 9.893 to 3 decimal places, test at the \(5 \%\) level of significance whether or not age and main use of their mobile phone are independent. State your hypotheses clearly.

3. A number of males and females were asked to rate their happiness under the headings "not happy", "fairly happy" and "very happy". The results are shown in the table below Stating your hypotheses, test at the \(5 \%\) level of significance, whether or not there is evidence of an association between happiness and gender. Show your working clearly.

1. (a) State two reasons why stratified sampling might be a more suitable sampling method than simple random sampling.
   (b) State two reasons why stratified sampling might be a more suitable sampling method than quota sampling.
2. A new drug to vaccinate against influenza was given to 110 randomly chosen volunteers. The volunteers were given the drug in one of 3 different concentrations, \(A , B\) and \(C\), and then were monitored to see if they caught influenza. The results are shown in the table below.

Test, at the \(10 \%\) level of significance, whether or not there is an association between catching influenza and the concentration of the new drug. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places.
(10)

4. A psychologist carries out a survey of the perceived body weight of 150 randomly chosen people. He asks them if they think they are underweight, about right or overweight. His results are summarised in the table below. The psychologist calculates two of the expected frequencies, to 2 decimal places, for a test of independence between perceived body weight and gender. These results are shown in the table below.

1. Complete the table of expected frequencies shown above.
2. Test, at the \(10 \%\) level of significance, whether or not perceived body weight is independent of gender. State your hypotheses clearly. The psychologist now combines the male and female data to test whether or not body weight types are chosen equally.
3. Find the smallest significance level, from the tables in the formula booklet, for which there is evidence of a preference.

7. A survey in a college was commissioned to investigate whether or not there was any association between gender and passing a driving test. A group of 50 male and 50 female students were asked whether they passed or failed their driving test at the first attempt. All the students asked had taken the test. The results were as follows. Stating your hypotheses clearly test, at the \(10 \%\) level, whether or not there is any evidence of an association between gender and passing a driving test at the first attempt.

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.