5.06a Chi-squared: contingency tables

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.

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. 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.

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.

6. \begin{figure}[h] \end{figure} The sketch in Figure 1 represents a target which consists of 4 regions formed from 4 concentric circles of radii \(4 \mathrm {~cm} , 7 \mathrm {~cm} , 9 \mathrm {~cm}\) and 10 cm . The regions are coloured as labelled in Figure 1.
A random sample of 100 children each choose a point on the target and their results are summarised in the table below. (b) Find the value of \(r\) and the value of \(s\). Henry obtained a test statistic of 6.188 and no groups were pooled.
(c) State what conclusion Henry should make about his claim. Phoebe believes that the children chose the region of the target according to colour. She believes that boys and girls would favour different colours and splits the original data by gender to obtain the following table. \section*{Observed frequencies} (d) State suitable hypotheses to test Phoebe's belief. Phoebe calculated the following expected frequencies to carry out a suitable test. \section*{Expected frequencies} (e) Show how the value of 25.35 was obtained. Phoebe carried out the test using 2 degrees of freedom and a \(10 \%\) level of significance. She obtained a test statistic of 1.411
(f) Explain clearly why Phoebe used 2 degrees of freedom.
(g) Stating your critical value clearly, determine whether or not these data support Phoebe's belief.

2. \begin{figure}[h] \end{figure} The pointer shown in Figure 1 is spun so that it comes to rest between 0 and 360 degrees.
Linda claims that it is equally likely to come to rest at any point between 0 and 360 degrees. She spins the pointer 100 times and her results are summarised in the table below. She calculates expected frequencies for some of the possible outcomes and these are also given in the table below.

1. Find the values of the missing expected frequencies \(a , b\) and \(c\).
2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not Linda's claim is supported by these data.

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.

3 The table shows the colour of hair and the colour of eyes of a sample of 750 people from a particular population.

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.

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?

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.

2 It is claimed that the way in which students voted at a particular general election was independent of their gender. In order to investigate this claim, 480 male and 540 female students who voted at this general election were surveyed. These students may be regarded as a random sample. The percentages of males and females who voted for the different parties are recorded in the table.

1. Complete the contingency table below.
2. Hence determine, at the \(1 \%\) level of significance, whether the way in which students voted at this general election was independent of their gender.

   	Conservative	Labour	Liberal Democrat	Other parties	Total
   Male					480
   Female					540
   Total					1020

3

1. Table 1 contains the observed frequencies, \(a , b , c\) and \(d\), relating to the two attributes, \(X\) and \(Y\), required to perform a \(\chi ^ { 2 }\) test. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 1}

   \cline { 2 - 4 } \multicolumn{1}{c|}{}	\(\boldsymbol { Y }\)	Not \(\boldsymbol { Y }\)	Total
   \(\boldsymbol { X }\)	\(a\)	\(b\)	\(m\)
   Not \(\boldsymbol { X }\)	\(c\)	\(d\)	\(n\)
   Total	\(p\)	\(q\)	\(N\)

   \end{table}
   1. Write down, in terms of \(m , n , p , q\) and \(N\), expressions for the 4 expected frequencies corresponding to \(a , b , c\) and \(d\).
   2. Hence prove that the sum of the expected frequencies is \(N\).
2. Andy, a tennis player, wishes to investigate the possible effect of wind conditions on the results of his matches. The results of his matches for the 2011 season are represented in Table 2. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 2}

   \cline { 2 - 4 } \multicolumn{1}{c|}{}	Windy	Not windy	Total
   Won	15	18	33
   Lost	12	5	17
   Total	27	23	50

   \end{table} Conduct a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to investigate whether there is an association between Andy's results and wind conditions.
   (8 marks)

2 A large estate agency would like all the properties that it handles to be sold within three months. A manager wants to know whether the type of property affects the time taken to sell it. The data for a random sample of properties sold are tabulated below.

1. Conduct a \(\chi ^ { 2 }\)-test, at the \(10 \%\) level of significance, to determine whether there is an association between the type of property and the time taken to sell it. Explain why it is necessary to combine two columns before carrying out this test.
2. The manager plans to spend extra money on advertising for one type of property in an attempt to increase the number sold within three months. Explain why the manager might choose:
   1. terraced properties;
   2. flats.
      (2 marks)

2 Syd, a snooker player, believes that the outcome of any frame of snooker in which he plays may be influenced by the time of day that the frame takes place. The results of 100 randomly selected frames of snooker, played by Syd, are recorded below. Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to test Syd's belief.
(10 marks)