Chi-squared test of independence

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)

4 It is claimed that the area within which a school is situated affects the age profile of the staff employed at that school. In order to investigate this claim, the age profiles of staff employed at two schools with similar academic achievements are compared. Academia High School, situated in a rural community, employs 120 staff whilst Best Manor Grammar School, situated in an inner-city community, employs 80 staff. The percentage of staff within each age group, for each school, is given in the table.

1. 1. Form the data into a contingency table suitable for analysis using a \(\chi ^ { 2 }\) distribution.
      (2 marks)
   2. Use a \(\chi ^ { 2 }\) test, at the \(1 \%\) level of significance, to determine whether there is an association between the age profile of the staff employed and the area within which the school is situated.
2. Interpret your result in part (a)(ii) as it relates to the 22-34 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.

2

1. The continuous random variable \(X\) has a rectangular distribution defined by the probability density function $$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u \\ 0 & \text { otherwise } \end{cases}$$ where \(u\) is a constant.
   1. Show that \(u = \frac { 10 } { \pi }\).
   2. Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of \(\pi\), values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   3. Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length \(\left( X + \frac { 10 } { \pi } \right)\).
2. A machine produces circular discs which have an area of \(Y \mathrm {~cm} ^ { 2 }\). The distribution of \(Y\) has mean \(\mu\) and variance 25 . A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be \(40.5 \mathrm {~cm} ^ { 2 }\). Calculate a 95\% confidence interval for \(\mu\). Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area. The table shows Emily's collected data, denoted by \(O _ { i }\), together with the corresponding expected frequencies, \(E _ { i }\), necessary for a \(\chi ^ { 2 }\) test.

   \multirow{2}{*}{}	\(\boldsymbol { \geq } \mathbf { 5 }\) GCSEs		\(\mathbf { 1 } \boldsymbol { \leqslant }\) GCSEs < \(\mathbf { 5 }\)		No GCSEs
   	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)
   Jolliffe College for the Arts	187	193.15	93	90.62	30	26.23
   Volpe Science Academy	175	184.43	97	86.52	24	25.05
   Radok Music School	183	183.81	78	86.23	34	24.96
   Bailey Language School	265	248.61	112	116.63	22	33.76

   Emily used these values to correctly conduct a \(\chi ^ { 2 }\) test at the \(1 \%\) level of significance.

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 town council wanted residents to apply for grants that were available for home insulation. In a trial, a random sample of 200 residents was encouraged, either in a letter or by a phone call, to apply for the grants. The outcomes are shown in the table.

1. The council believed that a phone call was more effective than a letter in encouraging people to apply for a grant. Use a \(\chi ^ { 2 }\)-test to investigate this belief at the \(5 \%\) significance level.
2. After the trial, all the residents in the town were encouraged, either in a letter or by a phone call, to apply for the grants. It was found that there was no association between the method of encouragement and the outcome. State, with a reason, whether a Type I error, a Type II error or neither occurred in carrying out the test in part (a).
   (2 marks)

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 In a particular town, a survey was conducted on a sample of 200 residents aged 41 years to 50 years. The survey questioned these residents to discover the age at which they had left full-time education and the greatest rate of income tax that they were paying at the time of the survey. The summarised data obtained from the survey are shown in the table.

1. Use a \(\chi ^ { 2 }\)-test, at the \(5 \%\) level of significance, to investigate whether there is an association between age when leaving education and greatest rate of income tax paid.
2. It is believed that residents of this town who had left education at a later age were more likely to be paying the higher rate of income tax. Comment on this belief.
   [0pt] [1 mark]

5 A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.

1. Use a \(\chi ^ { 2 }\)-test to investigate, at the \(10 \%\) significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine.
2. Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems. Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa.
   [0pt] [2 marks]

4. A group of 40 males and 40 females were asked which of three "Reality TV" shows they liked most - Watched, Stranded or One-2-Win. The results were as follows: Stating your hypotheses clearly, test at the \(10 \%\) level whether or not there is a significant difference in the preferences of males and females.

4. A hospital administrator is assessing staffing needs for its Accident and Emergency Department at different times of day. The administrator already has data on the number of admissions at different times of day but needs to know if the proportion of the cases that are serious remains constant. Staff are asked to assess whether each person arriving at Accident and Emergency has a "minor" or "serious" problem and the results for three different time periods are shown below. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the proportion of serious injuries being different at different times of day.
(11 marks)

6. Two schools in the same town advertise at the same time for new heads of English and History departments. The number of applicants for each post are shown in the table below. Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not there is evidence of the proportion of applicants for each job being different in the two schools.
(11 marks) Turn over

6. A survey found that of the 320 people questioned who had passed their driving test aged under twenty-five, 104 had been involved in an accident in the two years following their test. Of the 80 people in the survey who were aged twenty-five or over when they passed their test, 16 had been involved in an accident in the following two years.

1. Draw up a contingency table showing this information. It is desired to test whether the proportion of drivers having accidents within two years of passing their test is different for those who were aged under twenty-five at the time of passing their test than for those aged twenty-five or over.
2. 1. Stating your hypotheses clearly, carry out the test at the \(5 \%\) level of significance.
   2. Explain clearly why there is only one degree of freedom. It is found that 12 people who were aged under twenty-five when they took their test and had been involved in an accident in the following two years had been omitted from the information given.
3. Explain why you do not need to repeat the calculation to know the correct result of the test.
   (2 marks)

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).

5. A Policy Unit wished to find out whether attitudes to the European Union varied with age. It conducted a survey asking 200 individuals to which of three age groups they belonged and whether they regarded themselves as generally pro-Europe or Eurosceptic. The results are shown in the table below.

1. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether attitudes to Europe are associated with age.
   (11 marks)
   The survey also asked people if they voted at the last election. When the above test was repeated using only the results from those who had voted a value of 4.872 was calculated for \(\sum \frac { ( O - E ) ^ { 2 } } { E }\). No classes were combined.
2. Find if this value leads to a different result.

5 A student wants to know if there is a positive correlation between the amounts of two pollutants, sulphur dioxide and PM10 particulates, on different days in the area of London in which he lives; these amounts, measured in suitable units, are denoted by \(s\) and \(p\) respectively.
He uses a government website to obtain data for a random sample of 15 days on which the amounts of these pollutants were measured simultaneously. Fig. 5.1 is a scatter diagram showing the data. Summary statistics for these 15 values of \(s\) and \(p\) are as follows. \(\sum s _ { 1 } = 155.4 \quad \sum p = 518.9 \quad \sum s ^ { 2 } = 2322.7 \quad \sum p ^ { 2 } = 21270.5 \quad \sum s p = 6009.1\) \begin{figure}[h] \end{figure}

1. Explain why the student might come to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.
2. Find the value of Pearson's product moment correlation coefficient.
3. Carry out a test at the \(5 \%\) significance level to investigate whether there is positive correlation between the amounts of sulphur dioxide and PM10 particulates.
4. Explain why the student made sure that the sample chosen was a random sample. The student also wishes to model the relationship between the amounts of nitrogen dioxide \(n\) and PM10 particulates \(p\).
   He takes a random sample of 54 values of the two variables, both measured at the same times. Fig. 5.2 is a scatter diagram which shows the data, together with the regression line of \(n\) on \(p\), the equation of the regression line and the value of \(r ^ { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4a4d5816-5b53-49a1-b72f-f8bcf3b4e8bc-5_824_1230_495_258} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
5. Predict the value of \(n\) for \(p = 150\).
6. Discuss the reliability of your prediction in part (e).

4 A genetics researcher is investigating whether there is any association between natural hair colour and natural eye colour. A random sample of 800 adults is selected. Each adult can categorise their natural hair colour as blonde, brown, black or red and their natural eye colour as brown, blue or green.

1. Explain the benefit of using a random sample in this investigation. The data collected from the sample are summarised in Table 4.1. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 4.1}

   \multirow{2}{*}{Observed frequency}		Hair Colour
   		Blonde	Brown	Black	Red	Total
   \multirow{3}{*}{Eye Colour}	Brown	47	153	196	36	432
   	Blue	61	78	115	26	280
   	Green	19	22	31	16	88
   	Total	127	253	342	78	800

   \end{table} The researcher decides to carry out a chi-squared test.
2. Determine the expected frequencies for each eye colour in the blonde hair category. You are given that the test statistic is 28.62 to 2 decimal places.
3. Carry out the chi-squared test at the 10\% significance level. Table 4.2 shows the chi-squared contributions for some of the categories. The contributions for the categories relating to green eye colour have been deliberately omitted. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 4.2}

   Hair Colour \cline { 2 - 6 } Blonde Brown Black Red \multirow{3}{*}{ Eye Colour } Brown 6.791 1.964 0.694 0.889 \cline { 2 - 6 } Blue 6.162 1.257 0.185 0.062 \cline { 2 - 6 } Green \end{table} Calculate the chi-squared contribution for the green eye and blonde hair category. With reference to the values in Table 4.2, discuss what the data suggest about brown eye colour and blue eye colour for people with blonde hair. A different researcher, carrying out the same investigation, independently takes a different random sample of size 800 and performs the same hypothesis test, but at the 1\% significance level, reaching the same conclusion as the original test. By comparing only the significance level of the two tests, specify which test, the one at the 10\% significance level or the one at the 1\% significance level, provides stronger evidence for the conclusion. Justify your answer. OCR MEI Further Statistics Major 2022 June Q10 13 marks Standard +0.3 10 A scientist is researching dietary fat intake and cholesterol level. A random sample of 60 people is selected and their dietary fat intakes and cholesterol levels are measured. Dietary fat intakes are classified as low, medium and high, and cholesterol levels are classified as normal and high. The scientist decides to carry out a chi-squared test to investigate whether there is any association between dietary fat intake and cholesterol level. Tables \(\mathbf { 1 0 . 1 }\) and \(\mathbf { 1 0 . 2 }\) show the data and some of the expected frequencies for the test. \begin{table}[h] \multirow{2}{*}{} Dietary fat intake Low Medium High Total \multirow{2}{*}{Cholesterol level} Normal 9 18 5 32 High 3 13 12 28 Total 12 31 17 60 \captionsetup{labelformat=empty} \caption{Table 10.1} \end{table} \begin{table}[h] Expected frequency Dietary fat intake \cline { 3 - 5 } Low Medium High \multirow{2}{*}{ Cholesterol level } Normal 9.0667 \cline { 2 - 5 } High 7.9333 \captionsetup{labelformat=empty} \caption{Table 10.2} \end{table} Complete the table of expected frequencies in the Printed Answer Booklet. Determine the contribution to the chi-squared test statistic for people with normal cholesterol level and high dietary fat intake, giving your answer to \(\mathbf { 4 }\) decimal places. The contributions to the chi-squared test statistic for the remaining categories are shown in Table 10.3. \begin{table}[h] Dietary fat intake \cline { 2 - 5 } Low Medium High \multirow{2}{*}{ Cholesterol level } Normal 1.0563 0.1301 \cline { 2 - 5 } High 1.2071 0.1487 2.0846 \captionsetup{labelformat=empty} \caption{Table 10.3} \end{table} In this question you must show detailed reasoning. Carry out the test at the 5\% significance level. For each level of dietary fat intake, give a brief interpretation of what the data suggest about the level of cholesterol. OCR MEI Further Statistics Major 2023 June Q9 10 marks Standard +0.3 9 A cyclist who lives on an island suspects that car drivers with locally registered number plates allow more space when passing her than those with non-locally registered number plates. She decides to carry out a hypothesis test and so over a period of time selects a random sample of 250 cars which pass her. For each car she estimates whether the car driver allows at least the recommended 1.5 metres when passing her. The table shows the data which she collected. Where registered \cline { 3 - 4 } \multicolumn{2}{|c|}{} Local Non-local \multirow{2}{*}{ Passing distance } Under 1.5 m 12 11 \cline { 2 - 4 } At least 1.5 m 157 70 In this question you must show detailed reasoning. Carry out the test at the \(5 \%\) significance level to examine whether there is any association between where the car is registered and passing distance. A friend of the cyclist suggests that there may be a problem with the data, since the cyclist may have introduced some bias in estimating whether cars were allowing the recommended distance. Explain how any bias might have arisen. OCR MEI Further Statistics Major 2024 June Q9 13 marks Standard +0.3 9 A cyclist has 3 bicycles, a road bike, a gravel bike and an electric bike. She wishes to know if the bicycle which she is riding makes any difference to whether she reaches a speed of 25 mph or greater on a journey. She selects a random sample of 120 journeys and notes the bicycle and whether or not her maximum speed was 25 mph or greater. She decides to carry out a chisquared test to investigate whether there is any association between bicycle type and whether her maximum speed is 25 mph or greater. Tables 9.1 and 9.2 show the data and some of the expected frequencies for the test. \begin{table}[h] \captionsetup{labelformat=empty} \caption{Table 9.1} \multirow{2}{*}{} Bicycle Road Gravel Electric Total \multirow{2}{*}{Maximum speed} Less than 25 mph 2 21 19 42 25 mph or greater 13 47 18 78 Total 15 68 37 120 \end{table} \begin{table}[h] \captionsetup{labelformat=empty} \caption{Table 9.2} \multirow{2}{*}{Expected frequency} Bicycle Road Gravel Electric \multirow{2}{*}{Maximum speed} Less than 25 mph 12.95 25 mph or greater 24.05 \end{table} Complete the table of expected frequencies in the Printed Answer Booklet. Determine the contribution to the chi-squared test statistic for the Electric bicycle and maximum speed 25 mph or greater. Give your answer correct to 4 decimal places. The contributions to the chi-squared test statistic for the remaining categories are shown in Table 9.3. \begin{table}[h] \captionsetup{labelformat=empty} \caption{Table 9.3} \multirow{2}{*}{Contribution to the test statistic} Bicycle Road Gravel Electric \multirow{2}{*}{Maximum speed} Less than 25 mph 2.0119 0.3294 2.8264 25 mph or greater 1.0833 0.1774 \end{table} In this question you must show detailed reasoning. Carry out the test at the 5\% significance level. For each type of bicycle, give a brief interpretation of what the data suggest about maximum speed.