Standard 2×3 contingency table

A question is this type if and only if the data form a 2-row by 3-column (or 3-row by 2-column) contingency table requiring a chi-squared test of independence with 2 degrees of freedom, with no need to combine cells.

36 questions · Standard +0.1

5.06a Chi-squared: contingency tables
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CAIE Further Paper 4 2020 June Q1
6 marks Standard +0.3
1 Young children are learning to read using two different reading schemes, \(A\) and \(B\). The standards achieved are measured against the national average standard achieved and classified as above average, average or below average. For two randomly chosen groups of young children, the numbers in each category are shown in the table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Standard achieved
\cline { 2 - 4 } \multicolumn{1}{c|}{}Above averageAverageBelow average
Scheme \(A\)313522
Scheme \(B\)195043
Test at the \(5 \%\) significance level whether standard achieved is independent of the reading scheme used.
CAIE Further Paper 4 2020 June Q1
6 marks Standard +0.3
1 Two randomly selected groups of students, with similar ranges of abilities, take the same examination in different rooms. One group of 140 students takes the examination with background music playing. The other group of 210 students takes the examination in silence. Each student is awarded a grade for their performance in the examination and the numbers from each group gaining each grade are shown in the following table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Grade awarded
\cline { 2 - 4 } \multicolumn{1}{c|}{}ABC
Background music495140
Silence936849
Test at the 10\% significance level whether grades awarded are independent of whether background music is playing during the examination.
OCR S2 2011 January Q10
Moderate -0.5
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    OCR S2 2011 January Q12
    Moderate -0.5
    12
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    OCR S2 2011 January Q13
    Moderate -0.5
    13
    8
    		(continued)
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  • \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT
    OCR S3 2010 January Q7
    14 marks Standard +0.3
    7 A chef wished to ascertain her customers' preference for certain vegetables. She asked a random sample of 120 customers for their preferred vegetable from asparagus, broad beans and cauliflower. The responses, classified according to the gender of the customer, are shown in the table.
    1. Test, at the \(5 \%\) significance level, whether vegetable preference and gender are independent.
    2. Determine whether, at the \(10 \%\) significance level, the vegetables are equally preferred.
    OCR S3 2012 June Q7
    16 marks Standard +0.3
    7 A study was carried out into whether patients suffering from a certain respiratory disorder would benefit from particular treatments. Each of 90 patients who agreed to take part was given one of three treatments \(A\), \(B\) or \(C\) as shown in the table.
    Treatment\(A\)\(B\)\(C\)
    Number in group312534
    1. It is claimed that each patient was equally likely to have been given any of the treatments. Test at the \(5 \%\) significance level whether the numbers given each treatment are consistent with this claim.
    2. After 3 months the numbers of patients showing improvement for treatments \(A , B\) and \(C\) were 14, 18 and 25 respectively. By setting up a \(2 \times 3\) contingency table, test whether the outcome is dependent on the treatment. Use a \(5 \%\) significance level.
    3. If one of the treatments is abandoned, explain briefly which it should be. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
    CAIE FP2 2010 June Q10
    13 marks Standard +0.3
    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.
    CAIE FP2 2011 June Q6
    7 marks Standard +0.3
    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.
    Area 1Area 2Area 3
    Local bus service733630
    Road surfaces474420
    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.
    CAIE FP2 2015 June Q6
    8 marks Moderate -0.3
    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.
    Reliability
    \cline { 3 - 5 } \multicolumn{2}{|c|}{}GoodFairPoor
    \multirow{2}{*}{Supplier}\(A\)656333
    \cline { 2 - 5 }\(B\)514444
    Test, at the 5\% significance level, whether reliability is independent of supplier.
    CAIE FP2 2019 June Q8
    8 marks Standard +0.3
    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.
    Type of holiday
    \cline { 3 - 5 } \multicolumn{2}{|c|}{}Self-cateringHotelCruise
    \multirow{2}{*}{Salesman}\(A\)253821
    \cline { 2 - 5 }\(B\)282117
    Test at the 10\% significance level whether the type of holiday arranged is independent of the salesman.
    CAIE FP2 2017 November Q8
    8 marks Standard +0.3
    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.
    \cline { 3 - 5 } \multicolumn{2}{c|}{}Method of voting
    \cline { 3 - 5 } \multicolumn{2}{c|}{}PostTextEmail
    \multirow{2}{*}{Gender}Male101238
    \cline { 2 - 5 }Female52114
    Test, at the \(1 \%\) significance level, whether there is an association between method of voting and gender.
    Edexcel S3 2022 January Q4
    10 marks Standard +0.3
    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.
    \multirow{2}{*}{}Type of property
    BungalowFlatHouseTotal
    Sold within three months7294682
    Sold in more than three months9191543
    Total164861125
    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.
    Edexcel S3 2022 January Q4
    14 marks Standard +0.3
    1. A survey was carried out with students that had studied Maths, Physics and Chemistry at a college between 2016 and 2020. The students were divided into two groups \(A\) and \(B\).
      1. Explain how a sample could be obtained from this population using quota sampling.
      The students were asked which of the three subjects they enjoyed the most. The results of the survey are shown in the table.
      \multirow{2}{*}{}Subject enjoyed the most
      MathsPhysicsChemistryTotal
      Group A16101339
      Group B38131061
      Total542323100
    2. Test, at the \(5 \%\) level of significance, whether the subject enjoyed the most is independent of group. You should state your hypotheses, expected frequencies, test statistic and the critical value used for this test. The Headteacher discovered later that the results were actually based on a random sample of 200 students but had been recorded in the table as percentages.
    3. For the test in part (b), state with reasons the effect, if any, that this information would have on
      1. the null and alternative hypotheses,
      2. the critical value,
      3. the value of the test statistic,
      4. the conclusion of the test.
    Edexcel S3 2024 January Q1
    8 marks Standard +0.3
    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.
    \multirow{2}{*}{}Type of treatment
    NoneSulphurCopper sulphate
    No fungus present205548
    Fungus present1089
    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.
    Edexcel S3 2014 June Q5
    12 marks Standard +0.3
    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.
    \cline { 3 - 6 } \multicolumn{2}{|c|}{}
    \multirow{2}{*}{Total}
    \cline { 3 - 5 } \multicolumn{2}{|c|}{}TeaCoffeeHot Chocolate
    \multirow{2}{*}{Gender}Males57261194
    \cline { 2 - 6 }Females424717106
    Total997328200
    1. Test, at the \(5 \%\) significance level, whether or not there is an association between type of drink preferred and gender. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places.
    2. State what difference using a \(0.5 \%\) significance level would make to your conclusion. Give a reason for your answer.
    Edexcel S3 2016 June Q2
    12 marks Moderate -0.3
    2. A researcher investigates the results of candidates who took their driving test at one of three driving test centres. A random sample of 620 candidates gave the following results.
    \multirow{2}{*}{}Driving test centre\multirow{2}{*}{Total}
    \(\boldsymbol { A }\)BC
    \multirow{2}{*}{Result}Pass9911068277
    Fail108116119343
    Total207226187620
    1. Test, at the \(5 \%\) level of significance, whether there is an association between the results of candidates' driving tests and the driving test centre. State your hypotheses and show your working clearly. You should state your expected frequencies correct to 2 decimal places. The researcher decides to conduct a further investigation into the results of candidates' driving tests.
    2. State which driving test centre you would recommend for further investigation. Give a reason for your answer.
    Edexcel S3 2017 June Q2
    10 marks Standard +0.3
    2. A school uses online report cards to promote both hard work and good behaviour of its pupils. Each card details a pupil's recent achievement and contains exactly one of three inspirational messages \(A , B\) or \(C\), chosen by the pupil's teacher. The headteacher believes that there is an association between the pupil's gender and the inspirational message chosen. He takes a random sample of 225 pupils and examines the card for each pupil. His results are shown in Table 1. \begin{table}[h]
    \cline { 2 - 5 } \multicolumn{2}{c|}{}Inspirational message\multirow{2}{*}{Total}
    \cline { 3 - 5 } \multicolumn{2}{c|}{}\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)
    \multirow{2}{*}{
    Pupil's
    gender
    }    		Male253745107
    \cline { 2 - 6 }Female325036118
    Total578781225
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table} Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not there is evidence to support the headteacher's belief. Show your working clearly. You should state your expected frequencies correct to 2 decimal places.
    Edexcel S3 2021 October Q4
    11 marks Moderate -0.3
    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.
    VillageNumber of households
    A41
    B164
    C123
    D82
    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
      18-4950-69Older than 69
      Listen to LSB13016265
      Do not listen to LSB789862
      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.
    Edexcel S3 Specimen Q5
    10 marks Standard +0.3
    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.
    \backslashbox{Annual income}{Finances}WorseSameBetter
    Under £15 00014119
    £15000 and above172029
    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}
    Edexcel S3 2007 June Q2
    10 marks Standard +0.3
    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.
    Maths grade A or BMaths grade C or DMaths grade E or U
    English grade A or B252510
    English grade C to U153015
    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.
    Edexcel S3 2008 June Q2
    11 marks Standard +0.3
    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
    \cline { 3 - 4 } \multicolumn{2}{c|}{}Course
    \cline { 3 - 5 } \multicolumn{2}{c|}{}ArtsScienceHumanities
    EsuderBoy305035
    \cline { 2 - 5 }Girl402042
    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.
    Edexcel S3 2010 June Q5
    10 marks Standard +0.3
    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.
    Annual income FinancesWorseSameBetter
    Under \(\pounds 15000\)14119
    \(\pounds 15000\) and above172029
    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.
    Edexcel S3 2012 June Q4
    10 marks Standard +0.3
    1. Two breeds of chicken are surveyed to measure their egg yield. The results are shown in the table below.
    \backslashbox{Breed}{Egg yield}LowMediumHigh
    Leghorn225226
    Cornish14324
    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.
    Edexcel S3 2014 June Q3
    10 marks Standard +0.3
    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
    Happiness\multirow{2}{*}{Total}
    \cline { 3 - 5 } \multicolumn{2}{|c|}{}Not happyFairly happyVery happy
    \multirow{2}{*}{Gender}Female9433486
    \cline { 2 - 6 }Male13251654
    Total226850140
    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.