Chi Squared Tests for Contingency Tables (Edexcel A Level Further Maths: Further Statistics 1)

What is a chi-squared test using contingency tables?

• A chi-squared () using contingency tables is a hypothesis test used to test whether two variables are independent of each other
  • For example whether or not favourite music genre is independent of the age group of the listener 
  • This is sometimes called a  two-way test
• This is an example of a goodness of fit test
  • We are testing whether the data fits the modelling assumption that the variables are independent
• The chi-squared () distribution is used for this test
• You will use a contingency table
  • This is a two-way table that shows the observed frequencies for the different combinations of the two variables
  • A   contingency table has  rows and  columns
    • This does not include any rows or columns used to record the row, column and grand totals

Why might I have to combine rows or columns?

• The observed values are used to calculate expected values
  • These are the expected frequencies for each combination assuming that the variables are independent
    • Your calculator may be able to calculate these for you after you input the observed frequencies
    • Or else you can calculate them using the formula

• None of the expected values used to run the test can be less than 5
  
  • If one of the expected values is less than 5 then you will have to combine the corresponding row or column in the table of observed values with the adjacent row or column
  • The decision between row or column will be based on which seems the most appropriate
    • For example: if the two variables are age and favourite music genre then it is more appropriate to combine age groups than types of genre

What are the degrees of freedom?

• There will be a minimum number of expected values you would need to know in order to be able to calculate all the expected values
• This minimum number is called the degrees of freedom and is often denoted by
• For a test for independence with an contingency table
  •  
  • For example: If there are 5 rows and 3 columns then you only need to know 2 of the values in 4 of the rows as the rest can be calculated using the totals

What are the steps for a chi-squared test using contingency tables?

• STEP 1: Write the hypotheses
  • : Variable X and Variable Y are independent
  • : Variable X and Variable Y are not independent
  • The hypotheses should always be stated in the context of the question
  • Make sure you clearly write what the variables are and don’t just call them 'Variable X' and 'Variable Y'
• STEP 2: Calculate the expected frequencies
  • Use the formula

 

• •  You will need to combine rows or columns if any of the expected frequencies are less than 5
    • This process is described above
    • After combining, calculate the new expected frequencies for the modified table
  • You may also be able to enter the observed frequencies as a matrix in your calculator
    • Use the option for a 2-way test
    • Your calculator will calculate the matrix of expected frequencies
• STEP 3: Calculate the degrees of freedom for the test
  • For an contingency table (after combining)
    • Degrees of freedom is
• STEP 4: Calculate using the formula

   

  • • then you will also need to determine the appropriate critical value
    • use the 'Percentage Points of the Distribution' table in the exam formula booklet
  • If you entered the observed frequencies as a matrix in your calculator
    
    • then your calculator's 2-way test option will give you the test statistic and the associated p-value

• STEP 5: Decide whether there is evidence to reject the null hypothesis
  • Compare the statistic with the critical value you have determined
    • If  > critical value (or ) then there is sufficient evidence to reject 
    • If  < critical value (or ) then there is insufficient evidence to reject 
• STEP 6: Write your conclusion
  • If you reject H₀
    
    • Variable X and variable Y are not independent
  •  If you do not reject H₀
    
    • Variable X and variable Y are independent
  • Be sure to state your conclusion in the context of the question