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IB DP Maths: AI HL復(fù)習(xí)筆記4.11.2 Chi-squared Test for Independence

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月21日 下午5:45

Chi-Squared Test for Independence

What is a chi-squared test for independence?

  • 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
      • For example: if the two variables are hair colour and eye colour then the contingency table will show the frequencies of the different combinations

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 GDC can calculate these for you after you input the observed frequencies
  • The?expected values?must all be?bigger 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 matrix 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 TV genre then it is more appropriate to combine age groups than types of genre

What is the degree of freedom?

  • The?degree of freedom?refers to the?minimum number of expected values?you need to know in order to be able to calculate them all

What are the steps for a chi-squared test for independence?

  • STEP 1: Write the?hypotheses
    • H0?: Variable?X?is independent of variable?Y
    • H1?: Variable?X?is not independent of variable?Y
      • Make sure you clearly write what the variables are and don’t just call them?X?and?Y
  • STEP 2: Calculate the?degree of freedom?for the test
    • For an?m?×?n?contingency table
  • STEP 4: Decide whether there is evidence to reject the null hypothesis
    • EITHER compare the?χ2 statistic?with the given?critical value
      • If?χ2 statistic?>?critical value then?reject H0
      • If?χ2 statistic?<?critical value then?accept H0
    • OR compare the?p-value?with the given?significance level
      • If?p-value?<?significance level then?reject H0
      • If?p-value?>?significance level then?accept H0
  • STEP 5: Write your?conclusion
    • If you?reject H0
      • There is sufficient evidence to suggest that variable?X?is not independent of variable?Y
      • Therefore this suggests they are?associated
    • If you?accept H0
      • There is insufficient evidence to suggest that variable?X?is not independent of variable?Y
      • Therefore this suggests they are?independent

How do I calculate the chi-squared statistic?

  • You are?expected?to be able to use your?GDC?to calculate the?χ2 statistic by inputting the matrix of the observed frequencies
  • Seeing how it is done by hand might deepen your understanding but you are?not expected?to use this method
  • STEP 1: For each?observed frequency?Oi?calculate its?expected frequency?Ei
    • Assuming the variables are independent
    • You do not need to learn this formula as your GDC calculates it for you
  • To calculate the?p-value you would find the probability of a value being bigger than your?χ2 statistic using a?χ2 distribution with ν degrees of freedom

Exam Tip

Note for Internal Assessments (IA)

  • If you use a?χ2 test in your IA then beware that the outcome may not be accurate if there is only 1 degree of freedom
    • This means it is a 2 × 2 contingency table

Worked Example4-7-2-ib-ai-sl-chi-squared-ind-a-we-solutionb)Write down the number of degrees of freedom for this table.4-7-2-ib-ai-sl-chi-squared-ind-b-we-solution

4-7-2-ib-ai-sl-chi-squared-ind-c-we-solution

d)Write down the conclusion to the test. Give a reason for your answer.

4-7-2-ib-ai-sl-chi-squared-ind-d-we-solution

 

 

 

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