IB DP Maths: AI HL復習筆記4.11.2 Chi-squared Test for Independence
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
- This is a two-way table that shows the?observed frequencies?for the different combinations of the two variables
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
- These are the expected frequencies for each combination?assuming that the variables are independent
- 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
- The decision between row or column will be based on which seems the?most appropriate
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
- EITHER compare the?χ2 statistic?with the given?critical value
- 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
- If you?reject H0
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 Example
b)Write down the number of degrees of freedom for this table.
d)Write down the conclusion to the test. Give a reason for your answer.
轉載自savemyexams
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