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IB DP Maths: AI SL复习笔记4.7.2 Chi-squared Test for Independence

Category: IB课程, 教材笔记, 福利干货 Date: 2022年7月19日下午4:15

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

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 O_i calculate its expected frequency E_i
- Assuming the variables are independent
  - E_i = P(X = x) × P(Y = y) × Total
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:
- Any of the expected values are less than 5
- There is only 1 degree of freedom
  - This means it is a 2 × 2 contingency table
Note that none of these cases will occur in the exam

Worked Example

At a school in Paris, it is believed that favourite film genre is related to favourite subject. 500 students were asked to indicate their favourite film genre and favourite subject from a selection and the results are indicated in the table below.