The chi-square test of independence helps us figure out if two categorical variables are related. We use observed and expected frequencies to calculate a test statistic, which we compare to a critical value to make a decision.

This test is crucial for understanding relationships between variables in real-world scenarios. By analyzing contingency tables and interpreting results, we can draw meaningful conclusions about associations in our data.

Chi-Square Test of Independence

Test statistic calculation for independence

• Calculate chi-square test statistic using formula $\chi^2 = \sum \frac{(O - E)^2}{E}$
  • $O$ observed frequency in each cell of contingency table (actual counts)
  • $E$ expected frequency in each cell of contingency table (theoretical counts assuming independence)
• Determine expected frequency for each cell
  • Multiply row total by column total divide by grand total
  • Formula $E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$
• Calculate degrees of freedom $(r - 1)(c - 1)$
  • $r$ number of rows in contingency table (categories of one variable)
  • $c$ number of columns in contingency table (categories of other variable)

Interpretation of independence test results

• Test of independence assesses relationship between two categorical variables
• Null hypothesis ($H_0$) variables are independent (no association)
• Alternative hypothesis ($H_a$) variables are not independent (associated)
• Compare calculated chi-square test statistic to critical value from chi-square distribution table
  • Use degrees of freedom significance level (0.05) to find critical value
• If calculated chi-square test statistic > critical value reject null hypothesis
  • Sufficient evidence to suggest variables are not independent (associated)
• If calculated chi-square test statistic < critical value fail to reject null hypothesis
  • Insufficient evidence to suggest variables are associated (independent)

Chi-square analysis in real-world scenarios

• Identify categorical variables of interest in real-world scenario (gender, age group)
• Construct contingency table with observed frequencies for each category combination
• Calculate expected frequencies for each cell in contingency table
• Calculate chi-square test statistic using observed expected frequencies
• Determine degrees of freedom based on number of rows columns in contingency table
• Choose significance level (0.05) find critical value from chi-square distribution table
• Compare calculated chi-square test statistic to critical value make conclusion about variable relationship
  • If test statistic > critical value conclude variables are associated (preference, behavior)
  • If test statistic < critical value conclude insufficient evidence to suggest association between variables (independence)

Statistical Inference and Hypothesis Testing

• Chi-square test of independence is a form of statistical inference
• Uses sample data to draw conclusions about population parameters
• Follows hypothesis testing framework to make decisions about independence or association
• Relies on probability distribution (chi-square distribution) to determine critical values
• Chi-square test is a nonparametric test, making fewer assumptions about the underlying population distribution