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The Chi-square test of independence is a statistical method used to determine if there's a significant association between two categorical variables. It's crucial in fields like psychology, healthcare, and marketing, helping to inform decisions by analyzing the relationship between variables without assuming a normal distribution. The test involves calculating expected frequencies, test statistics, and interpreting results to assess the presence of an association.

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## Definition and Purpose

### Statistical Procedure

The Chi-square test of independence is a non-parametric test used to evaluate the relationship between two categorical variables

### Null and Alternative Hypotheses

Null Hypothesis

The null hypothesis states that there is no association between the variables

Alternative Hypothesis

The alternative hypothesis claims that there is an association between the variables

### Assumptions

The Chi-square test of independence relies on several assumptions, including a contingency table with categorical variables, mutually exclusive and exhaustive categories, and a sufficiently large sample size

## Calculation and Interpretation

### Expected Frequencies

The expected frequency for each cell in the contingency table is calculated based on the total frequencies for each row and column

### Chi-square Test Statistic

The Chi-square test statistic is calculated by comparing the observed and expected frequencies to determine the discrepancy between them

### Degrees of Freedom and Significance

The degrees of freedom are determined by the number of categories in each variable, and the significance of the test is determined by comparing the test statistic to the critical value at a specified significance level

## Applications

### Public Health

The Chi-square test of independence can be used in public health to compare rates of a health outcome across different exposure groups

### Marketing

In marketing, the test can help determine if there is an association between consumer preferences and demographic factors

### Education

Educational institutions can use the test to assess the relationship between student performance and teaching methods