The document outlines the chi-square test, a statistical method for assessing associations between categorical variables, including its purpose, assumptions, and three main types: goodness-of-fit, independence, and homogeneity tests. Each type serves a different role, such as evaluating a single categorical variable's distribution or examining relationships between multiple variables. Key assumptions include having independent observations, expected frequencies of at least 5, and a preference for large sample sizes.

1. CHI-SQUARE TEST
• INTRODUCTION
• TYPES

2. Introduction to Chi-Square Test
• Definition: A Chi-Square (χ²) test is a statistical test used to determine if there is
a significant association between categorical variables.
• Purpose: It is used to test hypotheses about the distribution of categorical
variables.
• Assumptions:
• Data is in the form of counts/frequencies.
• Observations should be independent.
• Expected frequencies should be greater than 5 in each cell.

3. Types of Chi-Square Tests
1. Chi-Square Goodness-of-Fit Test
2.Chi-Square Test of Independence
3.Chi-Square Test of Homogeneity

4. Chi-Square Goodness-of-Fit Test
• Purpose: Determines whether the observed distribution of a single
categorical variable matches an expected distribution.
• Null Hypothesis (H ):
₀ The observed frequencies follow the expected
distribution.
• Alternative Hypothesis (H ):
₁ The observed frequencies do not follow the
expected distribution.
• Example: Testing if a die is fair (each outcome has equal probability).

5. Chi-Square Test of Independence
• Purpose: Tests whether two categorical variables are independent or if there
is a significant association between them.
• Null Hypothesis (H ):
₀ The two variables are independent (no association).
• Alternative Hypothesis (H ):
₁ The two variables are dependent (there is an
association).
• Example: Testing whether gender is related to voting preference.

6. Chi-Square Test of Homogeneity
• Purpose: Compares the distribution of a categorical variable across two or
more different groups.
• Null Hypothesis (H ):
₀ The distribution of the categorical variable is the
same across all groups.
• Alternative Hypothesis (H ):
₁ The distribution of the categorical variable is
different across the groups.
• Example: Testing if different age groups prefer different types of music.

7. Comparing Goodness-of-Fit and Test of Independence
• Goodness-of-Fit:
• Focuses on one categorical variable.
• Compares observed vs expected distribution.
• Test of Independence:
• Focuses on the relationship between two categorical variables.
• Examines if there is an association between the variables.

8. Assumptions for Chi-Square Tests
• Expected Frequency Assumption: Each expected cell frequency should be
at least 5.
• Independence Assumption: Observations in each group should be
independent of each other.
• Size of Sample: Chi-Square tests are valid for large sample sizes (typically n
≥ 30).