## Answer

The Chi-Square test is a statistical test used to determine whether there is a significant association between categorical variables. It helps in assessing whether the observed frequencies in a contingency table differ significantly from the expected frequencies under a null hypothesis of independence or no effect. The Chi-Square test is used in various contexts, including:

### 1. **Testing for Independence**

**Purpose**: To determine if there is an association between two categorical variables.**Example**: Analyzing if there is a relationship between gender (male/female) and voting preference (candidate A/B/C) in a survey.

### 2. **Goodness of Fit**

**Purpose**: To assess how well observed data fit a specific theoretical distribution or model.**Example**: Testing if the observed distribution of a die roll follows the expected uniform distribution (i.e., each face has an equal probability).

### 3. **Homogeneity**

**Purpose**: To compare the distribution of a categorical variable across different populations or groups to check if they have the same distribution.**Example**: Comparing the preference for a product across different cities to see if the distribution of preferences is the same in each city.

### 4. **Testing for Fit of a Model**

**Purpose**: To evaluate how well a statistical model fits the observed data.**Example**: In genetic research, checking if the observed distribution of genotypes fits the expected Mendelian ratios.

### 5. **Evaluating Survey or Experimental Data**

**Purpose**: To analyze data collected from surveys or experiments to determine if there are significant patterns or associations.**Example**: Analyzing survey results to see if there is a significant difference in satisfaction levels between different demographic groups.

### Key Uses and Contexts

**Market Research**- Assessing consumer preferences and behavior based on categorical survey responses.

**Epidemiology**- Investigating associations between risk factors and health outcomes.

**Social Sciences**- Analyzing survey data to study relationships between demographic variables and various social indicators.

**Biological Sciences**- Evaluating genetic inheritance patterns and the distribution of traits in populations.