Inference for Categorical Data(KA/SAP/U14InferenceFor)

Unit 14: Inference for Categorical Data (Chi-Square Tests)

Topics Covered:

• Chi-square goodness-of-fit tests
• Chi-square tests for relationships

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1. Chi-square Goodness-of-Fit Tests

The chi-square goodness-of-fit test is used to determine whether a set of observed categorical data matches an expected distribution. It answers questions like: "Does a die appear to be fair?" or "Are the colors in a bag of candies distributed as claimed by the manufacturer?"

• Null hypothesis (H₀): The observed frequencies match the expected frequencies.
• Alternative hypothesis (H_a): The observed frequencies do not match the expected frequencies.
• Test statistic: \( \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \), where \(O_i\) is observed and \(E_i\) is expected count for category \(i\).
• Degrees of freedom: Number of categories minus 1.

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2. Chi-square Tests for Relationships (Independence/Association)

The chi-square test for independence (or association) is used to determine if there is a relationship between two categorical variables in a population. For example: "Is there an association between gender and voting preference?"

• Null hypothesis (H₀): The variables are independent (no association).
• Alternative hypothesis (H_a): The variables are associated (not independent).
• Test statistic: \( \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \), where \(O_{ij}\) is observed and \(E_{ij}\) is expected count for cell \((i,j)\).
• Degrees of freedom: (number of rows - 1) × (number of columns - 1).

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Visualizations

• Bar charts for observed vs. expected counts
• Interactive contingency tables
• Dynamic calculation of chi-square statistics