Chi-square Tests

Chi-square (χ²) tests are used to determine whether there is a significant association between categorical variables. They compare the observed frequencies in each category to the frequencies expected under the null hypothesis.

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1. When to Use Chi-square Tests

• Testing independence between gender and preference for a product

• Checking if a die is fair

• Analyzing survey results in contingency tables

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2. Types of Chi-square Tests

a. Test of Independence

Used to assess whether two categorical variables are associated.

• Example: Is there a relationship between smoking status (yes/no) and disease outcome (positive/negative)?

b. Goodness-of-fit Test

Used to test whether the distribution of a single categorical variable fits a specified distribution.

• Example: Do observed coin toss results match the expected 50/50 distribution?

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3. Assumptions of Chi-square Tests

• Observations are independent

• Categories are mutually exclusive

• Expected frequency in each cell should be ≥ 5 (for validity)

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4. Chi-square Test Formula

[ \chi^2 = \sum \frac{(O - E)^2}{E} ] Where:

• ( O ): Observed frequency

• ( E ): Expected frequency

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5. Interpreting Results

• Null hypothesis (H0): No association between variables (or observed distribution matches expected)

• Alternative hypothesis (Ha): There is an association (or distribution differs)

• If p-value < 0.05, reject H0

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6. Example (R Code)

# Test of independence
chisq.test(table(data$smoking, data$disease))

# Goodness-of-fit test
observed <- c(25, 30, 45)
expected <- c(33.3, 33.3, 33.3)
chisq.test(x = observed, p = expected/sum(expected))

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7. Summary

Chi-square tests are non-parametric and widely used for analyzing categorical data. Ensure assumptions are met and interpret the results in the context of the data.