PEARSON'S CHI-SQUARE TEST FOR INDEPENDENCE (p)

PEARSON'S CHI-SQUARE TEST FOR INDEPENDENCE (p)

  • Hypotheses

    • Null Hypothesis (H0): Variables are independent (carrying a weapon is not associated with age group).

    • Alternative Hypothesis (Ha): Variables are dependent (carrying a weapon depends on age group).

  • Chi-Square Calculation

    • Formula: χ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}

    • Where:

      • OO = observed frequency

      • EE = expected frequency

  • Expected Frequencies Calculation

    • Expected value formula: E=(row sum×column sum)table sumE = \frac{(\text{row sum} \times \text{column sum})}{\text{table sum}}

  • Degrees of Freedom

    • Formula: df=(r−1)(c−1)df = (r - 1)(c - 1)

    • Example: For this case, df=(3−1)(2−1)=2df = (3 - 1)(2 - 1) = 2

  • Significance Level

    • Significance level chosen: 0.05

    • Chi-square value determined: 28.42

    • Critical value: 5.99

    • If χ2≥critical value\chi^2 \geq \text{critical value}, reject H0

  • p-value Interpretation

    • If p-value < 0.05: reject null hypothesis, variables are dependent.

    • If p-value > 0.05: fail to reject null hypothesis, variables are independent.

  • Reporting Results

    • APA Style:

    • Example: "A chi-squared test of independence was performed… X2(2, N=1743) = 28.42, p < .001."

  • Assumptions for Chi-Square Test

    • Two categorical variables, mutually exclusive observations.

    • Expected cell frequency > 5; no more than 25% of cells should have an expected frequency < 5.

  • Checklist for Assumptions

    • Ensure assumptions are met before applying the test. Consider transformations or alternative tests if assumptions aren’t fulfilled.