Goodness of Fit Test

Goodness of Fit Test

  • Used to analyze proportions of relative frequencies on one variable with multiple levels.

  • Common applications: political party preferences, ethnic group representation.

Hypotheses

  • Null Hypothesis (H0): A variable follows a hypothesized distribution.

  • Alternative Hypothesis (H1): A variable does not follow a hypothesized distribution.

Example Scenario

  • Examined detentions granted by day of the week:

    • Monday: 50

    • Tuesday: 60

    • Wednesday: 40

    • Thursday: 47

    • Friday: 53

  • Total detentions = 250; expect equal distribution across 5 days (50 each).

Chi-square Calculation

  • Formula: X2=Σ(O−E)2EX² = \Sigma \frac{(O - E)²}{E}

    • $O$ = observed value

    • $E$ = expected value

  • Observed Frequencies and Expected Frequencies table generated.

Statistical Significance

  • Determine if p-value < 0.05

  • Degrees of freedom (df) = n - 1; where n = 5 days, df = 4

Decision Making

  • If p-value <= 0.05, reject H0; otherwise, fail to reject H0.

  • Example outcome: p-value = 0.359, conclusion: fail to reject H0.

    • Result example: "A chi-square test indicated no significant difference between detention counts across days (X² = 4.36, p = 0.36)."

Summary

  • Use goodness of fit test to assess categorical data distribution.

  • Calculate using chi-square formula.

  • Properly communicate results in write-ups.