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Understanding the Chi-Squared Test

Dec 3, 2024

Lecture Notes: Chi-Squared Test for Independence

Introduction

  • Section: 15D, Last section of the course
  • Topic: Chi-squared test for independence
  • Objective: Understand the chi-squared test to prepare for the final exam

Key Concepts

Independence

  • Definition: Independence means no association between variables.
  • Example: Sex at birth and eye color are independent.
  • Association: If there's a relationship, variables are not independent.

Chi-Squared Test of Independence

  • Purpose: Test whether two categorical variables are independent.

Example

  • Data Source: Pew Research Center
  • Variables:
    • Income level (Poor: <30,000, Middle: 30,000-74,999, Upper: 75,000+)
    • Education level
  • Two-way Table: Displays the relationship between income and education level

Null and Alternate Hypotheses

  • Null Hypothesis (H₀): The two variables are independent (no association)
  • Alternate Hypothesis (H₁): The two variables are not independent (there is an association)

Conditions for Chi-Squared Test

  1. Data Type: Two categorical variables measured on one sample
  2. Sample Quality: Data collected should represent a random and unbiased sample
  3. Sample Size: Each expected cell count should be at least 5

Calculating Expected Values

  • Formula:
    • Expected Value = (Row Total × Column Total) / Grand Total
  • Interpretation: If H₀ is true, expected count should be observed

Performing the Chi-Squared Test

  • Steps:
    • Enter data into a statistical software
    • Use software to calculate chi-squared test statistic
    • Determine p-value

Conclusion

  • Small p-value: Reject the null hypothesis (suggests an association)
  • Important Note: Chi-squared test only suggests association, not causation

Limitations

  • Test shows association, not causality or direction of association

Additional Concepts

  • Degrees of Freedom: (Number of rows - 1) × (Number of columns - 1)
  • Test Statistic Calculation: Sum of (Observed - Expected)² / Expected for each cell

Lurking Variables

  • Variables that can affect both variables being studied (e.g., family support)

Final Thoughts

  • Importance of understanding assumptions and methodology of chi-squared test
  • Prepare for final by reviewing these notes and completing homework assignments

This concludes your course section on chi-squared tests. Best of luck on your final exam!

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