In a chi-square test of independence, what does a p-value of 0.03 indicate when testing the association between smoking status and lung cancer diagnosis in a 2×2 contingency table?

## Interpretation of p-values in Chi-Square Tests **Key Point:** A p-value is NOT the probability that the null hypothesis is true. It is the probability of observing the test statistic (or more extreme) given that the null hypothesis IS true. ### Correct Definition of p-value **High-Yield:** p-value = P(observed data or more extreme | H₀ is true) In a chi-square test of independence: - **H₀** (null hypothesis): The two variables are independent (no association) - **H₁** (alternative hypothesis): The two variables are associated - **p = 0.03**: If smoking and lung cancer were truly independent, there is only a 3% chance of observing a chi-square statistic as extreme as (or more extreme than) what we observed ### Common Misinterpretations of p-values | Incorrect Interpretation | Why It's Wrong | |--------------------------|----------------| | "p = 0.03 means 3% chance H₀ is true" | p-value is conditional on H₀ being true, not the probability of H₀ | | "p = 0.03 means 97% chance H₁ is true" | p-value does not directly quantify the probability of the alternative hypothesis | | "p = 0.03 means clinical significance" | Statistical significance ≠ clinical significance; effect size matters | | "p = 0.03 means 3% chance of Type I error" | Type I error rate is α (e.g., 0.05), not the p-value itself | **Warning:** A statistically significant p-value (p < 0.05) does NOT prove causation. Chi-square tests only detect association, not causality. **Clinical Pearl:** In epidemiology, a significant chi-square test for smoking and lung cancer suggests an association, but confounders (age, occupational exposure) must be controlled to infer causation. **Mnemonic:** **PHAT** — p-value is Probability of data (or more extreme) given Hypothesis is true, not probability of Hypothesis being true.