IB Math AI SL — Expected, observed, hypotheses, chi-squared, goodness of fit, and t-test

Key Idea: The chi-squared test of independence (χ²) tests whether two categorical variables are related, or whether they are independent. You set up a contingency table of observed frequencies, calculate expected frequencies if the variables were independent, and compare. A small p-value means the variables are not independent — there is a statistically significant association.

✅ Hypothesis test structure

📊 GDC method

Example: 2×2 contingency table: Gender vs Preferred sport Observed: Male/Football=40, Male/Tennis=20, Female/Football=30, Female/Tennis=50. Grand total = 140. Row totals: Male=60, Female=80. Column totals: Football=70, Tennis=70. Expected (Male, Football) = (60 × 70)/140 = 30 GDC gives: χ² = 9.33, p = 0.0023. Since p < 0.05, reject H₀. Evidence that gender and sport preference are associated.

The conclusion must always be in context — name the two variables. 'Reject H₀' alone is not a full answer. The test only tells you that association exists — it does not say how strong or in which direction.

Paper 2 (GDC allowed): State both hypotheses in full before running the test. After: write χ², p-value, compare with α, and state conclusion with the variable names. Check expected frequencies: After running the test on GDC, view the expected matrix and verify all values ≥ 5. If not, state this as a limitation of the test.