1. STA291 Statistical Methods Lecture 24

2. Comparing Two Proportions Sample of 25,000 24-year-olds: Men: 84.9% diploma rate Women: 88.1% diploma rate Are women more likely to graduate high school than men, or are the differences due to random variation? Overall, of the sample had diplomas. Use this proportion later for significance testing*.

3. Notation 3 sample sizes for the first, second group sample no. successes for the respective groups sample proportions for the groups pooled estimate of the proportion estimated standard error of the difference of the sample proportions, CI estimated standard error of the difference of the sample proportions, signif. test of diffs

4. Comparing Two Proportions 100 ( 1 –  )% Confidence Interval for p 1 – p 2 :

5. Sample of 25,000 24-year-olds: For high school graduation, a 95% confidence interval for the true difference between women’s and men’s rates is: We can be 95% confident that women’s rates of having a HS diploma by 2000 were 2% to 4% higher than men’s. Comparing Two Proportions—CI example

6. Hypothesis Testing Assumptions: Null Hypothesis: Test statistic: 6 H 0 : p 1 – p 2 =  Independent, SRSs; Binary data; At least 10 per group

7. For high school graduation, a level  = 0.01 test for whether women’s and men’s rates differ is: We reject the null at  = 0.01, deciding the men’s and women’s graduation rates differ. CI far more informative. Comparing Two Proportions—Significance Test example H 0 : p 1 – p 2 =  H a : p 1 – p 2 ≠  with a p-value of ~1.1e-13 — not too surprising because ???

8. Looking back o Two-sample problem with categorical data o Notation, estimators o Different SE estimates for CI, test statistic (!) o CI o Significance testing