Confidence Intervals and Hypothesis Tests for Two Proportions

1. Confidence Intervals and Hypothesis Tests forTwo Proportions

2. CI for Two Proportions • We are interested in confidence intervals for the difference p1 – p2 between the unknown values of two population proportions

3. x ˆ = p 1 1 n 1 Point Estimator: • Two random samples are drawn from two populations. • The number of successes in each sample is recorded. • The sample proportions are computed. Sample 1 Sample size n1 Number of successes x1 Sample proportion Sample 2 Sample size n2 Number of successes x2 Sample proportion

4. Confidence Interval for p1 p2

5. Example: confidence interval for p1 – p2 • Estimating the cost of life saved • Two drugs are used to treat heart attack victims: • Streptokinase (available since 1959, costs $460) • t-PA (genetically engineered, costs $2900). • The maker of t-PA claims that its drug outperforms Streptokinase. • An experiment was conducted in 15 countries. • 20,500 patients were given t-PA • 20,500 patients were given Streptokinase • The number of deaths by heart attacks was recorded.

6. Example: confidence interval for p1 – p2(cont.) • Experiment results • A total of 1497 patients treated with Streptokinase died. • A total of 1292 patients treated with t-PA died. • Estimate the difference in the death rates when using Streptokinase and when using t-PA.

7. Example: confidence interval for p1 – p2(cont.) • Solution • The problem objective: Compare the outcomes of two treatments. • The data are nominal (a patient lived or died) • The parameter to be estimated is p1 – p2. • p1 = death rate with Streptokinase • p2 = death rate with t-PA

8. Example: confidence interval for p1 – p2(cont.) • Compute: Manually • Sample proportions: • The 95% confidence interval estimate is

9. Example: confidence interval for p1 – p2(cont.) • Interpretation • The interval (.0051, .0149) for p1 – p2 does not contain 0; it is entirely positive, which indicates that p1, the death rate for streptokinase, is greater than p2, the death rate for t-PA. • We estimate that the death rate for streptokinase is between .51% and 1.49% higher than the death rate for t-PA.

10. Example: 95% confidence interval for p1 – p2 The age at which a woman gives birth to her first child may be an important factor in the risk of later developing breast cancer. An international study conducted by WHO selected women with at least one birth and recorded if they had breast cancer or not and whether they had their first child before their 30th birthday or after. • The parameter to be estimated is p1 – p2. • p1 = cancer rate when age at 1st birth >30 • p2 = cancer rate when age at 1st birth <=30 We estimate that the cancer rate when age at first birth > 30 is between .05 and .082 higher than when age <= 30.

11. =0 HypothesisTests for p1 – p2 This test is appropriate when all counts are at least 10 (number of successes and number of failures in each sample).

12. Gastric Freezing Gastric freezing was once a treatment for ulcers. Patients would swallow a deflated balloon with tubes, and a cold liquid would be pumped for an hour to cool the stomach and reduce acid production, thus relieving ulcer pain. The treatment was shown to be safe, significantly reducing ulcer pain, and so widely used for years. A randomized comparative experiment later compared the outcome of gastric freezing with that of a placebo: 28 of the 82 patients subjected to gastric freezing improved, while 30 of the 78 in the control group improved. Conclusion: Do not reject H0. The gastric freezing was no better than a placebo (P-value 0.69), and this treatment was abandoned. ALWAYS USE A CONTROL! H0: pgf - pplacebo =0 Ha: pgf – pplacebo>0