1 Confidence Intervals for Two Proportions Section 6.1

2 Section 6.1 CI for Two Proportions We are interested in confidence intervals for the difference p 1 – p 2 between the unknown values of two population proportions

3 6.1 Confidence Intervals for the difference p 1 – p 2 between two population proportions In this section we deal with two populations whose data are qualitative. For nominal data we compare the population proportions of the occurrence of a certain event. Examples –Comparing the effectiveness of new drug versus older one –Comparing market share before and after advertising campaign –Comparing defective rates between two machines

4 Parameter and Statistic Parameter –When the data are qualitative, we can only count the occurrences of a certain event in the two populations, and calculate proportions. – The parameter we want to estimate is p 1 – p 2. Statistic –An unbiased estimator of p 1 – p 2 is (the difference between the sample proportions).

5 Sample 1 Sample size n 1 Number of successes x 1 Sample proportion Sample 1 Sample size n 1 Number of successes x 1 Sample proportion Two random samples are drawn from two populations. The number of successes in each sample is recorded. The sample proportions are computed. Sample 2 Sample size n 2 Number of successes x 2 Sample proportion Sample 2 Sample size n 2 Number of successes x 2 Sample proportion x n 1 1 ˆ  p 1 Point Estimator:

Large-sample CI for two proportions For two independent samples of sizes n 1 and n 2 with sample proportion of successes 1 and 2 respectively, an approximate level C confidence interval for p 1 – p 2 is Use this method when C is the area under the standard normal curve between −z* and z*.

7 Example: confidence interval for p 1 – p 2 p. 2 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.

8 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. Example: confidence interval for p 1 – p 2 (cont.)

9 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 p 1 – p 2. p 1 = death rate with Streptokinase p 2 = death rate with t-PA Example: confidence interval for p 1 – p 2 (cont.)

10 Compute: Manually –Sample proportions: –The 95% confidence interval estimate is Example: confidence interval for p 1 – p 2 (cont.)

11 Interpretation –The interval (.0051,.0149) for p 1 – p 2 does not contain 0; it is entirely positive, which indicates that p 1, the death rate for streptokinase, is greater than p 2, 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. Example: confidence interval for p 1 – p 2 (cont.)

12 Example: 95% confidence interval for p 1 – p 2 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 30 th birthday or after. CancerSample Size Age at First Birth > % Age at First Birth <= , % The parameter to be estimated is p1 – p2. p1 = cancer rate when age at 1 st birth >30 p2 = cancer rate when age at 1 st birth <=30 We estimate that the cancer rate when age at first birth > 30 is between.05 and.082 higher than when age <= 30.

Beware!! Common Mistake !!! A common mistake is to calculate a one-sample confidence interval for p   a one-sample confidence interval for p   and to then conclude that p  and p  are equal if the confidence intervals overlap. This is WRONG because the variability in the sampling distribution for from two independent samples is more complex and must take into account variability coming from both samples. Hence the more complex formula for the standard error.

INCORRECT Two single-sample 95% confidence intervals: The confidence interval for the rightie BA and the confidence interval for the leftie BA overlap, suggesting no significant difference between Ryan Howard’s ABILITY to hit right- handed pitchers and his ABILITY to hit left-handed pitchers. Rightie interval: (0.274, 0.366) HitsABphat(BA) Rightie Leftie Leftie interval: (0.170, 0.280)

Reason for Contradictory Result 15