Comparing Two Population Proportions

1. Comparing Two Population Proportions

2. Example #1 Population of Male Penn State Students Population of Female Penn State Students CI: What is the difference in pM, the proportion of males who abstain from alcohol, and pF, the proportion of females who do? HT: Is pM  pF? Or, equivalently, is pM - pF  0? Sample of 84 Male students Sample of 88 Female students Calculate sample difference.

3. As long as sample is “large” as defined by: • In first sample, more than 5 people have trait, and more than 5 people do not have trait; and • In second sample, more than 5 people have trait, and more than 5 people do not have trait. Inference for Two Proportions Confidence Interval Hypothesis Test

4. In Minitab with Raw Data • Select Stat. Select Basic Statistics. Select 2 Proportions… • Under “Samples in One Column,” put the response variable in Samples box, and put the group variable in Subscripts box. • Under Options…, specify the confidence level, the null and alternative hypotheses. • Select OK.

5. In Minitab with Summarized Data • Select Stat. Select Basic Statistics. Select 2 Proportions… • Click on button in front of “Summarized Data.” For each sample: In box labeled “Trials,” put n, the number in the sample. In box labeled “Successes,” put the number in the sample with the trait of interest. • Under Options…, do the same as previous. • Select OK.

6. Example #1 Test and Confidence Interval for Two Proportions Success = Yes gender X N Sample p M 17 84 0.202381 F 24 88 0.272727 Estimate for p(M) - p(F): -0.0703463 95% CI for p(M) - p(F): (-0.196998, 0.0563051) Test for p(M) - p(F) = 0 (vs not = 0): Z = -1.09 P-Value = 0.276 P-value is not small (P = 0.276), so cannot reject null hypothesis. Interval contains 0, so cannot conclude that proportions differ.

7. Example #2 Are “non-Greeks” more likely to recycle than “Greeks”? That is, HA: pnon > pgreek. Test and Confidence Interval for Two Proportions Success = 1 greek X N Sample p 0 45 72 0.625000 1 6 16 0.375000 Estimate for p(0) - p(1): 0.25 95% CI for p(0) - p(1): (-0.0122520, 0.512252) Test for p(0) - p(1) = 0 (vs > 0): Z = 1.87 P-Value = 0.031

8. An Aside to Example #2 Test and Confidence Interval for Two Proportions Success = 1 greek X N Sample p 0 45 72 0.625000 1 6 16 0.375000 Estimate for p(0) - p(1): 0.25 95% CI for p(0) - p(1): (-0.0122520, 0.512252) Test for p(0) - p(1) = 0 (vs not = 0): Z = 1.87 P-Value = 0.062 Note: Confidence intervals need only agree with two-sided P-values.

9. Example #3 Are off-campus residents less likely to recycle than on-campus residents? That is, HA: poff < pon. Test and Confidence Interval for Two Proportions Success = 1 oncampus X N Sample p 0 25 52 0.480769 1 26 35 0.742857 Estimate for p(0) - p(1): -0.262088 95% CI for p(0) - p(1): (-0.460600, -0.0635763) Test for p(0) - p(1) = 0 (vs < 0): Z = -2.59 P-Value = 0.005

10. Example #4 Are tattooed students more or less likely to own a bike than non-tattooed students? That is, HA: ptattooed  pnontattooed. Test and Confidence Interval for Two Proportions Success = 1 tattoo X N Sample p 0 17 82 0.207317 1 1 7 0.142857 Estimate for p(0) - p(1): 0.0644599 95% CI for p(0) - p(1): (-0.209212, 0.338132) Test for p(0) - p(1) = 0 (vs not = 0): Z = 0.46 P-Value = 0.644 * NOTE * The normal approximation may be inaccurate for small samples.

11. As always… • P-values and confidence intervals are only accurate if the assumptions are met. • Check to make sure you have large enough samples.