2. Inference about proportions

3. Example: One Proportion Population of students Sample of 175 students CI: What proportion (percentage) of students abstain from alcohol? HT: Is it more than 20%? How likely is it that sample proportion would be as extreme as 0.234 if p = 0.20?

4. Example: Two Proportions Population of Male Penn State Students Population of Female Penn State Students CI: Does p M, proportion of males who abstain from alcohol, differ from p F, proportion of females who do? HT: Is the difference in proportions (p M - p F ) less than 0? Sample of 84 Male students Sample of 88 Female students Calculate sample difference.

5. “Theory” for One Proportion Binary variable: person either has trait (“1”) or does not have trait (“0”). Then, since proportion is number with trait divided by total number, a proportion is basically an average of 0’s and 1’s. That is, (0 + 0 + 1 + … + 1 + 0 + 1)/n. Central Limit Theorem: For large samples, sample proportions will be at least approximately normally distributed.

6. Inference for One Proportion Confidence IntervalHypothesis Test As long as sample is “large” as defined by: More than 5 people are in sample with trait, and More than 5 people are in sample without trait. If n = sample size, and p 0 = value of proportion in null hypothesis

7. Example: CI for One Proportion What proportion of students abstain from alcohol? 0.23 ± 1.96  (0.23)(0.77)/175=0.23 ± 0.06 We can be 95% confident that between 17% and 29% of all students abstain from alcohol.

8. Example: HT for One Proportion Is the proportion who abstain more than 0.20? H 0 : p = 0.20 versus H A : p > 0.20 Z = (0.234 - 0.20)/  (0.20)(0.80)/175 = 1.13 p-value = P(Z > 1.13) = 0.129 Cannot conclude that p > 0.20

9. Example: Minitab Output for One Proportion Test and Confidence Interval for One Proportion Test of p = 0.2 vs p > 0.2 Success = 1 Variable X N Sample p 95.0 % CI Z P-Value abstain 41 175 0.234286 (0.172, 0.297) 1.13 0.128

10. Trick for quick estimate of 95% margin of error 95% margin of error is The largest the margin of error can be is when p-hat is ½. If p-hat is ½, then 95% margin of error is

11. Example 100 students sampled 25% of those sampled love almonds 95% margin of error is no larger than 1/sqrt(100) = 0.10 We can be 95% confident that between 15% and 35% of all students love almonds.

12. Inference for Two Proportions Confidence Interval 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.

13. Inference for Two Proportions (continued) Hypothesis Test

14. Example: Minitab CI for Two Proportions Test and Confidence Interval for Two Proportions Success = 1 gender X N Sample p 1 17 84 0.202381 2 24 88 0.272727 Estimate for p(1) - p(2): -0.0703463 95% CI for p(1) - p(2): (-0.196998, 0.0563051) Test for p(1) - p(2) = 0 (vs not = 0): Z = -1.09 P-Value = 0.276 Interval contains 0, so cannot conclude that percentages differ.

15. Example: Minitab HT for Two Proportions Test and Confidence Interval for Two Proportions Success = 1 gender X N Sample p 1 17 84 0.202381 2 24 88 0.272727 Estimate for p(1) - p(2): -0.0703463 95% CI for p(1) - p(2): (-0.196998, 0.0563051) Test for p(1) - p(2) = 0 (vs < 0): Z = -1.09 P-Value = 0.138 P-value not quite small enough to conclude proportions differ.

16. As always… P-values and confidence intervals are only accurate if the assumptions are met. Check to make sure you have large enough sample(s).