Inferences About Proportions of Two Groups Two-Sample Z Test for P Inferences About Proportions of Two Groups Each slide has its own narration in an audio file. For the explanation of any slide click on the audio icon to start it. Professor Friedman's Statistics Course by H & L Friedman is licensed under a  Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 

Testing the Difference Between Two Proportions Examples: You wish to compare the defective rates (a proportion) of two companies that supply the computer chips needed for your tablet computer. You want to compare the death rates for heart transplants at two hospitals. You want to compare the graduation rates of two high schools in the same area. Two Sample Z Test for Proportions

Testing the Difference Between Two Proportions We are trying to determine whether the observed difference between two proportions is statistically significant or just sampling error. We use a two-sample Z test to compare the proportions (P) of two groups. To use this test n1 and n2 should be large. Specifically, n1p1 ≥ 5 n1(1-p1) ≥ 5 n2p2 ≥ 5 n2(1-p2) ≥ 5 Two Sample Z Test for Proportions

The Two-Sample Z Test for P This Z test uses the normal approximation: Two Sample Z Test for Proportions

The Two-Sample Z Test for P Ps1= = the sample proportion in population 1 Ps2= = the sample proportion in population 2 Where X1 represents the # of “successes” in sample 1 X2 represents the # of “successes” in sample 2 A “success” is the outcome you are interested in, e.g., a defective part. n1 = sample size for group 1 n2 = sample size for group 2 is the pooled estimate of the population proportion: Two Sample Z Test for Proportions

Problem 1: Death Rates at Two Hospitals We would like to compare the death rates from liver transplants at two hospitals in similar areas. Hospital A: 77/100 died within 6 months Hospital B: 120/200 died within 6 months Are the death rates for the two hospitals statistically different? Test at α = .05. Two Sample Z Test for Proportions

Problem 1: Death Rates at Two Hospitals (cont’d) H0: P1=P2 H1: P1≠P2 Ps1 = 77/100 = .77; Ps2 = 120/200 = .60 Z= Reject H0 Two Sample Z Test for Proportions

Problem 2: Two Unemployment Rates Unemployment rates in two counties. Are they different? Two random samples taken: County A: 100/400 unemployed County B: 44/200 unemployed Test at α=.05 Pooled P= (100+44)/(400+200) = Two Sample Z Test for Proportions

Problem 2: Two Unemployment Rates (cont’d) H0: P1=P2 H1: P1≠P2 Ps1 = 100/400 = .25; Ps2 = 44/200 = .22 Z= Conclusion: Do not reject H0 Two Sample Z Test for Proportions

Problem 3: The Donner Party This is an actual study published by Donald Grayson. Dr. Grayson was interested in knowing whether the survival rate under conditions of starvation is different for men and women. The Donner party were traveling from Illinois to California and were caught in a huge blizzard. They were stranded for 6 months and had no food; they resorted to cannibalism to survive. The death rate for women was 10/34 and for men 30/53. We will conduct a test at the =.05 significance level to determine whether the death rates for men and women are statistically different from each other. Two Sample Z Test for Proportions

Problem 3: The Donner Party H0: P1=P2 H1: P1≠P2 Death rates: Ps1 = 10/34 = .294 (women) Ps2 = 30/53 = .566 (men) 𝑝 = (10+30) (34+53) = 40 87 = .46 Z = (.294 − .566) .46 (.54)( 1 34 + 1 53 ) = −.272 .1095 = -2.48 Conclusion: Reject H0. -2.48 is in the rejection region. The two population proportions are not the same. Women have a statistically higher survival rate under adverse conditions than do men. Possible reason given is that women have an additional layer of fat tissue which provides a fetus (and, of course, women) with sufficient nourishment in case of a famine. Two Sample Z Test for Proportions

Homework Practice, practice, practice. Do lots and lots of problems. You can find these in the online lecture notes and homework assignments. Two Sample Z Test