Inferences Concerning the Difference in Population Proportions (9.4) Previous sections (9.1,2,3): We compared the difference in the means (  1 -  2 ) of two different populations This section (9.4): We compare the difference in the proportions (p 1 – p 2 ) of two different populations

Assumptions and Defintions p 1 and p 2 represents the true proportion of individuals in population (1) and (2) who posses a certain characteristic. These are unknown parameters Let X & Y denote the number of individuals in a sample of the respective sizes m or n who posses this characteristic

Assumptions and Defintions Cont’d Assume that X~Bin(m,p 1 ) and Y~Bin(n,p 2 ). This implies that the population size is infinite (much larger than the sample size) or that we are sampling with replacement If the sample size is large, the Binomial distribution may be approximated by a Normal distribution (4.3)

Computing the Expectation and Variance of a Sample Proportion Let That is, estimate the proportion of individuals (p 1 and p 2 ) in populations (1) and (2) who posses a certain characteristic with the number of individuals in a sample who posses such divided by the size of the sample.

Computing the Expectation and Variance Cont’d

Normality of Sample Proportions

Test Procedures Only large sample test procedures are considered. Inference on small samples is difficult Further, we only consider null hypotheses of the form H o :p 1 -p 2 =0; we do not consider H o :p 1 -p 2 =  o where  o  0

Test Procedure Cont’d If H o is true, then p 1 =p 2 =p. Hence, under the null hypothesis

Test Procedure Cont’d Estimate p with a weighted average of the sample proportions for population (1) and (2)

Test Procedure Cont’d

Example – Polio Vaccine The Polio vaccine was tested on 401,974 children. Of those, were given the vaccine and 200,745 were given the placebo. Of the children who received the vaccine, 33 contacted polio, and of those who received the placebo, 110 contracted polio. Test whether the vaccine is effective at a 0.05 level of significance.

Large Sample C.I. for p 1 -p 2

Example – C.I. for Polio Vaccine

Inference Concerning Two Population Variances (9.5) Comparison of population variances based on the F distribution, not the normal or t distributions. The F distribution cannot assume a negative value F 1- , 1, 2 = 1/F , 2, 1

Development of Test Let Z 1 ~  2 1 and Z 2 ~  2 2, Then It follows that

Development of Test Cont’d So let It follows that

Hypothesis Test

Confidence Interval

Example Consider the following data on the tread-life of two competing brands of tires Test whether the population variances are equal at a 0.05 level of significance. Construct a 95% confidence interval on the ratio of these variances