Tests for Two Means – Normal Populations

1. Tests for Two Means – Normal Populations • Suppose we have two independent random samples from two different Normal populations and we are interested in comparing the means of these two populations. • The CLT is not necessary and hence the samples can be small. • We differentiate between three cases, variances of the populations are known, variances are unknown, and variances are unknown but equal…. week 12

2. Example week 12

3. Hypothesis Tests with Paired Data • A matched pairs study is a study in which, subjects are matched in pairs and the outcomes are compared within each matched pair. One situation calling for match pairs is when observations are taken on the same subjects, under different conditions. • A match pairs analysis is needed when there are two measurements or observations on each individual and we want to examine the difference. • For each individual (pair), we find the difference d between the measurements from that pair. Then we treat the d’sas one sample. week 12

4. If the sample size is large, the CLT applies and we perform the test for no difference between the treatments effect using the standard normal distribution. • If the sample size is small, the CLT do not apply, we need to assume the data is normal and test for no difference between the treatments effect using the t distribution. week 12

5. Example week 12

6. Tests for Two Binomial Proportions • Suppose X1, …, Xnare iid Bernoulli(p1) independent of Y1, …, Ym that are iid Bernoulli(p2). • Further, suppose that n and m are large. • The CLT applies to both sample proportions. • The test is conducted using standard normal distribution, using pooled variance estimate…. • The CI is constructed using the standard normal distribution, without pooling variance estimates… week 12

7. Example week 12

8. Test on Pairs of Variances • Suppose are iid independent of that are iid . • We are interested in testing versus a one sided or a two sided alternative… • Then… week 12

9. Example week 12