STA 291 Summer 2010

1. STA 291Summer 2010 Lecture 20 Dustin Lueker

2. Example • If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 5% level of significance? • Yes • No • Maybe STA 291 Summer 2010 Lecture 20

3. Example • If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 1% level of significance? • Yes • No • Maybe STA 291 Summer 2010 Lecture 20

4. Significance Test for a Proportion • Same process as with population mean • Value we are testing against is called p0 • Test statistic • P-value • Calculation is exactly the same as for the test for a mean • Sample size restrictions: STA 291 Summer 2010 Lecture 20

5. Testing Difference Between Two Population Proportions • Similar to testing one proportion • Hypotheses are set up like two sample mean test • H0:p1-p2=0 • Same as H0: p1=p2 • Test Statistic STA 291 Summer 2010 Lecture 20

6. Testing the Difference Between Means from Different Populations • Hypothesis involves 2 parameters from 2 populations • Test statistic is different • Involves 2 large samples (both samples at least 30) • One from each population • H0: μ1-μ2=0 • Same as H0: μ1=μ2 • Test statistic STA 291 Summer 2010 Lecture 20

7. Example • In the 1982 General Social Survey, 350 subjects reported the time spent every day watching television. The sample yielded a mean of 4.1 and a standard deviation of 3.3. • In the 1994 survey, 1965 subjects yielded a sample mean of 2.8 hours with a standard deviation of 2. • Set up hypotheses of a significance test to analyze whether the population means differ in 1982 and 1994 and test at α=.05 using the p-value method. STA 291 Summer 2010 Lecture 20

8. Correspondence Between Confidence Intervals and Tests • Constructing a confidence interval to do a hypothesis test with 2 samples works the same as it did when we were dealing with 1 sample • The confidence interval shows plausible values for the difference between the two means STA 291 Summer 2010 Lecture 20

9. Small Sample Tests for Two Means • Used when comparing means of two samples where at least one of them is less than 30 • Normal population distribution is assumed for both samples • Equal Variances • Both groups have the same variability • Unequal Variances • Both groups may not have the same variability STA 291 Summer 2010 Lecture 20

10. Small Sample Test for Two Means, Equal Variances • Test Statistic • Degrees of freedom • n1+n2-2 STA 291 Summer 2010 Lecture 20

11. Small Sample Confidence Interval for Two Means, Equal Variances • Degrees of freedom • n1+n2-2 STA 291 Summer 2010 Lecture 20

12. Small Sample Test for Two Means, Unequal Variances • Test statistic • Degrees of freedom STA 291 Summer 2010 Lecture 20

13. Small Sample Confidence Interval for Two Means, Unequal Variances STA 291 Summer 2010 Lecture 20

14. Method 1 (Equal Variances) vs. Method 2 (Unequal Variances) • How to choose between Method 1 and Method 2? • Method 2 is always safer to use • Definitely use Method 2 • If one standard deviation is at least twice the other • If the standard deviation is larger for the sample with the smaller sample size • Usually, both methods yield similar conclusions STA 291 Summer 2010 Lecture 20