STA 291 Spring 2010 Lecture 21 Dustin Lueker

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 Spring 2010 Lecture 21

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 Spring 2010 Lecture 21

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 Spring 2010 Lecture 21

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 Spring 2010 Lecture 21

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 Spring 2010 Lecture 21

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 Spring 2010 Lecture 21

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 Spring 2010 Lecture 21

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 Spring 2010 Lecture 21

Small Sample Test for Two Means, Equal Variances Test Statistic Degrees of freedom n1+n2-2 STA 291 Spring 2010 Lecture 21

Small Sample Confidence Interval for Two Means, Equal Variances Degrees of freedom n1+n2-2 STA 291 Spring 2010 Lecture 21

Small Sample Test for Two Means, Unequal Variances Test statistic Degrees of freedom STA 291 Spring 2010 Lecture 21

Small Sample Confidence Interval for Two Means, Unequal Variances STA 291 Spring 2010 Lecture 21

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 Spring 2010 Lecture 21