STA 291 Summer 2010

1. STA 291Summer 2010 Lecture 21 Dustin Lueker

2. 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 21

3. 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 21

4. Comparing Dependent Samples • Comparing dependent means • Example • Special exam preparation for STA 291 students • Choose n=10 pairs of students such that the students matched in any given pair are very similar given previous exam/quiz results • For each pair, one of the students is randomly selected for the special preparation (group 1) • The other student in the pair receives normal instruction (group 2) STA 291 Summer 2010 Lecture 21

5. Example (cont.) • “Matches Pairs” plan • Each sample (group 1 and group 2) has the same number of observations • Each observation in one sample ‘pairs’ with an observation in the other sample • For the ith pair, let Di = Score of student receiving special preparation – score of student receiving normal instruction STA 291 Summer 2010 Lecture 21

6. Comparing Dependent Samples • The sample mean of the difference scores is an estimator for the difference between the population means • We can now use exactly the same methods as for one sample • Replace Xi by Di STA 291 Summer 2010 Lecture 21

7. Comparing Dependent Samples • Small sample confidence interval Note: • When n is large (greater than 30), we can use the z-scores instead of the t-scores STA 291 Summer 2010 Lecture 21

8. Comparing Dependent Samples • Small sample test statistic for testing difference in the population means • For small n, use the t-distribution with df=n-1 • For large n, use the normal distribution instead (z value) STA 291 Summer 2010 Lecture 21

9. Example • Ten college freshman take a math aptitude test both before and after undergoing an intensive training course • Then the scores for each student are paired, as in the following table STA 291 Summer 2010 Lecture 21

10. Example STA 291 Summer 2010 Lecture 21

11. Example • Compare the mean scores after and before the training course by • Finding the difference of the sample means • Find the mean of the difference scores • Compare • Calculate and interpret the p-value for testing whether the mean change equals 0 • Compare the mean scores before and after the training course by constructing and interpreting a 90% confidence interval for the population mean difference STA 291 Summer 2010 Lecture 21

12. Reducing Variability • Variability in the difference scores may be less than the variability in the original scores • This happens when the scores in the two samples are strongly associated • Subjects who score high before the intensive training also tend to score high after the intensive training • Thus these high scores aren’t raising the variability for each individual sample STA 291 Summer 2010 Lecture 21