Lecture 22 Dustin Lueker

 Similar to testing one proportion  Hypotheses are set up like two sample mean test ◦ H 0 :p 1 -p 2 =0  Same as H 0 : p 1 =p 2  Test Statistic 2STA 291 Summer 2008 Lecture 21

 Hypothesis involves 2 parameters from 2 populations ◦ Test statistic is different  Involves 2 large samples (both samples at least 30)  One from each population  H 0 : μ 1 -μ 2 =0 ◦ Same as H 0 : μ 1 =μ 2 ◦ Test statistic 3STA 291 Summer 2008 Lecture 21

 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 4STA 291 Summer 2008 Lecture 21

 Test Statistic ◦ Degrees of freedom  n 1 +n STA 291 Summer 2008 Lecture 21

◦ Degrees of freedom  n 1 +n STA 291 Summer 2008 Lecture 21

 Test statistic  Degrees of freedom 7STA 291 Summer 2008 Lecture 21

8

9  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 2008 Lecture 21

 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) 10STA 291 Summer 2008 Lecture 21

 “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 i th pair, let D i = Score of student receiving special preparation – score of student receiving normal instruction 11STA 291 Summer 2008 Lecture 21

 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 X i by D i 12STA 291 Summer 2008 Lecture 21

 Small sample confidence interval Note: ◦ When n is large (greater than 30), we can use the z- scores instead of the t-scores 13STA 291 Summer 2008 Lecture 21

 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) 14STA 291 Summer 2008 Lecture 21

 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 15 Student Before After STA 291 Summer 2008 Lecture 21

16STA 291 Summer 2008 Lecture 21

 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 17 Student Before After STA 291 Summer 2008 Lecture 21

18 Output from Statistical Software Package SAS N 10 Mean 7 Std Deviation Tests for Location: Mu0=0 Test -Statistic p Value Student's t t Pr > |t| Sign M 4 Pr >= |M| Signed Rank S 25.5 Pr >= |S| STA 291 Summer 2008 Lecture 21

 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 dent to score high after the intensive training  Thus these high scores aren’t raising the variability for each individual sample 19STA 291 Summer 2008 Lecture 21