1. Lecture 22 Dustin Lueker

2.  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

3.  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

4.  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

5.  Test Statistic ◦ Degrees of freedom  n 1 +n 2 -2 5STA 291 Summer 2008 Lecture 21

6. ◦ Degrees of freedom  n 1 +n 2 -2 6STA 291 Summer 2008 Lecture 21

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

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9. 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

10.  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

11.  “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

12.  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

13.  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

14.  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

15.  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 Student12345678910 Before60734288667790635596 After70804094798693717097 STA 291 Summer 2008 Lecture 21

16. 16STA 291 Summer 2008 Lecture 21

17.  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 Student12345678910 Before60734288667790635596 After70804094798693717097 STA 291 Summer 2008 Lecture 21

18. 18 Output from Statistical Software Package SAS N 10 Mean 7 Std Deviation 5.24933858 Tests for Location: Mu0=0 Test -Statistic- -----p Value------ Student's t t 4.216901 Pr > |t| 0.0022 Sign M 4 Pr >= |M| 0.0215 Signed Rank S 25.5 Pr >= |S| 0.0059 STA 291 Summer 2008 Lecture 21

19.  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