1. Inferences About Means of Two Independent Samples Chapter 11 Homework: 1, 2, 3, 4, 6, 7

2. Hypotheses with 2 Independent Samples n Ch 11: select 2 independent samples l are they from same population? H 0 :  1 =  2 ~

3. Dependent & Independent Variables n Rest of course measure 2 variables l Until now: measured a single variable n Variables in an experiment n Dependent variable (DV) l *measured outcome of interest n Independent variable (IV) l value defines group membership l manipulated variable ~

4. Dependent & Independent Variables n True experiment l experimenter randomly assigns subjects to groups l at least 2 variables: DV & IV n e.g., Does the amount of sleep you get the night before an exam affect your performance? n Randomly assign groups l Group 1: 4 hours; Group 2: 8 hours ~

5. Dependent & Independent Variables n Dependent variable l test score n Independent variable l amount of sleep l 2 levels of IV: 4 & 8 hours 2 groups, n =10 per group ~

6. H 0 & H 1 n Does the amount of sleep you get the night before an exam affect your performance? n Directionality? H 0 :  1 =  2 or H 0 :  1 -  2 = 0 H 1 :  1   2 n no value for either specified l Group 1 scores = Group 2 scores ~

7. Experimental Outcomes n Do not expect to be exactly equal l sampling error n How much overlap allowed to accept H 0 l What size difference to reject? ~

8. The Test Statistic n Sample differences: X 1 - X 2 n test statistic = sample statistic - population parameter standard error of sample statistic l general form the same n Must use t test do not know  ~

9. The Test Statistic n test statistic = [df = n 1 + n 2 - 2] n Denominator l s X of difference between 2 means ~

10. The Test Statistic Since  1 -  2 = 0 test statistic = [df = n 1 + n 2 - 2]

11. The Test Statistic: Assumptions Assume:  1 =  2 Assume equal variance   1 =    2 does not require s 2 1  s 2 2 therefore,  1 =  2 n t test is robust l violation of assumptions l No large effect on probability of rejecting H 0 ~

12. Standard Error of (X 1 - X 2 ) n Distribution of differences: X 1 - X 2 l all possible combinations of 2 means l from same population n Compute standard error of difference between 2 means

13. s 2 pooled : Pooled Variance n Best estimate of variance of population s 2 1 is 1 estimate of  2 s 2 2 is a 2d estimate of same  2 l Pooling them gives a better estimate ~

14. Pooled Variance s 2 pooled is estimate of  2 pooled weighted average of 2 or more variances (  2 ) l Weight depends on sample size n Equal sample sizes: n 1 = n 2 ~

15. Example n What effect does the amount of sleep the night before an exam have on exam performance? n Dependent variable? n independent variable l Grp 1: 4 hrs sleep (n = 6) l Grp 1: 8 hrs sleep (n = 6) ~

16. Example 1. State Hypotheses H 0 :  1 =  2 H 1 :  1  2 2. Set criterion for rejecting H 0 : nondirectional  =.05 df = (n 1 + n 2 - 2) = (6 + 6 - 2) = 10 t CV.05 = + 2.228 ~

17. The Test Statistic n Sample differences: X 1 - X 2 n test statistic = sample statistic - population parameter standard error of sample statistic l general form the same n Must use t test do not know  ~

18. The Test Statistic n test statistic = [df = n 1 + n 2 - 2] n Denominator l s X of difference between 2 means ~

19. The Test Statistic Since  1 -  2 = 0 test statistic = [df = n 1 + n 2 - 2]

20. The Test Statistic: Assumptions Assume:  1 =  2 Assume equal variance   1 =    2 does not require s 2 1  s 2 2 therefore,  1 =  2 n t test is robust l violation of assumptions l No large effect on probability of rejecting H 0 ~

21. Standard Error of (X 1 - X 2 ) n Distribution of differences: X 1 - X 2 l all possible combinations of 2 means l from same population n Compute standard error of difference between 2 means

22. s 2 pooled : Pooled Variance n Best estimate of variance of population s 2 1 is 1 estimate of  2 s 2 2 is a 2d estimate of same  2 l Pooling them gives a better estimate ~

23. Example n What effect does the amount of sleep the night before an exam have on exam performance? n Dependent variable? n independent variable l Grp 1: 4 hrs sleep (n = 6) l Grp 1: 8 hrs sleep (n = 6) ~

24. Example 1. State Hypotheses H 0 :  1 =  2 H 1 :  1  2 2. Set criterion for rejecting H 0 : nondirectional  =.05 df = (n 1 + n 2 - 2) = (6 + 6 - 2) = 10 t CV.05 = + 2.228 ~

25. Example 3. select sample, compute statistics do experiment mean exam scores for each group l Group 1: X 1 = 15 ; s 1 = 4 l Group 2: X 2 = 17; s 2 = 3 n compute l s 2 pooled l s X 1 - X 2 l t obs ~

26. Example n compute s 2 pooled

27. Example n compute s X1- X2

28. Example n compute test statistic [df = n 1 + n 2 - 2]

29. Example 4. Interpret Is t obs beyond t CV ? If yes, Reject H 0. n Practical significance?

30. Pooled Variance: n 1  n 2 n Unequal sample sizes l weight each variance l bigger n ---> more weight

31. Pooled Variance n since df = n - 1

32. Example n What effect does the amount of sleep the night before an exam have on exam performance? n Dependent variable n independent variable l Grp 1: 4 hrs sleep (n = 6) l Grp 1: 8 hrs sleep (n = 7) ~

33. Example 1. State Hypotheses H 0 :  1 =  2 or  1 -  2 = 0 H 1 :  1  2 or  1 -  2  0 2. Set criterion for rejecting H 0 : nondirectional  =.05 df = (n 1 + n 2 - 2) = (6 + 7 - 2) = 11 t CV = + 2.201 ~

34. Example 3. select sample, compute statistics do experiment mean exam scores for each group l Group 1: X 1 = 14 ; s 1 = 3 l Group 2: X 2 = 19; s 2 = 2 n compute l s 2 pooled l s X 1 - X 2 l t obs ~

35. Example n compute s 2 pooled n compute s X1- X2 n compute test statistic [df = n 1 + n 2 - 2]

36. Example 4. Interpret Is t obs beyond t CV ? If yes, Reject H 0. n Practical significance?