Inferences About Means of Two Independent Samples

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

2. Hypotheses with 2 Independent Samples • Ch 11: select 2 independent samples • are they from same population? • Is difference due to chance?

3. Experimental Method • True experiment • subjects randomly assigned to groups • at least 2 variables • Dependent variable (DV) • measured outcome of interest • Independent variable (IV) • value defines group membership • manipulated variable ~

4. Experiment: Example • Does the amount of sleep the night before an exam affect exam performance? • Randomly assign groups • Group 1: 8 hours; Group 2: 4 hours ~

5. Example: Variables • Dependent variable • test score • Independent variable • amount of sleep • 2 levels of IV: 8 & 4 hours ~

6. Experimental Outcomes • Do not expect to be exactly equal • sampling error • How much overlap allowed to accept H0 • What size difference to reject? ~

7. The Test Statistic • Sample statistic: X1 - X2 • general form test statistic = sample statistic - population parameter standard error of sample statistic • Must use t test • do not know s~

8. The Test Statistic • test statistic = [df = n1 + n2 - 2] • Denominator • Standard error of difference between 2 means ~

9. [df = n1 + n2 - 2] The Test Statistic • Because m1 - m2 = 0 test statistic =

10. The Test Statistic: Assumptions • Assume: m1 = m2 • Assume equal variance • s21 = s22 • does not require s21= s22 • t test is robust • violation of assumptions • No large effect on probability of rejecting H0 ~

11. Standard Error of (X1 - X2) • Distribution of differences: X1 - X2 • all possible combinations of 2 means • from same population • Compute standard error of difference between 2 means

12. s2pooled : Pooled Variance • Best estimate of variance of population • s21 is 1 estimate of s2 • s22 is a 2d estimate of same s2 • Pooling them gives a better estimate ~

13. Pooled Variance • Weighted average of 2 or more variances • s2 • Weight depends on sample size • Equal sample sizes: n1 = n2 ~

14. Example • Does the amount of sleep the night before an exam affect exam performance? • Grp 1: 8 hrs sleep (n = 6) • Grp 2: 4 hrs sleep (n = 6) ~

15. Example 1. State Hypotheses H0: m1 = m2 H1: m1¹ m2 2. Set criterion for rejecting H0: directionality: nondirectional a = .05 df = (n1 + n2 - 2) = (6 + 6 - 2) = 10 tCV.05 = + 2.228 ~

16. Example : Nondirectional 3. select sample, compute statistics do experiment mean exam scores for each group • Group 1: X1 = 20 ; s1= 4 • Group 2: X2 = 14; s2= 3 • compute • s2pooled • s X1- X2 • tobs~

17. Example : Nondirectional • compute s2pooled • compute

18. Example : Nondirectional • compute test statistic [df = n1 + n2 - 2]

19. Example : Nondirectional 4. Interpret Is tobsbeyond tCV? If yes, Reject H0. • Practical significance?

20. Pooled Variance: n1¹n2 • Unequal sample sizes • weight each variance • bigger n ---> more weight

21. Example: Directional Hypothesis • One-tailed test • Do students who sleep a full 8 hrs the night before an exam perform better on the exam than students who sleep only 4 hrs? • Grp 1: 8 hrs sleep (n = 6) • Grp 2: 4 hrs sleep (n = 6) ~

22. Example : Directional 1. State Hypotheses H0: m1 m2 H1: m1> m2 2. Set criterion for rejecting H0: directionality: directional a = .05 df = (n1 + n2 - 2) = (6 + 6 - 2) = 10 tCV = + 1.812 ~

23. Example : Directional 3. select sample, compute statistics do experiment mean exam scores for each group • Group 1: X1 = 20 ; s1= 4 • Group 2: X2 = 14; s2= 3 • compute • s2pooled • s X1- X2 • tobs~

24. Example: Directional 4. Interpret Is tobsbeyond tCV? If yes, Reject H0. • Practical significance?