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

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

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

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

Example: Variables n Dependent variable l test score n Independent variable l amount of sleep l 2 levels of IV: 8 & 4 hours ~

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? ~

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

n Denominator l Standard error of difference between 2 means ~ The Test Statistic n test statistic = [df = n 1 + n 2 - 2]

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

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

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

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 ~

Pooled Variance n Weighted average of 2 or more variances   2 l Weight depends on sample size n Equal sample sizes: n 1 = n 2 ~

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

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

Example : Nondirectional 3. select sample, compute statistics do experiment mean exam scores for each group l Group 1: X 1 = 20 ; s 1 = 4 l Group 2: X 2 = 14; s 2 = 3 n compute l s 2 pooled l s X 1 - X 2 l t obs ~

Example : Nondirectional n compute s 2 pooled n compute

n compute test statistic Example : Nondirectional [df = n 1 + n 2 - 2]

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

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

Example: Directional Hypothesis n One-tailed test n 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? l Grp 1: 8 hrs sleep (n = 6) l Grp 2: 4 hrs sleep (n = 6) ~

Example : Directional 1. State Hypotheses H 0 :  1   2 H 1 :  1  2 2. Set criterion for rejecting H 0 : directionality: directional  =.05 df = (n 1 + n 2 - 2) = ( ) = 10 t CV = ~

Example : Directional 3. select sample, compute statistics do experiment mean exam scores for each group l Group 1: X 1 = 20 ; s 1 = 4 l Group 2: X 2 = 14; s 2 = 3 n compute l s 2 pooled l s X 1 - X 2 l t obs ~

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