COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman

1. COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick Chapter : 12 Analysis of Variance: ANOVA

2. Key Terms: Don’t Forget Notecards • Factors (p. 388) • Levels (p. 388) • Testwise Alpha Level (p. 391) • Experimentwise Alpha Level (p. 391) • Error Term (p. 394) • Post Hoc Tests or Post Tests (p. 416)

3. ANOVA Notation • k is used to identify the number of treatment conditions • n is used to identify the number of scores in each treatment condition • N is used to identify the total number scores in the entire study • N= kn, when samples are the same size • T stands for treatment total and is calculated by ∑X, which equals the sum of the scores for each treatment condition • G stands for the sum of all scores in a study (Grand Total) • Calculate by adding up all N scores or by adding treatment total (G=∑T) • You will also need SS and M for each sample, and ∑X2 for the entire set of all scores.

4. Formulas • F-ratio: • SStotal: • SSwithin: • SSbetween: • SSbetween: • dftotal: • dfwithin: or • dfbetween:

5. More Formulas • MSwithin: • MSbetween: • Tukey’s HSD: • Scheffe Test: • Effect Size:

6. Hypothesis Testing with ANOVA • Question 1: A psychologist studied three computer keyboard designs. Three samples of individuals were given material to type on a particular keyboard, and the number of errors committed by each participant was recorded. The data are as follows: N = G = ΣX2 = T = SS = T = SS = T = SS =

7. Hypothesis Testing with ANOVA • Question 1: Are these data sufficient to conclude that there are significant differences in typing performance among the three keyboard designs? Set alpha at α = 0.05 N = 15 G = 60 ΣX2 = 356 T = 5 SS = 12 T = 25 SS = 20 T = 30 SS = 14

8. Hypothesis Testing with ANOVA • Question 1 Answer: • Step 1: State the hypothesis. • H0: μ1 = μ2 = μ3 (Type of keyboard has no effect) • H1: At least one of the treatment means is different.

9. Hypothesis Testing with ANOVA • Question 1 Answer: • Step 2: Locate the critical region • For this problem df = 2,12 and the critical value for α = 0.05 is F = 3.88. If F-ratio ≤ Fcritical (3.88), then fail to reject H0. If F-ratio > Fcritical (3.88), then reject H0.

10. Hypothesis Testing with ANOVA • Question 1 Answer: • Step 3: Perform the analysis. • or

11. Hypothesis Testing with ANOVA F = 9.14

12. Hypothesis Testing with ANOVA • Question 1 Answer: • Step 4: Make a decision • F-ratio (9.14) > Fcritical (3.88). Therefore, we reject H0. The type of keyboard used has a significant effect on the number of errors committed. If F-ratio ≤ Fcritical (3.88), then fail to reject H0. If F-ratio > Fcritical (3.88), then reject H0.

13. Computing Effect Size for ANOVA • Question 2: Compute effect size (η2), the percentage of variance explained, for the data that were analyzed in Question 1.

14. Computing Effect Size for ANOVA • Question 2 Answer:

15. Post Hoc Tests • Question 3: For the data used in Question 1, perform a post hoc test to determine which mean differences are significant and which are not. Use both Tukey’s HSD and the Scheffe Test.

16. Post Hoc Tests: Tukey’s HSD • Question 3 Answer: • Find q. q = 3.77 (Table B.5, p.708) • Thus, the mean difference between any two samples must be at least 3.23 to be significant. • Find the means for each treatment.

17. Post Hoc Tests: Tukey’s HSD • Question 3 Answer: • , Treatment A is significantly different than Treatment B. • , Treatment A is significantly different than Treatment C. • , Treatment B is not significantly different than Treatment C. HSD = 3.23

18. Post Hoc Tests: Scheffe Test • Question 3 Answer: • First, compute SSbetween for Treatments A and B. • Now, find MSbetween. • For df(2,12) and α = 0.05, the critical region for F is 3.88. Therefore our obtained F-ratio is in the critical region, and we must conclude that these data show a significant difference between treatment A and treatment B. Notice: G is equal to the total of Treatments A and B, not A, B, and C. Similarly, N is equal to nA + nB. For dfbetween, use k-1.

19. Post Hoc Tests: Scheffe Test • Question 3 Answer: • First, compute SSbetween for Treatments A and C. • Now, find MSbetween. • For df (2,12) and α = 0.05, the critical region for F is 3.88. Therefore our obtained F-ratio is in the critical region, and we must conclude that these data show a significant difference between treatment A and treatment C. Notice: G is equal to the total of Treatments A and C, not A, B, and C. Similarly, N is equal to nA + nC. For dfbetween, use k-1.

20. Post Hoc Tests: Scheffe Test • Question 3 Answer: • First, compute SSbetween for Treatments B and C. • Now, find MSbetween. • For df (2,12) and α = 0.05, the critical region for F is 3.88. Therefore our obtained F-ratio is not in the critical region, and we must conclude that these data show no significant difference between treatment B and treatment C. Notice: G is equal to the total of Treatments B and C, not A, B, and C. Similarly, N is equal to nB + nC. For dfbetween, use k-1.

21. Assumptions for ANOVA • Question 4: What three assumptions are required for ANOVA?

22. Assumptions for ANOVA • Question 4 Answer: • The observations within each sample must be independent. • The populations from which the samples are selected must be normal. • The populations from which the samples are selected must have equal variances (homogeneity of variance).