2. Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2016 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays

4. By the end of lecture today 3/28/16 Analysis of Variance (ANOVA) Project 3

5. Before next exam (April 8 th ) Please read chapters 1 - 11 in OpenStax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

6. On class website: Please Complete Online Module Homework #22 ANOVA Due: Wednesday, March 30 th Homework

7. Everyone will want to be enrolled in one of the lab sessions Labs continue this week, Completing ANOVAs Project 3

8. “SS” = “Sum of Squares” “df” = degrees of freedom Sample Standard Deviation = s = Sample Variance = s 2 = Remember, you should know these two formulas by heart

9. Study Type 2: t-test Study Type 3: One-way Analysis of Variance (ANOVA) Comparing more than two means We are looking to compare two means Prep Project 3 Prep Project 3

10. Single Independent Variable comparing more than two groups Study Type 3: One-way ANOVA Single Dependent Variable (numerical/continuous) Ian was interested in the effect of incentives for girl scouts on the number of cookies sold. He randomly assigned girl scouts into one of three groups. The three groups were given one of three incentives and looked to see who sold more cookies. The 3 incentives were 1) Trip to Hawaii, 2) New Bike or 3) Nothing. This is an example of a true experiment Used to test the effect of the IV on the DV None New Bike Sales per Girl scout Trip Hawaii None New Bike Trip Hawaii Dependent variable is always quantitative Dependent variable is always quantitative In an ANOVA, independent variable is qualitative (& more than two groups) In an ANOVA, independent variable is qualitative (& more than two groups) Sales per Girl scout

11. Be careful you are not designing a Chi Square One-way ANOVA versus Chi Square None New Bike Sales per Girl scout Trip Hawaii This is an ANOVA None New Bike Total Number of Boxes Sold Trip Hawaii This is a Chi Square This is a Chi Square If this is just frequency you may have a problem These are means These are just frequencies

12. One-way ANOVA One-way ANOVAs test only one independent variable - although there may be many levels “Factor” = one independent variable “Level” = levels of the independent variable treatment condition groups “Main Effect” of independent variable = difference between levels Note: doesn’t tell you which specific levels (means) differ from each other A multi-factor experiment would be a multi-independent variables experiment Number of cookies sold Incentives None Bike Hawaii trip

13. Comparing ANOVAs with t-tests Similarities still include: Using distributions to make decisions about common and rare events Using distributions to make inferences about whether to reject the null hypothesis or not The same 5 steps for testing an hypothesis The three primary differences between t-tests and ANOVAS are: 1. ANOVAs can test more than two means 2. We are comparing sample means indirectly by comparing sample variances 3. We now will have two types of degrees of freedom t(16) = 3.0; p < 0.05 F(2, 15) = 3.0; p < 0.05 Tells us generally about number of participants / observations Tells us generally about number of groups / levels of IV

14. A girl scout troop leader wondered whether providing an incentive to whomever sold the most girl scout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. n = 5 x = 10 n = 5 x = 12 n = 5 x = 14 Troop 1 (nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 What is Independent Variable? How many groups? What is Dependent Variable? What is Dependent Variable? How many levels of the Independent Variable?

15. Hypothesis testing: Step 1: Identify the research problem Describe the null and alternative hypotheses Is there a significant difference in the number of cookie boxes sold between the girlscout troops that were given the different levels of incentive?

16. Hypothesis testing: Decision rule =.05 Critical F (2,12) =3.98 Degrees of freedom (between) = number of groups - 1 Degrees of freedom (within) = # of scores - # of groups = 3 - 1 = 2 = (15-3) = 12* *or = (5-1) + (5-1) + (5-1) = 12.

17. α =.05 Critical F (2,12) = 3.89 F (2,12) Appendix B.4 (pg.518)

18. ANOVA table ? dfMS F ? ? ? Source Between Within Total ? ? SS ? ? ? “SS” = “Sum of Squares” - will be given for exams - you can think of this as the numerator in a standard deviation formula 128 88 40

19. ANOVA table dfMSF ? ? ? Source Between Within Total SS ? ? ? Writing Assignment - ANOVA 1. Write formula for standard deviation of sample 2. Write formula for variance of sample 3. Re-write formula for variance of sample using the nicknames for the numerator and denominator 4. Complete this ANOVA table 128 88 40 SS df = MS

20. ? ? ? ANOVA table 128 dfMS F # groups - 1 # scores - number of groups # scores - 1 2 12 14 Source Between Within Total 88 40 SS ? ? ? ? ? ? “SS” = “Sum of Squares” - will be given for exams 3-1=2 15-3=12 15- 1=14

21. ANOVA table 128 df MS F 2 12 14 Source Between Within Total 88 40 SS MS between MS within SS within df within 20 7.33 SS between df between 88 12 =7.33 40 2 2 =20 20 7.33 =2.73 2.73 40 2 2 88 12 ? ? ? “SS” = “Sum of Squares” - will be given for exams

22. Make decision whether or not to reject null hypothesis 2.73 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Observed F = 2.73 Critical F (2,12) = 3.89 Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold F (2,12) = 2.73; n.s. The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14, the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s.

23. Main effect of incentive: Will offering an incentive result in more girl scout cookies being sold? If we have a “effect” of incentive then the means are significantly different from each other we reject the null we have a significant F p < 0.05 We don’t know which means are different from which …. just that they are not all the same To get an effect we want: Large “F” - big effect and small variability Small “p” - less than 0.05 (whatever our alpha is)

24. Let’s do same problem Using MS Excel A girlscout troop leader wondered whether providing an incentive to whomever sold the most girlscout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. Troop 1 (Nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 n = 5 x = 10 n = 5 x = 12 n = 5 x = 14

25. Let’s do one Replication of study (new data)

26. Let ’ s do same problem Using MS Excel

28. ANOVA table MS between MS within SS between df between 88 12 =7.33 40 2 2 =20 20 7.33 =2.73 “Sum of Squares” # groups - 1 # scores - # of groups # scores - 1 3-1=2 15-3=12 15- 1=14 SS within df within

29. F critical (is observed F greater than critical F?) P-value (is it less than.05?) No, so it is not significant Do not reject null No, so it is not significant Do not reject null No, so it is not significant Do not reject null No, so it is not significant Do not reject null “Sum of Squares”

30. Make decision whether or not to reject null hypothesis 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Observed F = 2.73 Critical F (2,12) = 3.89 Also p-value is not smaller than 0.05 so we do not reject the null hypothesis Step 6: Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold

31. Make decision whether or not to reject null hypothesis 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Observed F = 2.72727272 Critical F (2,12) = 3.88529 Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold F (2,12) = 2.73; n.s. The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14, the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s.

32. Homework

35. Type of major in school 4 (accounting, finance, hr, marketing) Grade Point Average 0.05 2.83 3.02 3.24 3.37

36. Homework 0.3937 0.1119 0.3937 / 0.1119 = 3.517 3.517 3.009 3 24 0.03 If observed F is bigger than critical F: Reject null & Significant! If p value is less than 0.05: Reject null & Significant! # groups - 1 # scores - number of groups # scores - 1 4-1=3 28 - 4=24 28 - 1=27

37. Homework Yes F (3, 24) = 3.517;p < 0.05 The GPA for four majors was compared. The average GPA was 2.83 for accounting, 3.02 for finance, 3.24 for HR, and 3.37 for marketing. An ANOVA was conducted and there is a significant difference in GPA for these four groups (F (3,24) = 3.52; p < 0.05).

38. Number of observations in each group Average for each group (We REALLY care about this one)

39. “SS” = “Sum of Squares” - will be given for exams Number of groups minus one (k – 1)  4-1=3 Number of people minus number of groups (n – k)  28-4=24

40. MS between MS within SS between df between SS within df within

43. Type of executive 3 (banking, retail, insurance) Hours spent at computer 0.05 10.8 8 8.4

44. 11.46 2 11.46 / 2 = 5.733 5.733 3.88 2 12 0.0179 If observed F is bigger than critical F: Reject null & Significant! If p value is less than 0.05: Reject null & Significant!

45. Yes F (2, 12)= 5.73; p < 0.05 The number of hours spent at the computer was compared for three types of executives. The average hours spent was 10.8 for banking executives, 8 for retail executives, and 8.4 for insurance executives. An ANOVA was conducted and we found a significant difference in the average number of hours spent at the computer for these three groups, (F (2,12) = 5.73; p < 0.05).

46. Number of observations in each group Just add up all scores Average for each group

47. “SS” = “Sum of Squares” - will be given for exams Number of groups minus one (k – 1)  3-1=2 Number of people minus number of groups (n – k)  15-3=12

48. MS between MS within SS between df between SS within df within