2. One-Way ANOVA ANOVA = Analysis of Variance This is a technique used to analyze the results of an experiment when you have more than two groups

3. Example You measure the number of days 7 psychology majors, 7 sociology majors, and 7 biology majors are absent from class You wonder if the average number of days each of these three groups was absent is significantly different from one another

4. Results X = 3.00X = 2.00X = 1.00

5. Hypothesis Alternative hypothesis (H 1 ) H 1: The three population means are not all equal

6. Hypothesis Null hypothesis (H 0 )  psych =  socio =  bio

7. Between and Within Group Variability Two types of variability Between –the differences between the mean scores of the three groups –The more different these means are, the more variability!

8. Results X = 3.00X = 2.00X = 1.00

9. Between Variability X = 3.00X = 2.00X = 1.00 S 2 =.66

10. Between Group Variability What causes this variability to increase? 1) Effect of the variable (college major) 2) Sampling error

11. Between and Within Group Variability Two types of variability Within –the variability of the scores within each group

12. Results X = 3.00X = 2.00X = 1.00

13. Within Variability X = 3.00X = 2.00X = 1.00 S 2 =.57S 2 =1.43S 2 =.57

14. Within Group Variability What causes this variability to increase? 1) Sampling error

15. Between and Within Group Variability Between-group variability Within-group variability

16. Between and Within Group Variability sampling error + effect of variable sampling error

17. Between and Within Group Variability sampling error + effect of variable sampling error Thus, if null hypothesis was true this would result in a value of 1.00

18. Between and Within Group Variability sampling error + effect of variable sampling error Thus, if null hypothesis was not true this value would be greater than 1.00

20. Calculating this Variance Ratio

23. Degrees of Freedom df between df within df total df total = df between + df within

24. Degrees of Freedom df between = k - 1 (k = number of groups) df within = N - k (N = total number of observations) df total = N - 1 df total = df between + df within

25. Degrees of Freedom df between = k - 1 3 - 1 = 2 df within = N - k 21 - 3 = 18 df total = N - 1 21 - 1 = 20 20 = 2 + 18

26. Sum of Squares SS Between SS Within SS total SS total = SS Between + SS Within

27. Sum of Squares SS total

28. Sum of Squares SS Within

29. Sum of Squares SS Between

30. Sum of Squares Ingredients:  X  X 2  T j 2 N n

31. To Calculate the SS

32. XX  X s = 21  X p = 14  X B = 7

33. XX  X s = 21  X p = 14  X B = 7  X = 42

34. X2X2  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7  X = 42

35. X2X2  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7  X = 42  X 2 = 116

36. T 2 = (  X) 2 for each group  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49  X = 42  X 2 = 116

37. Tj2Tj2  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49  X = 42  X 2 = 116  T j 2 = 686

38. N  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49  X = 42  X 2 = 116  T j 2 = 686 N = 21

39. n  X 2 s = 67  X 2 P = 38  X 2 B = 11  X s = 21  X p = 14  X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7

40. Ingredients  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7

41. Calculate SS  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7 SS total

42. Calculate SS  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7 SS total 116 42 21 32

43. Calculate SS SS Within  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7

44. Calculate SS SS Within  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7 116 686 7 18

45. Calculate SS SS Between  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7

46. Calculate SS SS Between  X = 42  X 2 = 116  T j 2 = 686 N = 21 n = 7 686 7 42 21 14

47. Sum of Squares SS Between SS Within SS total SS total = SS Between + SS Within

48. Sum of Squares SS Between = 14 SS Within = 18 SS total = 32 32 = 14 + 18

49. Calculating the F value

51. 14 2 7

52. Calculating the F value 7

53. 7 18 1

54. Calculating the F value 7 1 7

55. How to write it out

56. Significance Is an F value of 7.0 significant at the.05 level? To find out you need to know both df

57. Degrees of Freedom Df between = k - 1 (k = number of groups) df within = N - k (N = total number of observations)

58. Degrees of Freedom Df between = k - 1 3 - 1 = 2 df within = N - k 21 - 3 = 18 Page 390 Table F Df between are in the numerator Df within are in the denominator Write this in the table

59. Critical F Value F(2,18) = 3.55 The nice thing about the F distribution is that everything is a one-tailed test

60. Decision Thus, if F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0

61. Current Example F value = 7.00 F critical = 3.55 Thus, reject H 0, and accept H 1

62. Alternative hypothesis (H 1 ) H 1: The three population means are not all equal –In other words, psychology, sociology, and biology majors do not have equal class attendence –Notice: It does not say where this difference is at!!

63. How to write it out