1. Statistics
ANOVA

2. Difference Between Means
We use the t-test when we have paired data because it is more powerful We could use it for other two-group comparisons, but we usually use another analysis:

3. Difference Between Means
Comparing Several Group's Means: “ANOVA” “Analysis of Variance”

4. Difference Between Means
t-tests can only be used for comparing two groups ANOVA can be used to compare two or more groups

5. Difference Between Means
A paired t-test is more powerful A non-paired t-test is THE SAME as an ANOVA (ANOVA’s Excel output page is better)

6. Difference Between Means
“ANOVA” stands for “ANalysis Of VAriance”

7. Difference Between Means
The analysis assigns the variability in the data to: the difference between the groups the difference between individuals

8. Difference Between Means
Sir Ronald Aylmer Fisher

9. Difference Between Means
ANOVA sheet Salaries for Criminal Justice Jobs

10. Difference Between Means
There are four classifications of jobs: probation, administration, correctional and patrol We want to compare the salaries to see if they are the same

11. What would be a good value for α?
ANOVA
PROJECT QUESTION
What would be a good value for α?

12. α = .05 What would be a good level of practical significance?
ANOVA
PROJECT QUESTION
α = .05 What would be a good level of practical significance?

13. ANOVA
PROJECT QUESTION
What is Ha?

14. μprobation ≠ μadministration ≠ μcorrectional ≠ μpatrol
ANOVA
PROJECT QUESTION
Alternative hypothesis Ha:
There are differences in the salaries of the four job classifications:
μprobation ≠ μadministration ≠ μcorrectional ≠ μpatrol
What is H0?

15. μprobation = μadministration = μcorrectional = μpatrol
ANOVA
PROJECT QUESTION
Null (no difference) hypothesis H0:
There is no difference in salaries for the four job classifications:
μprobation = μadministration = μcorrectional = μpatrol

16. Difference Between Means
Our strategy: We hope to disprove H0 and thereby to prove Ha

17. Why can’t you use a t-test for this data?
ANOVA
PROJECT QUESTION
Why can’t you use a t-test for this data?

18. Difference Between Means
use: Data Data Analysis ANOVA: Single Factor

19. Don’t include “Years”

20. Difference Between Means
The first output table is just descriptive statistics:

21. Difference Between Means
The second table is the ANOVA table … EEK!

22. Difference Between Means
Don’t panic! Just look at the P-value

23. Difference Between Means
The P-value is the likelihood of our pattern of differences in the means IF H0 was TRUE

24. Difference Between Means
The probability is 1.99 E-5 or ( %) Is that very likely?

25. ANOVA
PROJECT QUESTION
We said we would reject H0 if it was only 0.05 (5%) likely to be true Can we reject H0?

26. Difference Between Means
Remember – Reject the null hypothesis if the statistic is smaller than 0.05

27. Difference Between Means
YES! If we are willing to be wrong in rejecting H0 5% of the time, % is a whole lot less likely to be wrong!

28. Was the difference practically significant?
ANOVA
PROJECT QUESTION
Was the difference practically significant?

29. What is your conclusion?
ANOVA
PROJECT QUESTION
What is your conclusion?

30. Questions?

31. Difference Between Means
For the CJ job classifications, we rejected H0 and concluded the salaries are different

32. Difference Between Means
But… Are they all different, or is just one different or two or …

33. Difference Between Means
Do a HI-Lo-Close Confidence Interval graph!

34. Difference Between Means

35. Difference Between Means

36. Which means are different?
ANOVA
PROJECT QUESTION
Which means are different?

37. Difference Between Means
Now do you see why we’ve been doing Hi-Lo-Close graphs once a week since we learned them?

38. Difference Between Means
BTW: pre-Excel, this comparison used to be REALLY hard to do!

39. Difference Between Means
Yay Excel!

40. What would happen if you had a bigger sample size?
ANOVA
PROJECT QUESTION
What would happen if you had a bigger sample size?

41. Difference Between Means
PROJECT QUESTION
What would happen if you had a bigger sample size? You would be able to show more statistically significant differences

42. ANOVA
PROJECT QUESTION

43. Difference Between Means
t-tests and ANOVAs are designed to be VERY powerful for small sample sizes

44. Difference Between Means
That’s why we include a level of practical significance

45. Difference Between Means
Similar to previous tests, the P comes from a standardized F-distribution:

46. Difference Between Means
Because “z” and “t” are based on 𝒙 , they have similar shapes F is based on a variance, so it is in squared units!

47. Questions?

48. ANOVA
The ANOVA we just did is called a “One Factor ANOVA”

49. ANOVA
The ANOVA we just did is called a “One Factor ANOVA” Because there is only one category (type of job) – called a “factor”

50. ANOVA
You can have as many factors as you want

51. ANOVA
Excel can handle 2 factors

52. 2 Factor ANOVA
This one is tricky – you have to have the same number of observations in each category
Educ Level HS
Assoc
Bachelor's M
$ 15,000
$ 25,000
$ 35,000
$ 14,000
$ 24,000
$ 34,000 F
$ 12,000
$ 32,000
$ 43,000

53. 2 Factor ANOVA
With only one observation it’s called “without replication”
Educ Level HS
Assoc
Bachelor's M
$ 15,000
$ 25,000
$ 35,000
$ 14,000
$ 24,000
$ 34,000 F
$ 12,000
$ 32,000
$ 43,000

54. 2 Factor ANOVA
With only one observation it’s called “without replication”
With more than one, it’s called “with replication”
Educ Level HS
Assoc
Bachelor's M
$ 15,000
$ 25,000
$ 35,000
$ 14,000
$ 24,000
$ 34,000 F
$ 12,000
$ 32,000
$ 43,000

55. 2 Factor ANOVA An ANOVA table: ANOVA Source of Variation SS df MS F p
Total
1,214,916,667 11
Gender
52,083,333 1
7.35 4%
Educ Level
960,166,667 2
480,083,333
67.78 0%
Interaction
160,166,667
80,083,333
11.31 1%
Within
42,500,000 6
7,083,333

56. Questions?