1. Welcome to .
Week 11 Thurs .
MAT135 Statistics

2. In-Class Project
What is your favorite diamond simulant? (1 most-5 least) What is your favorite white gem?

3. Hypothesis Tests
How all the inferential tests work:

4. Hypothesis Tests
How all the inferential tests work: Excel/your calculator calculates a probability that you would get the data you got if the null hypothesis were true

5. Hypothesis Tests If it’s ≤ α-level, reject H0
How all the inferential tests work:
Excel/your calculator calculates a probability that you would get the data you got if the null hypothesis were true
If it’s ≤ α-level, reject H0

6. The probability is .03 Do you reject H0?
HYPOTHESIS TESTS
PROJECT QUESTION
The probability is .03 Do you reject H0?

7. HYPOTHESIS TESTS
PROJECT QUESTION
The probability is .03 Do you reject H0? Yup! As long as your α wasn’t .01

8. Hypothesis Tests
CELEBRATE!

9. Remember – Reject the hypothesis if the statistic is smaller than 0.05
Hypothesis Tests
Remember – Reject the hypothesis if the statistic is smaller than 0.05

10. HYPOTHESIS TESTS
PROJECT QUESTION
Reject H0 ! Conclusion?

11. Reject H0 ! Conclusion: The batches do taste different!
HYPOTHESIS TESTS
PROJECT QUESTION
Reject H0 ! Conclusion: The batches do taste different!

12. HYPOTHESIS TESTS
PROJECT QUESTION
Which tastes better?

13. HYPOTHESIS TESTS
PROJECT QUESTION
For tests of differences between means, what would happen if you had a bigger sample size?

14. Hypothesis Tests
The p value comes from a standardized t-distribution:

15. Questions?

16. 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:

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

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

19. 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)

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

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

22. Difference Between Means
Sir Ronald Aylmer Fisher

23. Difference Between Means
Salaries for Criminal Justice Jobs

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

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

26. 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?

27. ANOVA
PROJECT QUESTION
What is Ha?

28. 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?

29. 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

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

31. 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?
Probation
Admin
Correctional
Patrol
$26,834
$54,780
$41,216
$64,632
$50,748
$63,447
$23,101
$26,782
$39,766
$63,687
$27,957
$28,697
$23,079
$55,653
$53,316
$30,732
$45,883
$59,299
$32,747
$52,670
$51,482
$63,063
$21,339
$36,893

32. Clear your TI83/4 data fields
ANOVA
PROJECT QUESTION
Clear your TI83/4 data fields
Probation
Admin
Correctional
Patrol
$26,834
$54,780
$41,216
$64,632
$50,748
$63,447
$23,101
$26,782
$39,766
$63,687
$27,957
$28,697
$23,079
$55,653
$53,316
$30,732
$45,883
$59,299
$32,747
$52,670
$51,482
$63,063
$21,339
$36,893

33. ANOVA
PROJECT QUESTION
Put “Probation” data in L1, “Admin” data in L2, “Correctional” in L3 and “Patrol” in L4
Probation
Admin
Correctional
Patrol
$26,834
$54,780
$41,216
$64,632
$50,748
$63,447
$23,101
$26,782
$39,766
$63,687
$27,957
$28,697
$23,079
$55,653
$53,316
$30,732
$45,883
$59,299
$32,747
$52,670
$51,482
$63,063
$21,339
$36,893

34. “ANOVA” (at the bottom on my TI83)
PROJECT QUESTION
“STAT”
“TESTS”
“ANOVA” (at the bottom on my TI83)
ENTER
ANOVA(L1,L2,L3,L4) ENTER
Probation
Admin
Correctional
Patrol
$26,834
$54,780
$41,216
$64,632
$50,748
$63,447
$23,101
$26,782
$39,766
$63,687
$27,957
$28,697
$23,079
$55,653
$53,316
$30,732
$45,883
$59,299
$32,747
$52,670
$51,482
$63,063
$21,339
$36,893

35. ANOVA
PROJECT QUESTION
Poof!
Done!
I got:
One-way ANOVA F= p= Factor df=3 SS= MS= Error df=20 SS= MS= Sxp=

36. ANOVA
PROJECT QUESTION
What are
we looking for?
One-way ANOVA F= p= Factor df=3 SS= MS= Error df=20 SS= MS= Sxp=

37. ANOVA
PROJECT QUESTION
What is our decision?
One-way ANOVA F= p= Factor df=3 SS= MS= Error df=20 SS= MS= Sxp=

38. ANOVA
PROJECT QUESTION
Reject H0!
One-way ANOVA F= p= Factor df=3 SS= MS= Error df=20 SS= MS= Sxp=

39. ANOVA
PROJECT QUESTION
What is our conclusion?

40. ANOVA
PROJECT QUESTION
What is your conclusion? We conclude there is a significant difference between the average pay of the CJ job categories

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

42. Questions?

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

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

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

46. Difference Between Means
From Excel:
Probation
Admin
Correctional
Patrol
Upper 95% CI
49,571
63,284
43,207
52,532
Lower 95% CI
29,693
56,692
23,351
27,603
Mean
39,632
59,988
33,279
40,068

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

48. Is the difference practically significant?
ANOVA
PROJECT QUESTION
Is the difference practically significant?

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

50. Difference Between Means
Yay Excel!

51. Difference Between Means
PROJECT QUESTION
What would happen if you had a bigger sample size?

52. 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

53. Difference Between Means
PROJECT QUESTION

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

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

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

57. 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!

58. Questions?

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

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

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

62. ANOVA
Excel can handle 2 factors

63. 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

64. 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

65. 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

66. 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

67. You survived! Turn in your homework! Don’t forget your homework
due next week!
Have a great
rest of the week!