1. Statistics
t-Test

2. Hypothesis Tests
So far, we have tested: Does µ = a value (Confidence Intervals)

3. Hypothesis Tests
Today we test whether means from different samples are different

4. No Difference?

5. Difference Between Means
Remember the guy who drew the “leptokutic/platykurtic” pics?

6. Difference Between Means
William Sealy Gosset

7. Difference Between Means
Comparisons of means of two groups are called “t-Tests”

8. Difference Between Means
There are a variety of t-Tests Tests between two unrelated groups, each with its own treatment Tests between two related (paired) groups, each with its own treatment

9. t-Test
PROJECT QUESTION
Suppose you have two stat classes (A and B) taught by different teachers You want to know if grades on a standardized test are the same for the two groups

10. t-Test
PROJECT QUESTION
What’s first?

11. t-Test
PROJECT QUESTION
Step 1: Set the α-level

12. t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2:

13. Step 1: α-level = 0.05 Step 2: Set Ha
t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Set Ha

14. Step 1: α-level = 0.05 Step 2: Set Ha (they are different) Ha: μA ≠ μB
t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Set Ha (they are different) Ha: μA ≠ μB

15. Step 1: α-level = 0.05 Step 2: Ha: μA ≠ μB Step 3: Set H0
t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Ha: μA ≠ μB Step 3: Set H0

16. t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Ha: μA ≠ μB Step 3: Set H0 (they are the same) H0: μA = μB

17. Step 1: α-level = 0.05 Step 2: Ha: μA ≠ μB Step 3: H0: μA = μB
t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Ha: μA ≠ μB Step 3: H0: μA = μB

18. Is this a one-tail or two tail t-Test?
PROJECT QUESTION
Is this a one-tail or two tail t-Test?

19. Is this a one-tail or two tail t-Test? two-tail
PROJECT QUESTION
Is this a one-tail or two tail t-Test? two-tail

20. t-Test
PROJECT QUESTION
Now what?

21. t-Test
PROJECT QUESTION
Now what? Data!

22. t-Test
PROJECT QUESTION
Open the spreadsheet Note that you don’t need to have the same number in each group!
Class A
Class B 86 68 85 84 90 74 93 80 81 61 73 94 96 97

23. Do a t-Test: Two-Sample Assuming Unequal Variances
PROJECT QUESTION
Do a t-Test: Two-Sample Assuming Unequal Variances

24. t-Test
PROJECT QUESTION
Tah-Dah!

25. t-Test
PROJECT QUESTION
But… what does it mean?

26. Here are the means for Class A and Class B
t-Test
PROJECT QUESTION
Here are the means for Class A and Class B

27. t-Test
PROJECT QUESTION
…and their variances

28. t-Test
PROJECT QUESTION
…and sample sizes

29. t-Test
PROJECT QUESTION
What is your p-value?

30. t-Test
PROJECT QUESTION
What is your p-value? 9.96%

31. t-Test
PROJECT QUESTION
Do you reject H0?

32. Do you reject H0? Nope: p is bigger than alpha
t-Test
PROJECT QUESTION
Do you reject H0? Nope: p is bigger than alpha

33. What is your conclusion?
t-Test
PROJECT QUESTION
What is your conclusion?

34. t-Test
PROJECT QUESTION
What is your conclusion? We have been unable to show that scores from the two classes are different

35. Questions?

36. “Tailed” Tests
A two-tailed test will reject H0 either if the experimental values we get are too high or too low

37. “Tailed” Tests
α is split between the upper and lower tails

38. “Tailed” Tests
A one-tailed test will reject H0 only on the side we think is likely to be true

39. “Tailed” Tests
You will be able to reject H0 more often for a one-tailed test – if you pick the right tail!

40. 1-Tailed t-Test
PROJECT QUESTION
For our two classes, suppose there were student complaints about the teacher in class B
Class A
Class B 86 68 85 84 90 74 93 80 81 61 73 94 96 97

41. 1-Tailed t-Test
PROJECT QUESTION
If we want to show class B has poorer scores than class A, what is our alternate hypothesis?
Class A
Class B 86 68 85 84 90 74 93 80 81 61 73 94 96 97

42. Step 1: α-level = 0.05 Step 2: Ha: μA > μB
1-Tailed t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Ha: μA > μB

43. 1-Tailed t-Test
PROJECT QUESTION
If we want to show class B has poorer scores than class A, what is our null hypothesis?
Class A
Class B 86 68 85 84 90 74 93 80 81 61 73 94 96 97

44. Step 1: α-level = 0.05 Step 2: Ha: μA > μB Step 3: H0: μA ≤ μB
1-Tailed t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Ha: μA > μB Step 3: H0: μA ≤ μB

45. Is this a one-tail or two tail t-Test?
PROJECT QUESTION
Is this a one-tail or two tail t-Test?

46. Is this a one-tail or two tail t-Test? one-tail
PROJECT QUESTION
Is this a one-tail or two tail t-Test? one-tail

47. t-Test
PROJECT QUESTION
What is your p-value?

48. t-Test
PROJECT QUESTION
What is your p-value? 4.98%

49. t-Test
PROJECT QUESTION
Do you reject H0?

50. Do you reject H0? YES! p is smaller than alpha
t-Test
PROJECT QUESTION
Do you reject H0? YES! p is smaller than alpha

51. What is your conclusion?
t-Test
PROJECT QUESTION
What is your conclusion?

52. What is your conclusion? Class B has a lower score than class A
t-Test
PROJECT QUESTION
What is your conclusion? Class B has a lower score than class A

53. “Tailed” Tests
So, if you pick the tail correctly, the 1-tail test will be more powerful (reject H0 more often) that a 2-tail test

54. Questions?

55. Difference Between Means
The P comes from a standardized t-distribution:

56. Hypothesis Tests
We want our test to be powerful enough to reject a false null hypothesis

57. Hypothesis Tests
To do that, we take advantage of the Law of Large Numbers and take a large sample size

58. Hypothesis Tests
Mr. Gosset designed his t-test to be very powerful for small sample sizes He had only a limited number of trained tasters, so he would not be able to increase his sample size to increase the power of his tests

59. Hypothesis Tests
SO… A large sample size can lead to finding statistically significant differences that are SO tiny they have no practical significance

60. Hypothesis Tests
So… we also set a level of practical significance (the minimum difference that would actually be a REAL difference)

61. Hypothesis Tests
For our car mpg test, we found our car got 27 mpg while the manufacturer stated we should get 30 mpg

62. Hypothesis Tests
We set a personal level of practical significance for the test

63. PRACTICAL SIGNIFICANCE
PROJECT QUESTION
What would be your level of practical significance for stat test scores?

64. PRACTICAL SIGNIFICANCE
PROJECT QUESTION
What would be your level of practical significance for stat test scores? Did our observed difference (87.9 vs 79.5) meet your level of practical significance?

65. Questions?

66. Difference Between Means
To compare two paired group's means, use a: “Paired t-test”

67. Difference Between Means
“Paired”: two measurements on the same person measurements on twins two measurements for the same date, etc

68. Difference Between Means
Open up the spreadsheet

69. Difference Between Means
Mr Gosset has two batches of stout: F5_10 and N25_5 Tasters are tasting each to see if they are the same Mr Gosset thinks they are not!

70. Difference Between Means
These are paired data because each analyst tasted each of the batches – two measurements BY the same person

71. Difference Between Means
H0: (what we want to disprove) There is no difference in the flavor of the two batches Ha: (what we want to prove) There is a difference in the

72. Difference Between Means
He hopes to disprove H0 and thereby to prove Ha

73. Paired t-Test
PROJECT QUESTION
Step 1:

74. Paired t-Test
PROJECT QUESTION
Step 1: Set the α-level

75. Step 1: Set the α-level = 0.05 Step 2:
Paired t-Test
PROJECT QUESTION
Step 1: Set the α-level = 0.05 Step 2:

76. Paired t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Set the practically-significant difference

77. Paired t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: Set the practically-significant difference = Mr Gosset says: 2 rating points Step 3:

78. Paired t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: practically-significant difference = 2 rating points Step 3: Set Ha:

79. Paired t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: practically-significant difference: 2 rating points Step 3: Set Ha (they are different) Ha: μF5_10 ≠ μN25_5 or: Ha: μF5_10 - μN25_5 ≠ 0

80. Paired t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: practically-significant difference = 2 rating points Step 3: Ha: μF5_10 ≠ μN25_5 Step 4: Set H0:

81. Paired t-Test
PROJECT QUESTION
Step 1: α-level = 0.05 Step 2: practically-significant difference = 2 rating points Step 3: Ha: μF5_10 ≠ μN25_5 Step 4: Set H0 (they are the same) H0: μF5_10 = μN25_5 or: H0: μF5_10 - μN25_5 = 0

82. Paired t-Test
PROJECT QUESTION
Because each taster tastes each of the two batches, the data are paired

83. use: Data/Data Analysis/ t-Test: Paired Two Sample for Means
Paired t-Test
PROJECT QUESTION
use: Data/Data Analysis/ t-Test: Paired Two Sample for Means

84. Paired t-Test
PROJECT QUESTION

85. Paired t-Test
PROJECT QUESTION
POOF! OK… now what?

86. Paired t-Test
PROJECT QUESTION
If you had any reason to believe that one batch would be better than the other you would use: P(T<=t) one tail

87. We didn’t, so you would use: P(T<=t) two tail
Paired t-Test
PROJECT QUESTION
We didn’t, so you would use: P(T<=t) two tail

88. The probability is .03 Do you reject H0?
Paired t-Test
PROJECT QUESTION
The probability is .03 Do you reject H0?

89. Difference Between Means
How all the inferential tests work:

90. Difference Between Means
How all the inferential tests work: Excel calculates a probability that you would get the data you got if the null hypothesis were true

91. Difference Between Means
How all the inferential tests work: Excel calculates a probability that you would get the data you got if the null hypothesis were true If it’s ≤ α-level, reject H0

92. The probability is .03 Do you reject H0?
Paired t-Test
PROJECT QUESTION
The probability is .03 Do you reject H0?

93. The probability is .03 Do you reject H0? Yup!
Paired t-Test
PROJECT QUESTION
The probability is .03 Do you reject H0? Yup!

94. Difference Between Means
CELEBRATE!

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

96. Paired t-Test
PROJECT QUESTION
Reject H0 ! Conclusion?

97. Reject H0 ! Conclusion: The batches do taste different!
Paired t-Test
PROJECT QUESTION
Reject H0 ! Conclusion: The batches do taste different!

98. Do we need a Multiple Comparison Graph?
Paired t-Test
PROJECT QUESTION
Do we need a Multiple Comparison Graph?

99. Paired t-Test
PROJECT QUESTION
Which tastes better?

100. Questions?

101. Paired t-Test
PROJECT QUESTION
For tests of differences between means, what would happen if you had a smaller sample size?

102. Paired t-Test
PROJECT QUESTION
What is the minimum sample size you would need for this dataset to provide a significant result?

103. t-Tests
Try the problems on the following sheets!

104. Questions?