2. No class on Wednesday 11/1 No class on Friday 11/3
Remember
No class on Wednesday 11/1
No class on Friday 11/3

3. Example
You just created a “Smart Pill” and you gave it to 150 subjects. Below are the results you found. Do people who take your “Smart Pill” have significantly ( = .05) greater IQ scores than the average IQ population ( = 100)?
X = 103
s = 14.4

4. Step 1: Write out Hypotheses
Alternative hypothesis
H1: sample > 100
Null hypothesis
H0: sample < or = 100

5. Step 2: Calculate the Critical t
N = 150
df = 149
 = .05
tcrit = 1.645

6. Step 3: Draw Critical Region
tcrit = 1.645

7. Step 4: Calculate t observed
tobs = (X - ) / Sx

8. Step 4: Calculate t observed
tobs = (X - ) / Sx
Sx = S / N

9. Step 4: Calculate t observed
tobs = (X - ) / Sx
1.18=14.4 / 150

10. Step 4: Calculate t observed
tobs = (X - ) / Sx
2.54 = ( ) / 1.18
1.18=14.4 / 150

11. Step 5: See if tobs falls in the critical region
tcrit = 1.645

12. Step 5: See if tobs falls in the critical region
tcrit = 1.645
tobs = 2.54

13. Step 6: Decision If tobs falls in the critical region:
Reject H0, and accept H1
If tobs does not fall in the critical region:
Fail to reject H0

14. Step 7: Put answer into words
We reject H0 and accept H1.
The average IQ of the people who took your “Smart Pill” is statistically greater ( = .05) than the average IQ of the population.

15. Practice
You create a program that reduces aggressiveness of children. After completing your program you give 15 children the “Aggressive Test” which has an average score of  = Below are the results you found. Did your program significantly ( = .05) reduce the aggressiveness of children?
X = 14.5
s = 2.77

16. Step 1: Write out Hypotheses
Alternative hypothesis
H1: sample < 15.6
Null hypothesis
H0: sample > or = 15.6

17. Step 2: Calculate the Critical t
N = 15
df = 14
 = .05
tcrit =

18. Step 3: Draw Critical Region
tcrit =

19. Step 4: Calculate t observed
tobs = (X - ) / Sx

20. Step 4: Calculate t observed
tobs = (X - ) / Sx
-1.53 = (14.5 – 15.6) / .72
.72 = 2.77/ 15

21. Step 5: See if tobs falls in the critical region
tcrit =

22. Step 5: See if tobs falls in the critical region
tcrit =
tobs = -1.53

23. Step 6: Decision If tobs falls in the critical region:
Reject H0, and accept H1
If tobs does not fall in the critical region:
Fail to reject H0

24. Step 7: Put answer into words
We fail to reject H0
The average aggressiveness score of the children who took your program is not statistically less then ( = .05) the average aggressiveness of the population.

26. So far. . . We have been doing hypothesis testing with a single sample
We find the mean of a sample and determine if it is statistically different than the mean of a population

27. Basic logic of research\

28. Start with two equivalent groups of subjects

29. Treat them alike except for one thing

30. See if both groups are different at the end

31. Notice
This means that we need to see if two samples are statistically different from each other
We can use the same logic we learned earlier with single sample hypothesis testing

32. Example
You just invented a “magic math pill” that will increase test scores.
You give the pill to 4 subjects and another 4 subjects get no pill
You then examine their final exam grades

33. Hypothesis Two-tailed
Alternative hypothesis
H1: pill = nopill
In other words, the means of the two groups will be significantly different
Null hypothesis
H0: pill = nopill
In other words, the means of the two groups will not be significantly different

34. Hypothesis One-tailed
Alternative hypothesis
H1: pill > nopill
In other words, the pill group will score higher than the no pill group
Null hypothesis
H0: pill < or = nopill
In other words, the pill group will be lower or equal to the no pill group

35. For current example, lets just see if there is a difference
Alternative hypothesis
H1: pill = nopill
In other words, the means of the two groups will be significantly different
Null hypothesis
H0: pill = nopill
In other words, the means of the two groups will not be significantly different

36. Results
Pill Group 5 3 4
No Pill Group 1 2 4 3

37. Remember before. . . Step 2: Calculate the Critical t
df = N -1

38. Now Step 2: Calculate the Critical t
df = N1 + N2 - 2
df = = 6
 = .05
t critical = 2.447

39. Step 3: Draw Critical Region
tcrit =
tcrit = 2.447

40. Remember before. . . Step 4: Calculate t observed
tobs = (X - ) / Sx

41. Now Step 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2

42. Now Step 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2

43. Now Step 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2
X1 = 3.75
X2 = 2.50

44. Now Step 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2

45. Standard Error of a Difference
Sx1 - x2
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 = Sx12 + Sx22

46. Results
Pill Group 5 3 4
No Pill Group 1 2 4 3

47. Standard Deviation
S = -1

48. Standard Deviation Pill Group 5 3 4 No Pill Group 1 2 4 3 X2= 10

49. Standard Deviation Pill Group 5 3 4 No Pill Group 1 2 4 3 X2= 10

50. Standard Deviation Pill Group 5 3 4 No Pill Group 1 2 4 3 X2= 10
Sx= .48
Sx= . 645

51. Standard Error of a Difference
Sx1 - x2
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 = Sx12 + Sx22

52. Standard Error of a Difference
Sx1 - x2
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 = (.48)2 + (.645)2

53. Standard Error of a Difference
Sx1 - x2
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 = (.48)2 + (.645)2
= .80

56. Standard Error of a Difference Raw Score Formula
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 =

57. X1= 15
X12= 59
N1 = 4
X2= 10
X22= 30
N2 = 4
Sx1 - x2 =

58. X1= 15
X12= 59
N1 = 4
X2= 10
X22= 30
N2 = 4
Sx1 - x2 = 10 15

59. Sx1 - x2 = X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 10 15

60. Sx1 - x2 = X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 10 15
4 (4 - 1)

61. Sx1 - x2 = X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 10 15
56.25 30 25 4 4 12

62. .80 = X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 10 15 59
56.25 30 25
7.75 4 4 12

63. Now Step 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2
Sx1 - x2 = .80
X1 = 3.75
X2 = 2.50

64. Now Step 4: Calculate t observed
Sx1 - x2 = .80
X1 = 3.75
X2 = 2.50

65. Now Step 4: Calculate t observed
1.56 = ( ) / .80
Sx1 - x2 = .80
X1 = 3.75
X2 = 2.50

66. Step 5: See if tobs falls in the critical region
tcrit =
tcrit = 2.447

67. Step 5: See if tobs falls in the critical region
tcrit =
tcrit = 2.447
tobs = 1.56

68. Step 6: Decision If tobs falls in the critical region:
Reject H0, and accept H1
If tobs does not fall in the critical region:
Fail to reject H0

69. Step 7: Put answer into words
We fail to reject H0.
The final exam grades of the “pill group” were not statistically different ( = .05) than the final exam grades of the “no pill” group.

72. Practice You wonder if people like Pepsi better than Coke ( = .05)
You give Pepsi to 5 people and Coke to 5 people.
You ask them to rate on a 1 to 5 scale how much they liked their soda.

73. Results
Pepsi 4 5 3
Coke 4 3 2

74. Hypotheses Alternative hypothesis Null hypothesis
H1: Pepsi > Coke
Null hypothesis
H0: Pepsi = or < Coke

75. Step 2: Calculate the Critical t
df = N1 + N2 - 2
df = = 8
 = .05
One-tailed
t critical = 1.860

76. Step 3: Draw Critical Region
tcrit = 1.860

77. Now Step 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2

78. Sx1 - x2 = X1= 21 X12= 91 N1 = 5 X1 = 4.2 X2= 15 X22= 49
5 (5 - 1)

79. .58 = X1= 21 X12= 91 N1 = 5 X1 = 4.2 X2= 15 X22= 49 N2 = 5
.58 = 15 21 91 49 5 5
5 (5 - 1)

80. Step 4: Calculate t observed
2.07 = ( ) / .58
Sx1 - x2 = .58
X1 = 4.2
X2 = 3

81. Step 5: See if tobs falls in the critical region
tcrit =
tobs = 2.07

82. Step 6: Decision If tobs falls in the critical region:
Reject H0, and accept H1
If tobs does not fall in the critical region:
Fail to reject H0

83. Step 7: Put answer into words
We Reject H0, and accept H1
People like Pepsi significantly ( = .05) better than Coke.