2. Practice Does drinking milkshakes affect (alpha = .05) your weight?
To see if milkshakes affect a persons weight you collected data from 5 sets of twins. You randomly had one twin drink water and the other twin drank milkshakes. After 3 months you weighed them.

3. Results Water Twin A 186 Twin B 200 Twin C 190 Twin D 162 Twin E 175
Milkshakes
195
202
196
165
183

4. Hypothesis Two-tailed
Alternative hypothesis
H1: water = milkshake
Null hypothesis
H0: water = milkshake

5. Step 2: Calculate the Critical t
N = Number of pairs
df = N - 1
5 - 1 = 4
 = .05
t critical = 2.776

6. Step 3: Draw Critical Region
tcrit =
tcrit = 2.776

7. Step 4: Calculate t observed
tobs = (X - Y) / SD

8. (D) -9 -2 -6 -3 -8
D = -28
D2 =194
N = 6
-28
3.04 =
194 5
5 - 1

9. Step 4: Calculate t observed
tobs = (X - Y) / SD
1.36=3.04 / 5
N = number of pairs

10. Step 4: Calculate t observed
-4.11 = (182.6 – 188.2) / 1.36
X = 182.6
Y = 188.2
SD = 1.36

11. Step 5: See if tobs falls in the critical region
tcrit =
tcrit = 2.776
tobs = -4.11

12. 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

13. Step 7: Put answer into words
Reject H0, and accept H1
Milkshakes significantly ( = .05) affect a persons weight.

15. Six Easy Steps for an ANOVA
1) State the hypothesis
2) Find the F-critical value
3) Calculate the F-value
4) Decision
5) Create the summary table
6) Put answer into words

16. Example
Want to examine the effects of feedback on self-esteem. Three different conditions -- each have five subjects
1) Positive feedback
2) Negative feedback
3) Control
Afterward all complete a measure of self-esteem that can range from 0 to 10.

17. Example:
Question: Is the type of feedback a person receives significantly (.05) related their self-esteem?

18. Results

19. Step 1: State the Hypothesis
H1: The three population means are not all equal
H0: pos = neg = cont

20. Step 2: Find F-Critical Step 2.1
Need to first find dfbetween and dfwithin
dfbetween = k (k = number of groups)
dfwithin = N - k (N = total number of observations)
dftotal = N - 1
Check yourself
dftotal = dfbetween + dfwithin

21. Step 2: Find F-Critical Step 2.1
Need to first find dfbetween and dfwithin
dfbetween = (k = number of groups)
dfwithin = (N = total number of observations)
dftotal = 14
Check yourself
14 =

22. Step 2: Find F-Critical Step 2.2 Look up F-critical using table F

23. Step 2: Find F-Critical Step 2.2 Look up F-critical using table F

24. Step 3: Calculate the F-value
Has 4 Sub-Steps
3.1) Calculate the needed ingredients
3.2) Calculate the SS
3.3) Calculate the MS
3.4) Calculate the F-value

25. Step 3.1: Ingredients X
X2
Tj2 N n

26. Step 3.1: Ingredients

27. X
X = 85
Xp = 40
Xn = 25
Xc = 20

28. X2 X = 85 X2 = 555 Xp = 40 Xn = 25 Xc = 20 X2n = 135 X2c = 90

29. T2 = (X)2 for each group X = 85 X2 = 555 Xp = 40 Xn = 25 Xc = 20
T2n = 625
T2c = 400
T2p = 1600

30. Tj2 X = 85 X2 = 555 Tj2 = 2625 Xp = 40 Xn = 25 Xc = 20
T2n = 625
T2c = 400
T2p = 1600

31. N X = 85 X2 = 555 Tj2 = 2625 N = 15 Xp = 40 Xn = 25 Xc = 20
T2n = 625
T2c = 400
T2p = 1600

32. n X = 85 X2 = 555 Tj2 = 2625 N = 15 n = 5 Xp = 40 Xn = 25
Xc = 20
X2n = 135
X2c = 90
X2p = 330
T2n = 625
T2c = 400
T2p = 1600

33. Step 3.2: Calculate SS SStotal X = 85 X2 = 555 Tj2 = 2625 N = 15

34. Step 3.2: Calculate SS 85 73.33 555 15 SStotal X = 85 X2 = 555
Tj2 = 2625
N = 15
n = 5
Step 3.2: Calculate SS
SStotal 85
73.33
555 15

35. Step 3.2: Calculate SS SSWithin X = 85 X2 = 555 Tj2 = 2625 N = 15

36. Step 3.2: Calculate SS 2625 30 555 5 SSWithin X = 85 X2 = 555
Tj2 = 2625
N = 15
n = 5
Step 3.2: Calculate SS
SSWithin
2625 30
555 5

37. Step 3.2: Calculate SS SSBetween X = 85 X2 = 555 Tj2 = 2625 N = 15

38. Step 3.2: Calculate SS 43.33 2625 85 5 15 SSBetween X = 85 X2 = 555
Tj2 = 2625
N = 15
n = 5
Step 3.2: Calculate SS
SSBetween
43.33
2625 85 5 15

39. Step 3.2: Calculate SS
Check!
SStotal = SSBetween + SSWithin

40. Step 3.2: Calculate SS
Check!
73.33 =

41. Step 3.3: Calculate MS

42. Step 3.3: Calculate MS
43.33
21.67 2

43. Calculating this Variance Ratio

44. Step 3.3: Calculate MS 30
2.5 12

45. Step 3.4: Calculate the F value

46. Step 3.4: Calculate the F value
21.67
8.67
2.5

47. Step 4: Decision If F value > than F critical
Reject H0, and accept H1
If F value < or = to F critical
Fail to reject H0

48. Step 4: Decision If F value > than F critical
Reject H0, and accept H1
If F value < or = to F critical
Fail to reject H0

49. Step 5: Create the Summary Table

50. Step 6: Put answer into words
Question: Is the type of feedback a person receives significantly (.05) related their self-esteem?
H1: The three population means are not all equal
The type of feedback a person receives is related to their self-esteem

52. Practice
You are interested in comparing the performance of three models of cars. Random samples of five owners of each car were used. These owners were asked how many times their car had undergone major repairs in the last 2 years.

53. Results

54. Practice
Is there a significant (.05) relationship between the model of car and repair records?

55. Step 1: State the Hypothesis
H1: The three population means are not all equal
H0: V = F = G

56. Step 2: Find F-Critical Step 2.1
Need to first find dfbetween and dfwithin
Dfbetween = (k = number of groups)
dfwithin = (N = total number of observations)
dftotal = 14
Check yourself
14 =

57. Step 2: Find F-Critical Step 2.2
Look up F-critical using table F on pages
F (2,12) = 3.88

58. Step 3.1: Ingredients
X = 60
X2 = 304
Tj2 = 1400
N = 15
n = 5

59. Step 3.2: Calculate SS SStotal X = 60 X2 = 304 Tj2 = 1400 N = 15

60. Step 3.2: Calculate SS 60 64 304 15 SStotal X = 60 X2 = 304
Tj2 = 1400
N = 15
n = 5
Step 3.2: Calculate SS
SStotal 60 64
304 15

61. Step 3.2: Calculate SS SSWithin X = 60 X2 = 304 Tj2 = 1400 N = 15

62. Step 3.2: Calculate SS 1400 24 304 5 SSWithin X = 60 X2 = 304
Tj2 = 1400
N = 15
n = 5
Step 3.2: Calculate SS
SSWithin
1400 24
304 5

63. Step 3.2: Calculate SS SSBetween X = 60 X2 = 304 Tj2 = 1400 N = 15

64. Step 3.2: Calculate SS 40 1400 60 5 15 SSBetween X = 60 X2 = 304
Tj2 = 1400
N = 15
n = 5
Step 3.2: Calculate SS
SSBetween 40
1400 60 5 15

65. Step 3.2: Calculate SS
Check!
SStotal = SSBetween + SSWithin

66. Step 3.2: Calculate SS
Check!
64 =

67. Step 3.3: Calculate MS

68. Step 3.3: Calculate MS 40 20 2

69. Calculating this Variance Ratio

70. Step 3.3: Calculate MS 24 2 12

71. Step 3.4: Calculate the F value

72. Step 3.4: Calculate the F value20 10 2

73. Step 4: Decision If F value > than F critical
Reject H0, and accept H1
If F value < or = to F critical
Fail to reject H0

74. Step 4: Decision If F value > than F critical
Reject H0, and accept H1
If F value < or = to F critical
Fail to reject H0

75. Step 5: Create the Summary Table

76. Step 6: Put answer into words
Question: Is there a significant (.05) relationship between the model of car and repair records?
H1: The three population means are not all equal
There is a significant relationship between the type of car a person drives and how often the car is repaired

79. Conceptual Understanding
Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05.

80. Conceptual Understanding
Fcrit = 3.18
Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05.
Fcrit (2, 57) = 3.15

82. Conceptual Understanding
Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?

83. Conceptual Understanding
Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?
F ratio will approach 1.00 when the null hypothesis is true
F ratio will be greater than 1.00 when the null hypothesis is not true

84. Conceptual Understanding
Without computing the SS within, what must its value be? Why?

85. Conceptual Understanding
The SS within is 0. All the scores within a group are the same (i.e., there is NO variability within groups)

87. Practice
You are interested in comparing the taste preference of four sodas. Subjects drank one of four sodas and asked to rank it on a 1-10 scale of how much they liked the taste.

88. Results

89. Practice
Is there a significant (.01) difference in how much these sodas were liked?

90. Step 1: State the Hypothesis
H0: P = C = R =M
H1: The four population means are not all equal

91. Step 2: Find F-Critical Step 2.1
Need to first find dfbetween and dfwithin
Dfbetween = (k = number of groups)
dfwithin = (N = total number of observations)
dftotal = 15
Check yourself
15 =

92. Step 2: Find F-Critical Step 2.2
Look up F-critical using table F on pages
F (3,12) = 5.95

93. Step 3.1: Ingredients
X = 121
X2 = 937
Tj2 = 3671
N = 16
n = 4

94. Step 3.2: Calculate SS SStotal X = 121 X2 = 937 Tj2 = 3671 N = 16

95. Step 3.2: Calculate SS 121 21.94 937 16 SStotal X = 121 X2 = 937
Tj2 = 3671
N = 16
n = 4
Step 3.2: Calculate SS
SStotal
121
21.94
937 16

96. Step 3.2: Calculate SS SSWithin X = 121 X2 = 937 Tj2 = 3671 N = 16

97. Step 3.2: Calculate SS 3671 19.25 937 4 SSWithin X = 121 X2 = 937
Tj2 = 3671
N = 16
n = 4
Step 3.2: Calculate SS
SSWithin
3671
19.25
937 4

98. Step 3.2: Calculate SS SSBetween X = 121 X2 = 937 Tj2 = 3671 N = 16

99. Step 3.2: Calculate SS 2.69 3671 121 4 16 SSBetween X = 121 X2 = 937
Tj2 = 3671
N = 16
n = 4
Step 3.2: Calculate SS
SSBetween
2.69
3671
121 4 16

100. Step 3.2: Calculate SS
Check!
SStotal = SSBetween + SSWithin

101. Step 3.2: Calculate SS
Check!
21.94 =

102. Step 3.3: Calculate MS

103. Step 3.3: Calculate MS
2.69
.90 3

104. Calculating this Variance Ratio

105. Step 3.3: Calculate MS
19.25
1.60 12

106. Step 3.4: Calculate the F value

107. Step 3.4: Calculate the F value
.90
.56
1.60

108. Step 4: Decision If F value > than F critical
Reject H0, and accept H1
If F value < or = to F critical
Fail to reject H0

109. Step 4: Decision If F value > than F critical
Reject H0, and accept H1
If F value < or = to F critical
Fail to reject H0

110. Step 5: Create the Summary Table

111. Step 6: Put answer into words
Question: Is there a significant (.01) difference in how much these sodas were liked?
H0: P = C = R =M
There was not a significant difference in how much these sodas were liked. On average, they were all liked equally as well.

112. Practice
Odometers measure automobile mileage. Suppose 12 cars drove exactly 10 miles and the following mileage figures were recorded. Determine if, on average, the odometers were accurate (Alpha = .05).
9.8, 10.1, 10.3, 10.2, 9.9, 10.4, 10.0, 9.9, 10.3, 10.0, 10.1, 10.2

113. One-sample t-test H1 = Mean not equal to 10 H0 = Mean = 10
t critical (11) = 2.201
t obs = 1.86
The odometers did not record a siginficantly different distance than what was driven.

114. Practice
A research study was conducted to examine the differences between older and younger adults on perceived life satisfaction. A pilot study was conducted to examine this hypothesis. Ten older adults (over the age of 70) and ten younger adults (between 20 and 30) were given a life satisfaction test with high scores indicative of high life satisfaction. Determine if age has an affect (.05) on life satisfaction.
Older Adults
Younger Adults
Mean = 44.5
Mean = 28.1
S = 8.68
S = 8.54

115. tobs = 4.257
tcrit = 2.101
Age does seem to “effect” life satisfaction.

116. Review Compare one sample to another value One sample t-test
Compare two independent samples to each other
Two sample independent t-test (equal or unequal n)
Compare two dependent samples to each other
Two sample dependent t-test
Compare two or more independent samples to each other
One-way ANOVA