So far. . .

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One-Way ANOVA

• ANOVA Analysis of Variance
• This is a technique used to analyze the results
  of an experiment when you have more than two
  groups

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Example

• You measure the number of days 7 psychology
  majors, 7 sociology majors, and 7 biology majors
  are absent from class
• You wonder if the average number of days each of
  these three groups was absent is significantly
  different from one another

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Results
X 3.00
X 2.00
X 1.00
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Hypothesis

• Alternative hypothesis (H1)
• H1 The three population means are not all equal

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Hypothesis

• Null hypothesis (H0)
• ?psych ?socio ?bio

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Between and Within Group Variability

• Two types of variability
• Between
• the differences between the mean scores of the
  three groups
• The more different these means are, the more
  variability!

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Results
X 3.00
X 2.00
X 1.00
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Between Variability
S2 .66
X 3.00
X 2.00
X 1.00
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Between Group Variability

• What causes this variability to increase?
• 1) Effect of the variable (college major)
• 2) Sampling error

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Between and Within Group Variability

• Two types of variability
• Within
• the variability of the scores within each group

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Results
X 3.00
X 2.00
X 1.00
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Within Variability
S2 .57
S2 1.43
S2 .57
X 3.00
X 2.00
X 1.00
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Within Group Variability

• What causes this variability to increase?
• 1) Sampling error

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Between and Within Group Variability

• Between-group variability
• Within-group variability

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Between and Within Group Variability

• sampling error effect of variable
• sampling error

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Between and Within Group Variability

• sampling error effect of variable
• sampling error
• Thus, if null hypothesis was true this would
  result in a value of 1.00

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Between and Within Group Variability

• sampling error effect of variable
• sampling error
• Thus, if null hypothesis was not true this value
  would be greater than 1.00

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Calculating this Variance Ratio
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Calculating this Variance Ratio
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Calculating this Variance Ratio
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Degrees of Freedom

• dfbetween
• dfwithin
• dftotal
• dftotal dfbetween dfwithin

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Degrees of Freedom

• dfbetween k - 1 (k number of groups)
• dfwithin N - k (N total number of
  observations)
• dftotal N - 1
• dftotal dfbetween dfwithin

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Degrees of Freedom

• dfbetween k - 1 3 - 1 2
• dfwithin N - k 21 - 3 18
• dftotal N - 1 21 - 1 20
• 20 2 18

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Sum of Squares

• SSBetween
• SSWithin
• SStotal
• SStotal SSBetween SSWithin

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Sum of Squares

• SStotal

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Sum of Squares

• SSWithin

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Sum of Squares

• SSBetween

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Sum of Squares

• Ingredients
• ?X
• ?X2
• ?Tj2
• N
• n

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To Calculate the SS
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?X
?Xs 21
?Xp 14
?XB 7
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?X
?X 42
?Xs 21
?Xp 14
?XB 7
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?X2
?X 42
?Xs 21
?Xp 14
?XB 7
?X2s 67
?X2P 38
?X2B 11
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?X2
?X 42 ?X2 116
?Xs 21
?Xp 14
?XB 7
?X2s 67
?X2P 38
?X2B 11
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T2 (?X)2 for each group
?X 42 ?X2 116
?Xs 21
?Xp 14
?XB 7
?X2s 67
?X2P 38
?X2B 11
T2P 196
T2B 49
T2s 441
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?Tj2
?X 42 ?X2 116 ?Tj2 686
?Xs 21
?Xp 14
?XB 7
?X2s 67
?X2P 38
?X2B 11
T2P 196
T2B 49
T2s 441
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N
?X 42 ?X2 116 ?Tj2 686 N 21
?Xs 21
?Xp 14
?XB 7
?X2s 67
?X2P 38
?X2B 11
T2P 196
T2B 49
T2s 441
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n
?X 42 ?X2 116 ?Tj2 686 N 21 n 7
?Xs 21
?Xp 14
?XB 7
?X2s 67
?X2P 38
?X2B 11
T2P 196
T2B 49
T2s 441
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Ingredients
?X 42 ?X2 116 ?Tj2 686 N 21 n 7
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Calculate SS
?X 42 ?X2 116 ?Tj2 686 N 21 n 7

• SStotal

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Calculate SS
?X 42 ?X2 116 ?Tj2 686 N 21 n 7

• SStotal

42
116
32
21
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Calculate SS
?X 42 ?X2 116 ?Tj2 686 N 21 n 7

• SSWithin

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Calculate SS
?X 42 ?X2 116 ?Tj2 686 N 21 n 7

• SSWithin

686
18
116
7
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Calculate SS
?X 42 ?X2 116 ?Tj2 686 N 21 n 7

• SSBetween

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Calculate SS
?X 42 ?X2 116 ?Tj2 686 N 21 n 7

• SSBetween

14
686
42
7
21
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Sum of Squares

• SSBetween
• SSWithin
• SStotal
• SStotal SSBetween SSWithin

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Sum of Squares

• SSBetween 14
• SSWithin 18
• SStotal 32
• 32 14 18

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Calculating the F value
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Calculating the F value
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Calculating the F value
14
7
2
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Calculating the F value
7
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Calculating the F value
7
18
1
18
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Calculating the F value
7
7
1
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How to write it out
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Significance

• Is an F value of 7.0 significant at the .05
  level?
• To find out you need to know both df

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Degrees of Freedom

• Dfbetween k - 1 (k number of groups)
• dfwithin N - k (N total number of
  observations)

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Degrees of Freedom

• Dfbetween k - 1 3 - 1 2
• dfwithin N - k 21 - 3 18
• Page 390 Table F
• Dfbetween are in the numerator
• Dfwithin are in the denominator
• Write this in the table

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Critical F Value

• F(2,18) 3.55
• The nice thing about the F distribution is that
  everything is a one-tailed test

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Decision

• Thus, if F value gt than F critical
• Reject H0, and accept H1
• If F value lt or to F critical
• Fail to reject H0

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Current Example

• F value 7.00
• F critical 3.55
• Thus, reject H0, and accept H1

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• Alternative hypothesis (H1)
• H1 The three population means are not all equal
• In other words, psychology, sociology, and
  biology majors do not have equal class attendence
• Notice It does not say where this difference is
  at!!

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How to write it out
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