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Published byLambert Barber Modified over 9 years ago
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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 = 3.00X = 2.00X = 1.00
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Hypothesis Alternative hypothesis (H 1 ) H 1: The three population means are not all equal
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Hypothesis Null hypothesis (H 0 ) 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 = 3.00X = 2.00X = 1.00
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Between Variability X = 3.00X = 2.00X = 1.00 S 2 =.66
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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 = 3.00X = 2.00X = 1.00
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Within Variability X = 3.00X = 2.00X = 1.00 S 2 =.57S 2 =1.43S 2 =.57
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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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Degrees of Freedom df between df within df total df total = df between + df within
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Degrees of Freedom df between = k - 1 (k = number of groups) df within = N - k (N = total number of observations) df total = N - 1 df total = df between + df within
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Degrees of Freedom df between = k - 1 3 - 1 = 2 df within = N - k 21 - 3 = 18 df total = N - 1 21 - 1 = 20 20 = 2 + 18
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Sum of Squares SS Between SS Within SS total SS total = SS Between + SS Within
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Sum of Squares SS total
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Sum of Squares SS Within
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Sum of Squares SS Between
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Sum of Squares Ingredients: X X 2 T j 2 N n
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To Calculate the SS
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XX X s = 21 X p = 14 X B = 7
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XX X s = 21 X p = 14 X B = 7 X = 42
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X2X2 X 2 s = 67 X 2 P = 38 X 2 B = 11 X s = 21 X p = 14 X B = 7 X = 42
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X2X2 X 2 s = 67 X 2 P = 38 X 2 B = 11 X s = 21 X p = 14 X B = 7 X = 42 X 2 = 116
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T 2 = ( X) 2 for each group X 2 s = 67 X 2 P = 38 X 2 B = 11 X s = 21 X p = 14 X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49 X = 42 X 2 = 116
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Tj2Tj2 X 2 s = 67 X 2 P = 38 X 2 B = 11 X s = 21 X p = 14 X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49 X = 42 X 2 = 116 T j 2 = 686
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N X 2 s = 67 X 2 P = 38 X 2 B = 11 X s = 21 X p = 14 X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49 X = 42 X 2 = 116 T j 2 = 686 N = 21
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n X 2 s = 67 X 2 P = 38 X 2 B = 11 X s = 21 X p = 14 X B = 7 T 2 s = 441 T 2 P = 196T 2 B = 49 X = 42 X 2 = 116 T j 2 = 686 N = 21 n = 7
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Ingredients X = 42 X 2 = 116 T j 2 = 686 N = 21 n = 7
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Calculate SS X = 42 X 2 = 116 T j 2 = 686 N = 21 n = 7 SS total
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Calculate SS X = 42 X 2 = 116 T j 2 = 686 N = 21 n = 7 SS total 116 42 21 32
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Calculate SS SS Within X = 42 X 2 = 116 T j 2 = 686 N = 21 n = 7
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Calculate SS SS Within X = 42 X 2 = 116 T j 2 = 686 N = 21 n = 7 116 686 7 18
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Calculate SS SS Between X = 42 X 2 = 116 T j 2 = 686 N = 21 n = 7
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Calculate SS SS Between X = 42 X 2 = 116 T j 2 = 686 N = 21 n = 7 686 7 42 21 14
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Sum of Squares SS Between SS Within SS total SS total = SS Between + SS Within
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Sum of Squares SS Between = 14 SS Within = 18 SS total = 32 32 = 14 + 18
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Calculating the F value
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14 2 7
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Calculating the F value 7
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7 18 1
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Calculating the F value 7 1 7
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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 Df between = k - 1 (k = number of groups) df within = N - k (N = total number of observations)
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Degrees of Freedom Df between = k - 1 3 - 1 = 2 df within = N - k 21 - 3 = 18 Page 390 Table F Df between are in the numerator Df within 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 > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0
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Current Example F value = 7.00 F critical = 3.55 Thus, reject H 0, and accept H 1
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Alternative hypothesis (H 1 ) H 1: 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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