Question

A social psychologist is interested in how optimism is related to life satisfaction. A sample of individuals categorized as optimistic were asked about past, present, and projected future satisfaction with their lives. Higher scores on the life satisfaction measure indicate more satisfaction. Below are the data. What can the psychologist conclude with α = 0.05?

past	present	future
22 24 27 26 28	27 28 29 30 30	21 27 26 28 29

a) What is the appropriate test statistic?
(na, one-way ANOVA ,within-subjects ANOVA, two-way ANOVA)  

b) Compute the appropriate test statistic(s) to make a decision about H₀.
critical value =   ; test statistic =   
Decision: ( Reject H0 or Fail to reject H0  )

c) Compute the corresponding effect size(s) and indicate magnitude(s).
η² =   ;  -(na, trivial effect ,small effect, medium effect, large effect)  


e) Conduct Tukey's Post Hoc Test for the following comparisons:
1 vs. 2: difference =   
1 vs. 3: difference =   

f) Conduct Scheffe's Post Hoc Test for the following comparisons:
1 vs. 3: test statistic =    
2 vs. 3: test statistic =   

Answer 1

a) appropriate test statistic : one-way ANOVA

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b)

	past	present	future	Total
Sum	127	144	131	402
Count	5	5	5	15
Mean, Sum/n	25.4	28.8	26.2
Sum of square, Ʃ(xᵢ-x̅)²	23.2	6.8	38.8

Number of treatment, k = 3

Total sample Size, N = 15

df(between) = k-1 = 2

df(within) = N-k = 12

df(total) = N-1 = 14

SS(between) = (Sum1)²/n1 + (Sum2)²/n2 + (Sum3)²/n3 - (Grand Sum)²/ N = 31.6

SS(within) = SS1 + SS2 + SS3 = 68.8

SS(total) = SS(between) + SS(within) = 100.4

MS(between) = SS(between)/df(between) = 15.8

MS(within) = SS(within)/df(within) = 5.7333

F = MS(between)/MS(within) = 2.7558

Critical value Fc = F.INV.RT(0.05, 2, 12) = 3.8853

Decision:

Fail to reject the null hypothesis.

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c) Eta, ɳ² = SS(between)/SS(total) = 31.6/100.4 = 0.3147

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d) At α = 0.05, k = 3, N-K = 12, Q value = 3.78

HSD = Q*√(MSW/n) = 3.78*√(5.7333/5) = 4.05

Comparison	Diff. = (xi - xj)	HSD	Results
x̅1-x̅2	-3.4	4.05	Means are not different
x̅1-x̅3	-0.8	4.05	Means are not different

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e) 1 vs 3 test statistic:

F' = (x̅1 - x̅3)²/(MSW*(1/n1 + 1/n3)) = (25.4 - 26.2)²/(5.7333*(1/5 + 1/5)) = 0.279

2 vs 3 test statistic:

F' = (x̅2 - x̅3)²/(MSW*(1/n2 + 1/n3)) = (28.8 - 26.2)²/(5.7333*(1/5 + 1/5)) = 2.948