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	24 27 30 28 29

a) What is the appropriate test statistic?
---Select--- 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:  ---Select--- Reject H0 Fail to reject H0

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

d) Make an interpretation based on the results.

At least one of the temporal means is different on life satisfaction.None of the temporal means is different on life satisfaction.    

Answer 1

a) . A sample of individuals categorized as optimistic were asked about past, present, and projected future satisfaction with their lives. Here the individuals are not different among the groups they are same hence this design is called within-subjects ANOVA.

b)

Summary of Data
	Treatments
	1	2	3	Total
N	5	5	5	15
∑X	127	144	138	409
Mean	25.4	28.8	27.6	27.267
∑X2	3249	4154	3830	11233
Std.Dev.	2.4083	1.3038	2.3022	2.4044

ANOVA Table

Source of variation	SS	df	MS	F
Between	29.733	2	14.867	18.979
Within	51.2	12	4.267
Error	6.267	8	0.783
Subjects	44.933	4	11.233
Total	80.933	14

Test statistic F = 18.979

Critical value F_0.05,2,8 = 4.4590

Since test statistic is greater than critical value we reject null hypothesis.

c) Effect size = η² = SS_between/SST = 29.733 / 80.933 = 0.3674

effect size is large.

d) At least one of the temporal means is different on life satisfaction.