Question

      You are a clinical psychologist is interested in comparing the effectiveness of cognitive behavioral therapy, relaxation therapy, and no therapy (control group) for treating mild anxiety. An experiment is conducted in which 15 patients with mild anxiety were randomly assigned 5 each to the three groups. After 4 weeks, participants were given an anxiety questionnaire and their scores were recorded. The following data were analyzed.

	Cognitive Behavioral Therapy	Relaxation Therapy Group	Control Group
	5	6	7
	6	8	10
	8	10	9
	4	9	10
	7	7	9
Mean	Mean = 6	Mean = 8	Mean = 9
Grand Mean = 7.67	SS = 10	SS = 10	SS = 6

What is the F ratio for this problem?

What should the researcher conclude given the data above?

What is the effect size for this data?

Answer 1

treatment	A	B	C
count, ni =	5	5	5
mean , x̅ i =	6.000	8.00	9.000
grand mean , x̅̅ =	Σni*x̅i/Σni =	7.67
square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	2.778	0.111	1.778
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	13.889	0.556	8.889		23.33333333

SSwithin = 10+10+6=26

no. of treatment , k =   3
df between = k-1 =    2
N = Σn =   15
df within = N-k =   12
  
mean square between groups , MSB = SSB/k-1 =    11.6667
  
mean square within groups , MSW = SSW/N-k =    2.1667
  
F-stat = MSB/MSW =    5.3846

anova table
	SS	df	MS	F	p-value	F-critical
Between:	23.3333	2	11.667	5.385	0.0214	3.885
Within:	26.0000	12	2.167
Total:	49.3333	14
					α =	0.05

   Decision:   p-value<α , reject null hypothesis
      
conclusion :    there is enough evidence of significant mean difference among three treatments  

eta square ,effect size = SSbet/SST=   0.4730