Question

A group of psychologists want to examine whether different types of psychotherapy work better for adolescent depression. One group of n₁ = 4 adolescents were given Cognitive-Behavioral therapy, a second group of n₂ = 4 adolescents were given Psychodynamic therapy, and a third group of n₃ = 4 adolescents were placed on a Waitlist Control condition. At the end of 10-weeks, each participant comnpletes a depression screening (higher scores mean more depressed). The data are below.

Psychotherapy Type
Cognitive-Behavioral	Psychodynamic	Waitlist Control
4	3	6
4	5	7
3	5	4
2	3	7

(a) Calculate the appropriate degrees of freedom and the critical F with α = 0.01. (Use 2 decimals)

F-critical =

(b) Calculate all the parts of an Independent Measures ANOVA to determine whether there are any significant differences among the three treatments. (Use 2 decimal places.)

Source	SS	df	MS	F
Between
Within
Total

(c) Write the two-sentence results as you would write it for the research literature (you do not need to include the table).

Answer 1

count, ni =	4	4	4
mean , x̅ i =	3.250	4.00	6.00
std. dev., si =	0.957	1.155	1.414
sample variances, si^2 =	0.917	1.333	2.000
total sum	13	16	24		53
grand mean , x̅̅ =	Σni*x̅i/Σni =	4.42

square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	1.361	0.174	2.507
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	5.444	0.694	10.028		16.16666667
SS(within ) = SSW = Σ(n-1)s² =	2.750	4.000	6.000		12.7500

no. of treatment , k =   3
df between = k-1 =    2
N = Σn =   12
df within = N-k =   9
  
mean square between groups , MSB = SSB/k-1 =    8.0833
  
mean square within groups , MSW = SSW/N-k =    1.4167
  
F-stat = MSB/MSW =    5.7059
P value =   0.0251

b)

	SS	df	MS	F	p-value	F-critical
Between:	16.17	2	8.08	5.71	0.0251	8.02
Within:	12.75	9	1.42
Total:	28.92	11
					α =	0.01
			conclusion :	p-value>α , do not reject null hypothesis

(a)

df between = k-1 =    2
N = Σn =   12
df within = N-k =   9

F-critical = 8.02

c)

conclusion :    p-value>α , do not reject null hypothesis        

at least one the mean is different.

THANKS

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