A clinical psychologist is interested in comparing the effectiveness of short term relaxation and cognitive-behavioral therapy in treating mild depression. An experiment is conducted in which 15 patients with mild depression are randomly selected and assigned 5 each to a relaxation therapy group, a cognitive/behavioral therapy group, and an attention placebo group. Therapy is administered until the patient is judged no longer depressed or until 10 treatment sessions have elapsed. The following data is obtained. Scores are the number of sessions for each patient. SS _between = _________. A) 27.53 B) 26.13 C) 23.67 D) 28.60

Question

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To find SSB, we first need to calculate the grand mean, which is the mean of all the scores.

Grand mean (X̄) = (5+6+7+8+8+8+9+9+9+10+1+2+2+2+3)/15 = 5.6

Next, we calculate the sum of squares within groups (SSW) for each group using the formula:

SSW = Σ(Xi - X̄i)^2

where Xi is the score of the ith observation in the group and X̄i is the mean of the ith group.

Relaxation group:

X̄1 = (5+6+7+8+8)/5 = 6.8

SSW1 = (5-6.8)^2 + (6-6.8)^2 + (7-6.8)^2 + (8-6.8)^2 + (8-6.8)^2 = 10.4

Cognitive/Behavioral group:

X̄2 = (8+9+9+9+10)/5 = 9

SSW2 = (8-9)^2 + (9-9)^2 + (9-9)^2 + (9-9)^2 + (10-9)^2 = 2

Attention placebo group:

X̄3 = (1+2+2+2+3)/5 = 2

SSW3 = (1-2)^2 + (2-2)^2 + (2-2)^2 + (2-2)^2 + (3-2)^2 = 4

Total sum of squares within groups:

SSW = SSW1 + SSW2 + SSW3 = 16.4

Now, we can calculate the sum of squares between groups (SSB) using the formula:

SSB = Σni(X̄i - X̄)^2

where ni is the sample size of the ith group and X̄ is the grand mean.

SSB = 5(6.8-5.6)^2 + 5(9-5.6)^2 + 5(2-5.6)^2 = 52.8

Finally, we can calculate the mean square between (MSB) and the mean square within (MSW) to perform an F-test:

MSB = SSB / dfb

dfb = k - 1 = 3 - 1 = 2 (where k is the number of groups)

MSB = 52.8 / 2 = 26.4

MSW = SSW / dfw

dfw = N - k = 15 - 3 = 12 (where N is the total sample size)

MSW = 16.4 / 12 = 1.367

F = MSB / MSW = 26.4 / 1.367 = 19.3

Using an F-table with dfb = 2 and dfw = 12 and a significance level of 0.05, the critical F-value is 3.89. Since our calculated F-value (19.3) is greater than the critical F-value (3.89), we can reject the null hypothesis and conclude that there is a significant difference between the effectiveness of the three therapies.

Therefore, the answer is B) 26.13.

A Clinical Psychologist Is Interested in Comparing the Effectiveness of Short