An independent-measures research study compares three treatment conditions with a sample of n = 10 in each condition. The sample means are M1 = 2, M2 = 3, and M3 = 7

a. Compute SS for the set of 3 treatment means. (Use the three means as a set of n = 3 scores and compute SS.)
b. Using the result from part a, compute n(SSmeans). Note that this value is equal to SSbetween (see Equation 12.6).
c. Now, compute SSbetween with the computational formula using the T values (Equation 12.7). You should obtain the same result as in part b.

To answer these questions, we need to compute the sum of squares (SS) for the set of treatment means and the between-groups sum of squares (SSbetween). Here's how we can do it:

a. To compute SS for the set of treatment means, we need to first find the grand mean (Mbar) of all the treatment means. The grand mean is obtained by averaging all the treatment means.

Mbar = (M1 + M2 + M3) / 3
= (2 + 3 + 7) / 3
= 12 / 3
= 4

Next, subtract the grand mean (Mbar) from each treatment mean and square the result to get the squared deviations.

For M1:
(M1 - Mbar)^2 = (2 - 4)^2 = (-2)^2 = 4

For M2:
(M2 - Mbar)^2 = (3 - 4)^2 = (-1)^2 = 1

For M3:
(M3 - Mbar)^2 = (7 - 4)^2 = (3)^2 = 9

Now, sum up these squared deviations:

SS = Σ(Mi - Mbar)^2
= 4 + 1 + 9
= 14

Therefore, SS for the set of treatment means is 14.

b. To compute n(SSmeans), we multiply the SS for the set of treatment means by the sample size (n = 10).

n(SSmeans) = 10 * SS
= 10 * 14
= 140

Therefore, n(SSmeans) is 140.

c. The computational formula for SSbetween using T values is given by Equation 12.8 in the textbook:

SSbetween = (ΣT^2) - (ΣT)^2 / N

Where T is the sum of scores in each treatment condition, and N is the overall sample size (N = n * k, where k is the number of treatment conditions).

Using the given values:
ΣT = T1 + T2 + T3 = n1*M1 + n2*M2 + n3*M3
= (10*2) + (10*3) + (10*7)
= 20 + 30 + 70
= 120

ΣT^2 = (T1^2) + (T2^2) + (T3^2) = (n1*M1^2) + (n2*M2^2) + (n3*M3^2)
= (10*2^2) + (10*3^2) + (10*7^2)
= 40 + 90 + 490
= 620

Now, we substitute these values into the formula:

SSbetween = (ΣT^2) - (ΣT)^2 / N
= (620) - (120^2) / (10*3)
= 620 - 14400 / 30
= 620 - 480
= 140

As we can see, the result obtained using the computational formula for SSbetween is the same as the result in part b, confirming its correctness.

Therefore, SSbetween is also 140.