Psychology 406 Assignment #11

A social psychologist was interested in problem solving carried out cooperatively by small groups of individuals. The theory upon which these experiments were based suggested that within a particular range of possible group sizes, the relationship between group size and average time to solution for a particular kind of problem would be linear and negative: the larger the group, the less time on the average should it take for the problem to be solved. To check on this theory, the psychologist decided to form a set of experimental groups, ranging in size from groups consisting of 1 individual each, five groups consisting of 2 individuals, five groups each of 3, and so on until there were six different and nonoverlapping sets of five groups each, the last consisting of five groups each of size 6. Each individual subject participated in one and only one group in the study.

Furthermore, five different problems were available for use with these groups. These problems had been scaled in difficulty, from 5 for the most difficult to 1 for the least difficult. To see if scaling correlated with the time it took the groups to solve these problems, and also as a way to reduce error variance, the experimenter decided to give all five problems within each set of groups of the same size, with each group receiving one problem assigned at random. Thus, every combination of problem difficulty and size was represented by exactly one group.

Each experimental group then solved its assigned problem, and the time taken for them to do so was noted; this time to completion was used as the dependent variable.

The data that resulted from this experiment are as follows, where Y = time to completion, X_{1} = group size, X_{2} = difficulty:

Y X_{1} X_{2}
26 1 1
28 1 2
30 1 3
29 1 4
32 1 5
22 2 1
23 2 2
24 2 3
22 2 4
22 2 5
23 3 1
24 3 2
24 3 3
26 3 4
27 3 5
18 4 1
20 4 2
20 4 3
21 4 4
19 4 5
21 5 1
25 5 2
23 5 3
22 5 4
25 5 5
16 6 1
18 6 2
16 6 3
18 6 4
20 6 5

In answering the question to follow, you should find the following summary information helpful: (all summations are assumed to be from i = 1 to 30, where n = 30):

∑ Y_{i} = 684; ∑ Y_{i}^{2} = 16058; ∑ X_{i1} = 105; ∑ X_{i1}^{2} = 455; ∑ X_{i2} = 90; ∑ X_{i2}^{2} = 330; ∑ Y_{i}X_{i1} = 2243; ∑ Y_{i}X_{i2} = 2090; ∑ X_{i1}X_{i2} = 315;

In computing the various expression you will need in answering the questions, make sue you carry a lot of decimal places until you construct the final results.

a) Find b and the necessary intermediate results, i.e., (X′X)^{−1} and X′Y.

b) Construct the ANOVA table and test H_{0} : β_{1} = 0 and β_{2} = 0 simultaneously.

c) Construct 95% confidence intervals for β_{0}, β_{1}, and β_{2}, and test the hypotheses that H_{0} : β_{0} = 0; H_{0} : β_{1} = 0; H_{0} : β_{2} = 0.

d) Construct a 95% confidence interval for the true mean value of Y when X_{1} = 4 and X_{2} = 3.

e) Construct a 95% prediction interval on Y for a new observation when X_{1} = 4 and X_{2} = 3.

f) Obtain the standardized regression coefficients corresponding to β_{1} and β_{2}.

g) Find the correlations between Y and X_{1}; Y and X_{2}; and X_{1} and X_{2}.

h) Find the correlations of X_{1} with Xb and of X_{2} with Xb. Is there a nice relationship to what you found in (g)? Any explanation?

i) Calculate R^{2} and adjusted R^{2}.

j) What is the relation between R^{2} and the sum of the squared correlations between Y and X_{1} and between Y and X_{2}? Any explanation?

k) If one included a third term in the regression equation of the form, X_{3} = X_{1}X_{2}, what particular substantive concerns would be addressed?

l) Would it be possible to find a Sum of Squares for Pure Error given the way the data were collected? If not, how could you redesign the experiment so a Pure Error term could be constructed?