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Psychology 406
Assignment \#11
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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.
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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.
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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.
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The data that resulted from this experiment are as follows, where Y
= time to completion, $\mathrm{X}_{1}$ = group size,
$\mathrm{X}_{2}$ = difficulty:
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\begin{tabular}{ccc}
Y & $\mathrm{X}_{1}$ & $\mathrm{X}_{2}$ \\
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 \\
\end{tabular}
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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):
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$\sum \mathrm{Y}_{i}$ = 684;
$\sum \mathrm{Y}_{i}^{2}$ = 16058;
$\sum \mathrm{X}_{i1}$ = 105;
$\sum \mathrm{X}_{i1}^{2}$ = 455;
$\sum \mathrm{X}_{i2}$ = 90;
$\sum \mathrm{X}_{i2}^{2}$ = 330;
$\sum \mathrm{Y}_{i} \mathrm{X}_{i1}$ = 2243;
$\sum \mathrm{Y}_{i} \mathrm{X}_{i2}$ = 2090;
$\sum \mathrm{X}_{i1} \mathrm{X}_{i2}$ = 315;
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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.
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a) Find $\mathbf{b}$ and the necessary intermediate results, i.e.,
$(\mathbf{X}'\mathbf{X})^{-1}$ and $\mathbf{X}'\mathbf{Y}$.
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b) Construct the ANOVA table and test $\mathrm{H}_{0}: \beta_{1} =
0 \ \mathrm{and} \ \beta_{2} = 0$ simultaneously.
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c) Construct 95\% confidence intervals for $\beta_{0}$, $\beta_{1}$,
and $\beta_{2}$, and test the hypotheses that $\mathrm{H}_{0}:
\beta_{0} = 0$; $\mathrm{H}_{0}: \beta_{1} = 0$; $\mathrm{H}_{0}:
\beta_{2} = 0$.
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d) Construct a 95\% confidence interval for the true mean value of Y
when $\mathrm{X}_{1}$ = 4 and $\mathrm{X}_{2}$ = 3.
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e) Construct a 95\% prediction interval on Y for a new observation
when $\mathrm{X}_{1}$ = 4 and $\mathrm{X}_{2}$ = 3.
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f) Obtain the standardized regression coefficients corresponding to
$\beta_{1}$ and $\beta_{2}$.
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g) Find the correlations between Y and $\mathrm{X}_{1}$; Y and
$\mathrm{X}_{2}$; and $\mathrm{X}_{1}$ and $\mathrm{X}_{2}$.
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h) Find the correlations of $\mathrm{X}_{1}$ with
$\mathbf{X}\mathbf{b}$ and of $\mathrm{X}_{2}$ with
$\mathbf{X}\mathbf{b}$. Is there a nice relationship to what you
found in (g)? Any explanation?
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i) Calculate $\mathrm{R}^{2}$ and adjusted $\mathrm{R}^{2}$.
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j) What is the relation between $\mathrm{R}^{2}$ and the sum of the
squared correlations between Y and $\mathrm{X}_{1}$ and between Y
and $\mathrm{X}_{2}$? Any explanation?
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k) If one included a third term in the regression equation of the
form, $\mathrm{X}_{3} = \mathrm{X}_{1}\mathrm{X}_{2}$, what
particular substantive concerns would be addressed?
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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?
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