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Psychology 407
Assignment \#6
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1) To assess the effects of anxiety on test performance, three sets
of instructions to examinees were used: (1) instructions designed to
reduce anxiety, (2) neutral (standard) directions, or (3) directions
designed to produce anxiety. Students were classified by sex, then
randomly assigned to one of three test conditions: $A_{1}$
(anxiety-reducing), $A_{2}$ (neutral), $A_{3}$ (anxiety-producing).
Within each level of factor $A$, 10 boys and 10 girls took a
standardized verbal ability test ($B_{1}$) and 10 boys and 10 girls
took a standardized math ability test ($B_{2}$) previously
calibrated to be of the same difficulty level as the verbal test.
Assume fact $C$ is gender ($C_{1}$ and $C_{2}$ are boys and girls in
some order).
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The resulting means based on the 10 observations per cell are given
below along with the various sums of squares:
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\footnotesize
\begin{flushleft}
\begin{tabular}{cccccccccccc}
$A_{1} B_{1} C_{1}$ & $A_{1} B_{1} C_{2}$ & $A_{1} B_{2} C_{1}$ &
$A_{1} B_{2} C_{2}$& $A_{2} B_{1} C_{1}$ & $A_{2} B_{1} C_{2}$ &
$A_{2} B_{2} C_{1}$ & $A_{2} B_{2} C_{2}$ & $A_{3} B_{1} C_{1}$ &
$A_{3} B_{1} C_{2}$ & $A_{3} B_{2} C_{1}$ & $A_{3} B_{2} C_{2}$ \\
12.7 & 14.0 & 16.0 & 10.5 & 13.0 & 13.2 & 12.6 & 10.2 &
12.1 & 16.3 & 12.2 & 13.2 \\
\end{tabular}
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\begin{tabular}{cc}
& Sum of Squares \\
A & 34.20 \\
B & 36.30 \\
C & 1.20 \\
AB & 15.20 \\
AC & 122.60 \\
BC & 132.30 \\
ABC & 25.80 \\
Error & 2947.50
\end{tabular}
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i) Complete the appropriate ANOVA table and carry out the relevant
significance tests.
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ii) Interpret the results of the 3-factor interaction test in (i)
(whether significant or not) by graphing sets of specific 2-factor
interactions. In how many ways (i.e., sets of specific 2-factor
interactions) could you do this?
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iii) Given the results of (i), explain what is occurring
substantively by graphing appropriate sets of means (e.g., think
specific effects and interactions).
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iv) Indicate schematically how this three-way ANOVA could be
rephrased as a regression model and indicate explicitly what the
design matrix would look like. (You only need to provide
representative rows in the design matrix that would be repeated for
all of the 10 observations in each cell).
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2) An experiment is performed in the twenty elementary schools of a
large school district. Ten of the schools are randomly designated
to be the sites for adoption of an innovative science curriculum,
``Science: A Process Approach (SAPA)''. The SAPA material were
bought and placed in the ten schools; teachers were trained to use
them. The other ten elementary schools continue to use the
district's traditional textbook-based science curriculum. After two
years of study in the respective programs, sixth-grade pupils in all
twenty schools are given the Science test (a 45-item measure of
scientific methods, reasoning, and knowledge) of the Sequential
Tests of Educational Progress (STEP). Each student's score is
expressed as a percentage. There are 50 to 120 sixth-grade pupils
in each school; but since the school itself (along with its
teachers, administrators, surrounding neighborhoods, and the like)
was randomly designated as either SAPA or Traditional (the two
experimental conditions, E versus C), the school is the experimental
unit. The twenty schools' means of sixth-grade pupils STEP-Science
scores will be used as the observational unit in the statistical
analysis. The data collected in the experiment are reproduced below
(denoted as Y) along with average Scholastic Aptitude Scores (X)
obtained prior to the treatment.
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\begin{tabular}{cc|cc}
E Schools & & C Schools & \\ \hline
X & Y & X & Y \\ \hline
105.7 & 77.63\% & 101.2 & 64.10\% \\
100.3 & 74.13 & 97.6 & 43.67 \\
94.3 & 67.20 & 96.4 & 50.40\\
108.7 & 78.23 & 109.6 & 84.33 \\
93.1 & 57.93 & 94.0 & 44.93\\
96.7 & 57.65 & 105.4 & 71.43 \\
106.9 & 83.30 & 102.4 & 71.10 \\
100.3 & 73.90 & 100.6 & 44.57 \\
86.5 & 45.90 & 104.2 & 68.23 \\
96.1 & 64.83 & 112.6 & 68.47 \\ \hline
Means: 98.86 & 68.07 & 102.40 & 61.12 \\
\end{tabular}
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\noindent In terms of Kutner et al.'s notation, you may find the
following quantities helpful.
$\mathrm{SSTO}_{Y}$ = 3266.24
$\mathrm{SSTR}_{Y}$ = 241.30
$\mathrm{SS}_{Y}$ = 3024.94
$\mathrm{SSTO}_{X}$ = 796.48
$\mathrm{SSTR}_{X}$ = 62.66
$\mathrm{SSE}_{X}$ = 733.82
$\mathrm{SPTO}$ = 1145.97
$\mathrm{SPTR}$ = $-$122.91
$\mathrm{SPE}$ = 1268.88
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i) Carry out two separate one-way ANOVAs on X and Y. Interpret.
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ii) Carry out an analysis-of-covariance on Y treating X as the
covariate. Interpret any differences from what you found in (i).
Graph the data (X by Y) and impose the within-group regression lines
of Y on X.
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iii) What are the ``adjusted'' means; interpret these in terms of
the plot in (ii).
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iv) Schematically indicate how the regression approach could be used
to carry out the analysis-of-covariance. Relate the various
adjusted mean squares to what you would obtain in comparing certain
Full and Reduced models.
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v) Discuss how you would evaluate the assumption of equal
within-group slopes using the regression.
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vi) If one were interested in comparing adjusted means
\emph{post-hoc} (even though there are only two here, and, thus, it
is redundant to do so), what standard error of the difference
between the two means would be used? Compute it.
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