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Psychology 406
Assignment \#9
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This assignment is based on the data given in Assignment \#8 and
requires the results and information from \#8. Treat variable 1 as
the dependent variable Y and variable 2 as the independent variable
X.
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\emph{Correlational Model}:
Suppose the pairs of values given by X and Y represent independent
observations from a bivariate normal distribution.
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a) Test $\mathrm{H}_{0}: \rho_{xy} = 0$ versus $\mathrm{H}_{1}:
\rho_{xy} \ne 0$ using the t-statistic.
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b) Test $\mathrm{H}_{0}: \rho_{xy} = .8$ versus $\mathrm{H}_{1}:
\rho_{xy} \ne .8$ using Fisher's Z-transformation.
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c) Using Fisher's Z-transformation, find a 95\% confidence interval
for $\rho_{xy}$.
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\emph{Regression Model}:
Suppose the data on X and Y are assumed to follow the usual linear
regression model with the X variable fixed.
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d) Construct a 95\% confidence interval on $\beta_{1}$ and test the
hypothesis that $\mathrm{H}_{0}: \beta_{1} = 0$ versus
$\mathrm{H}_{1}: \beta_{1} \ne 0$ at $\alpha$ = .05. What is the
relation of this latter test to that done in (a).
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e) Construct a 95\% confidence interval on $\beta_{0}$.
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f) Construct a 95\% confidence interval on the ``true mean'' for Y
given that X = 33.
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g) Construct a 95\% prediction interval for a new observation with
an X value of 33.
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h) Partition the Total Sum of Squares for Y into the Regression and
Error Sum of Squares. Construct the analysis-of-variance table and
carry out the appropriate F-test for testing $\mathrm{H}_{0}:
\beta_{1} = 0$ versus $\mathrm{H}_{1}: \beta_{1} \ne 0$. What is
the relation of the latter to what was done in (a) and (d)?
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i) Partition the Error Sum of Squares into the Sum of Squares for
Pure Error and for lack of fit. Construct the analysis-of-variance
table for assessing lack of fit and carry out the appropriate
F-test.
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