\documentclass{article}
\setlength{\oddsidemargin}{0.0in}
 \setlength{\textwidth}{6.5in}
\setlength{\textheight}{8.5in}
 \setlength{\topmargin}{0.0in}


\usepackage{graphicx}

\begin{document}
\Large

\begin{center}

        Psychology 406

        Assignment \#9


\end{center}

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.

\bigskip

\emph{Correlational Model}:

Suppose the pairs of values given by X and Y represent independent
observations from a bivariate normal distribution.

\smallskip

a) Test $\mathrm{H}_{0}: \rho_{xy} = 0$ versus $\mathrm{H}_{1}:
\rho_{xy} \ne 0$ using the t-statistic.

\smallskip

b) Test $\mathrm{H}_{0}: \rho_{xy} = .8$ versus $\mathrm{H}_{1}:
\rho_{xy} \ne .8$ using Fisher's Z-transformation.

\smallskip

c) Using Fisher's Z-transformation, find a 95\% confidence interval
for $\rho_{xy}$.

\bigskip

\emph{Regression Model}:

Suppose the data on X and Y are assumed to follow the usual linear
regression model with the X variable fixed.

\smallskip

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).

\smallskip

e) Construct a 95\% confidence interval on $\beta_{0}$.

\smallskip

f) Construct a 95\% confidence interval on the ``true mean'' for Y
given that X = 33.

\smallskip

g) Construct a 95\% prediction interval for a new observation with
an X value of 33.

\smallskip

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)?

\smallskip

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.




\end{document}
