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\begin{flushright}
Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
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\vspace{5mm}

\begin{center}   
{\Large \textbf{STA 441s 2024 Quiz 6}}\\
\vspace{3 mm}
\end{center}

\begin{enumerate} 

\item (2 points) \label{parallel} In your analysis of the Diet Data, you fit a model in which the explanatory variables were Diet, and weight before starting the diet. There were no interactions. You want to know whether, controlling for weight before, there is any difference in the results for Diet~2 and Diet~3. 
    \begin{enumerate}
        \item Fill in the table below. Do not correct for multiple testing.
\begin{center}               
\begin{tabular}{|c|c|c|c|} \hline
$t$, $\chi^2$ or $F$ Statistic  & ~~~~~$p$-value~~~~~  & Reject $H_0$? & Statistically Significant? \\
 (a number)   &      (a number or range)      & (Yes or No)             & (Yes or No)                \\
\hline
  &  & & \\ 
  &  & & \\ \hline
\end{tabular}
\end{center} \vspace{1mm}
On your printout, circle the test statistic and write ``Question~\ref{parallel}" beside it.  
        \item In plain, non-statistical language, what do you conclude? \vspace{40mm}
    \end{enumerate} % End equal slopes part

\item (2 points) \label{interaction} It is quite possible that which diet is more effective (or more exactly, the relative effectiveness of the diets) might depend on the initial weight of the person. You tested this hypothesis. 
    \begin{enumerate} 
        \item Please fill in the table below.
\begin{center}               
\begin{tabular}{|c|c|c|c|} \hline
$t$, $\chi^2$ or $F$ Statistic  & ~~~~~$p$-value~~~~~  & Reject $H_0$? & Statistically Significant? \\
 (a number)   &      (a number or range)      & (Yes or No)             & (Yes or No)                \\
\hline
  &  & & \\ 
  &  & & \\ \hline
\end{tabular}
\end{center} \vspace{1mm}
On your printout, circle the test statistic and write ``Question~\ref{interaction}" beside it.  
        \item In plain, non-statistical language, what do you conclude? \vspace{40mm}
    \end{enumerate} % End test of interaction.

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\begin{tabular}{ccc}
$y = \beta_0 + \beta_1 x_1 + \cdots + \beta_{p-1} x_{p-1} + \epsilon$ 
& $SST=SSR+SSE$ &
$R^2 = \frac{SSR}{SST}$
\\
$a = \frac{R^2_F - R^2_R}{1-R^2_R}$ 
& $F = \left( \frac{n-p}{s} \right) \left( \frac{a}{1-a} \right)$ & 
$a = \frac{sF}{n-p+sF}$
\\
\end{tabular}
\renewcommand{\arraystretch}{1.0}



\item (2 points) Based on your model the Diet Data  with equal slopes (Question~\ref{parallel}), what proportion of the \emph{remaining} variation in weight after 6 weeks is explained by weight before the experiment, once you allow for Diet? Hint: $F=t^2$. Please show some work below. You'll need a calculator. \textbf{Circle your final answer}. \vspace{40mm}

\item (4 points) Hens (female chickens) are randomly assigned to one of three different feed types: $A$, $B$ or $C$. The response variable is the mean weight of the eggs they lay, based on 100 eggs from each chicken. Hen's age is a covariate. Assume that the relationship between age and expected egg weight can be approximated by a straight line over the range of the data.
    \begin{enumerate} 
        \item Write a regression equation that assumes the lines relating age and expected egg weight have the \emph{same slope} for each feed type. Use \emph{cell means coding}. That's indicator dummy variables with no intercept.
\vspace{5mm}
        
 $E(Y|\mathbf{x}) = $       \vspace{10mm}
        

        \item Make a table with three rows, one for each feed type. Make columns showing how your dummy variables are defined. Add another, wider column showing expected egg weight for each feed type. The \emph{symbols} for the dummy variables for feed type will not appear in this last column.  \vspace{50mm}
        \item Controlling for hen's age, is egg weight related to the type of feed?  Give the null hypothesis, using symbols from your regression equation. \vspace{20mm}
    \end{enumerate}


\end{enumerate} % End of all questions



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