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\centerline{\Large \bf Final Exam} \vskip6pt
\centerline{\sc Due on US EST Jun 8 Midnight (Taiwan Time: noon Jun 9)} \vskip24pt

{\bf Please finish 3 questions out of all 4 questions}

Please email your solution (scanned handwritten solution or typed solution) to my email address with subject "Final exam submission of NCKU course"
\begin{enumerate}
%\item KNNL Problem 2.53

\item An agriculture study tries to find the relationship between crop yield and the usage of fertilizer. Suppose it is already scientifically justified that
\begin{equation}\label{eq1}
\mbox{The crop yield in \bf{one-acre} farm}  = \beta_0+\beta_1\cdot (\mbox{concentration level(\%) of fertilizer}) + \epsilon,
\end{equation}
where $\epsilon$ is normally distributed with a constant variance $\sigma^2$.

Five farms is used for this experiment to estimate the regression coefficients, and data table is:
\begin{center}
  \begin{tabular}[h]{|c|c|c|c|c|c|}\hline
   Farm size (acre)& 2 & 3 & 5 & 1 & 4\\\hline
   x-concentration level (\%)  & 1.0 & 1.25 & 1.50 & 1.75 & 2.00\\\hline
   y-total crop yield in this farm (ton) &7&10&17&3&18\\\hline
  \end{tabular}
\end{center}
\begin{itemize}
\item[a.]Given fact (\ref{eq1}), what is the distribution of crop yield $y$ of a farm with a size of $a$ acres, given $x$ level concentration of fertilizer? For simplicity, we assume independence among the crop yield of each acre area in this farm.

\vskip 2in
\item[b.]Since the ordinary square estimation is not optimal in this case, please describe how to estimate the parameter $\beta_0$ and $\beta_1$ via {\bf weighted least square}. (No actual calculation is required).
\end{itemize}

\newpage
\item An experiment was conducted to assess the yield of a manufacturing process in a chemical factory. 
Seven 2-level factors (denoted by $A$, $B$, $\dots$, $G$) are considered in the experiment. It is known that factorial effects of order 3 or higher
are negligible. The research decide to design a $2^{7-3}$ fractional factorial, defined by $E=ABC$, $F=BCD$, $G=ABCD$.

% Question with parts
\begin{itemize}
 \item How many runs (observations) is need for this experiment? Can we have observation for
 $(A,B,C,D,E,F,G)=(-, +,+,-,-,+,-)$?
 \vfill

 \item What is the complete defining relation of this design, and what is the
  wordlength pattern and corresponding resolution of this experiment?
 \vfill
\newpage

 \item Please find out the {\bf non-negligible}  factorial effects that is aliased with main effect G, and 
 {\bf non-negligible}  factorial effects that is aliased with interaction effect DF.
 \vfill
 
 \item If the estimated main effect for G is 20, how would you interpret this estimation value?
 \vfill
 
\end{itemize}

\newpage

\item  Let $\beta_1$, $\beta_2$, $\beta_3$ and $\beta_4$ be the interior angles of a convex quadrilateral, so that $\sum\beta_i=360$ degrees. Suppose we have available observations $Y_i$, $i=1,2,3,4$ such that $Y_i\sim N(\beta_i; \sigma^2)$ independently, with unknown $\sigma^2$. Can you think of an
   F-test to test the null hypothesis that the quadrilateral is a rectangle?\\
Hint 1: view $\beta_4$ as $360-\beta_1-\beta_2-\beta_3$ rather than a new parameter. Hint 2: consider the general linear F test, i.e., full model vs reduced model.

\newpage

\item Let $X_1$ be a random sample from pdf
$f(x|\theta)=2x/\theta^2$, $x\in(0,\theta)$.
Construct a $1-\alpha$ confidence interval for $\theta$ based on $X_1$.\\
HINT: $\theta$ is a scale parameter, and $X_1/\theta$ has a fixed distribution.

\vfill

\end{enumerate}

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