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Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

Directions: MAKE SURE TO COPY YOUR ANSWERS TO A SEPARATE SHEET FOR SENDING
ME AN ELECTRONIC COPY LATER.

{\bf 1.} (20) The courtroom analogy used in our text describing the
philosophy underlying significance testing is
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

{\bf 2.} (15) Consider the beta distribution family, Sec.\ 6.6.5.  It
has two parameters, $\alpha$ and $\beta$.  In fitting such a model to
our data, we would come up with estimates of these two parameters,
$\widehat{\alpha}$ and $\widehat{\beta}$.  Fill in the blank with a term
from our course:  The standard deviations of $\widehat{\alpha}$ and
$\widehat{\beta}$ are called their
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

{\bf 3.} (15) Suppose we wish to construct an (approximate) 80\%
confidence interval.  What number should we use instead of 1.96?  Your
answer must consist of an R call.

{\bf 4.} (20) For various distribution families, R provides the functions
`d', `p', `q' and `r'.  Give nonsimulation R code that computes (11.9),
using an appropriate one of these functions.  For full credit, your code
should not use loops.

{\bf 5.}  Let $W$ denote the weight of some kind of item.  Unknown to
us, in the populaiton $f_W(t) = 2(1-t)$ for $t$ in (0,1), 0 elsewhere.

\begin{itemize}

\item [(a)] (15) Find the population value $P(W > 0.2)$.

\item [(b)] (15) We take a random sample $W_1,...,W_{100}$, and calculate
$\overline{W} = \sum_{i=1}^n W_i ~/~ 100$.  Find the exact value of
$Var(\overline{W})$.  An expression of the form $a - b^2$ MUST appear in
your electronic answer.

\end{itemize}

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{\bf Solutions:}

{\bf 1.} innocent until proven guilty

{\bf 2.} standard errors

{\bf 3}

\begin{lstlisting}
qnorm(0.90)
\end{lstlisting}

{\bf 4}

\begin{lstlisting}
1 - pbinom(7,10,0.5)
\end{lstlisting}

{\bf 5.a} 

$$
\int_{0.2}^{1} 2(1-t) ~ dt = 0.64
$$ 

{\bf 5.b} $Var(\overline{W}) = \sigma^2/100$.  So compute $EW = 1/3$ and
$E(W^2)$ = 1/6.  Our answer is then
$(1/6 - (1/3)^2) / 100$.


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