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

Directions: {\bf \Large Work only on this sheet} (on both sides, if
needed); do not turn in any supplementary sheets of paper. There is
actually plenty of room for your answers, as long as you organize
yourself BEFORE starting writing.

{\bf \Large Unless otherwise stated, give numerical answers as
expressions, e.g. $\frac{2}{3} \times 6 - 1.8$.  Do NOT use
calculators.}

{\bf 1.} (35) Consider the lottery ticket example on pp.147ff. Suppose
500 tickets are sold, and you have data on 8 of them. Continue to assume
sampling with replacement.  Fill in the blank:  The probability that the
MLE is exactly equal to the true value of c is 1 -
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.

{\bf 2.} In an analysis published on the Web (Sparks {\it et al},
Disease Progress over Time, {\it The Plant Health Instructor}, 2008,
the following R output is presented:

\begin{Verbatim}[fontsize=\relsize{-2}]
> severity.lm <- lm(diseasesev~temperature,data=severity)
> summary(severity.lm)
Coefficients:
        Estimate Std. Error t value Pr(>|t|)   
(Intercept)  2.66233    1.10082   2.418  0.04195 * 
temperature  0.24168    0.06346   3.808  0.00518 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
\end{Verbatim}

Fill in the blanks:

\begin{itemize}

\item [(a)] (15) The model here is 

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ =
$\beta_0$ + $\beta_1$ 
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\item [(b)] (15) The two null hypotheses being tested here are
$H_0:$ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ and
$H_0:$ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_. 



\end{itemize}

{\bf 3.} Let X denote the number we obtain when we roll a single die
once.  Let $G_X(s)$ denote the generating function of X.

\begin{itemize}

\item [(a)] (20) Find $G_X(s)$.  

\item [(b)] (15) Suppose we roll the die 5 times, and let T denote the
total number of dots we get from the 5 rolls.  Find $G_T(s)$.

\end{itemize}

{\bf Solutions:}

{\bf 1.}

\begin{equation}
1 - \left ( \frac{499}{500} \right )^8
\end{equation}

{\bf 2a.} mean, diseasesev, temperature (the word {\it mean} is crucial)

{\bf 2b.} 
$H_0: \beta_0 = 0$, $H_0: \beta_1 = 0$

{\bf 3a.}

\begin{equation}
G_X(s) = \sum_{i} s^i P(X = i) = \frac{1}{6} \cdot (s+s^2+s^3+s^4+s^5+s^6)
\end{equation}

{\bf 3b.}  Write $T = X_1 +...+ X_5$.  Then

\begin{eqnarray}
G_T(s) &=& G_{X_1+...+X_5}(s) \\ 
&=& G_{X_1}(s) ... G_{X_5}(s) \\
&=& [G_X(s)]^5 \\
&=& 
\left [ \frac{1}{6} \cdot (s+s^2+s^3+s^4+s^5+s^6) \right ]^5 
\end{eqnarray}

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