\documentclass[twocolumn]{article}

\setlength{\oddsidemargin}{-0.5in}
\setlength{\evensidemargin}{-0.5in}
\setlength{\topmargin}{0.0in}
\setlength{\headheight}{0in}
\setlength{\headsep}{0in}
\setlength{\textwidth}{7.0in}
\setlength{\textheight}{9.5in}
\setlength{\parindent}{0in}
\setlength{\parskip}{0.05in}
\setlength{\columnseprule}{0.3pt}
\usepackage{fancyvrb}
\usepackage{relsize}
\usepackage{hyperref}

\begin{document}

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.} (15) What is the relation of the value of $s^2$ printed out by R's
{\bf var()} function to the value I use?  (Assume neither is 0.)
(i) My value is larger.  (ii) R's value is larger.  (iii) They are
equal.  (iv) The {\bf var()} function has no relation to my $s^2$; they
just have similar names.

{\bf 2.} Consider the R code on p.125. 

\begin{itemize}

\item [(a)] (15) Of all the variables in that code, which one---if
any---corresponds to the ``number of lines in the notebook''?

\item [(b)] (15) Which R expression in that code is a standard error?  And
for which variable in the code is that expression a standard error?

\end{itemize}

{\bf 3.} (5) Fill in the blank. The variable Y on p.131 is an example of
what is generally called a/an
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ variable.

{\bf 4.} (10) Consider Equation (4.16), p.122.  In each of the entries in
the table below, fill in either R for random, or NR for nonrandom:

\begin{tabular}{|r|r|}
\hline
quantity & R or NR? \\ \hline
$\overline{W}$ & \ \\ \hline
$s$ & \ \\ \hline
$\mu$ & \ \\ \hline
$n$ & \ \\ \hline 
\end{tabular}

(That quantity on the left, second line, is $\overline{W}$, ``W-bar,''
not so clear in the table.)

{\bf 5.} (10) Consider $\hat{p}$, the estimator of a population
proportion p, based on a sample of size n.  Give the expression for the
standard error of $\hat{p}$.

{\bf 6.} (15) The term {\bf random sample} means {\it with} replacement.
If it is {\it without} replacement, it is called a {\bf simple random
sample}.  Suppose we take a simple random sample of size 2 from a
population consisting of just three values, 66, 67 and 69.  Let
$\overline{X}$ denote the resulting sample mean.  Find
$p_{\overline{X}}(67.5)$.

{\bf 7.} (15) Suppose we have a random sample $W_1,...,W_n$, and we wish
to estimate the population mean $\mu$, as usual.  But we decide to place
double weight on $W_1$, so our estimator for $\mu$ is

\begin{equation}
U = \frac{2W_1+W_2+...+W_n}{n+1}
\end{equation}

Find E(U) and Var(U) in terms of $\mu$ and the population variance
$\sigma^2$.  Do reasonable algebraic simplification.

{\bf Solutions:}

{\bf 1.} (ii)

{\bf 2a.} {\bf numruns}

{\bf 2b.} {\bf s/sqrt(nreps)}

{\bf 3.} {\it indicator}

{\bf 4.} R, R, NR, NR

{\bf 5.} $\sqrt{\frac{1}{n} \hat{p} (1-\hat{p})}$

{\bf 6.}

$$
p_{\overline{X}}(67.5) = P(\overline{X}) = 67.5) = P(\textrm{get 66 then
69 or 69 then 66}) = 2 \cdot \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{3}
$$

{\bf 7.} $E(U) =  \mu$, $Var(U) = \frac{n+3}{(n+1)^2} \cdot \sigma^2$


\end{document}

