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\cfoot{Introduction to the Practice of Statistics using R: Chapter 3}
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\title{Introduction to the Practice of Statistics using R: \\
Chapter 3}

\author{
Nicholas J. Horton\thanks{Department of Mathematics, Amherst College, nhorton@amherst.edu} \and Ben Baumer
} 

\date{\today}

\begin{document}


\maketitle
\tableofcontents

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}
@

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gsub('^\\\\begin\\{alltt\\}\\s*|\\\\end\\{alltt\\}\\s*$', '', h)
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showOriginal=FALSE
showNew=TRUE
@ 

\section*{Introduction}

This document is intended to help describe how to undertake analyses introduced as examples in the Sixth Edition of \emph{Introduction to the 
Practice of Statistics} (2009) by David Moore, George McCabe and Bruce
Craig.
More information about the book can be found at \url{http://bcs.whfreeman.com/ips6e/}.  This
file as well as the associated \pkg{knitr} reproducible analysis source file can be found at
\url{http://www.math.smith.edu/~nhorton/ips6e}.

This work leverages initiatives undertaken by Project MOSAIC (\url{http://www.mosaic-web.org}), an NSF-funded effort to improve the teaching of statistics, calculus, science and computing in the undergraduate curriculum. In particular, we utilize the 
\pkg{mosaic} package, which was written to simplify the use of R for introductory statistics courses. A short summary of the R needed to teach introductory statistics can be found in the mosaic package vignette (\url{http://cran.r-project.org/web/packages/mosaic/vignettes/MinimalR.pdf}).

To use a package within R, it must be installed (one time), and loaded (each session). The package can be installed using the following command:
<<install_mosaic,eval=FALSE>>=
install.packages('mosaic')               # note the quotation marks
@
The {\tt \#} character is a comment in R, and all text after that on the 
current line is ignored.

Once the package is installed (one time only), it can be loaded by running the command:
<<load_mosaic,eval=FALSE>>=
require(mosaic)
@
This
needs to be done once per session.

We also set some options to improve legibility of graphs and output.
<<eval=TRUE>>=
trellis.par.set(theme=col.mosaic())  # get a better color scheme for lattice
options(digits=3)
@

The specific goal of this document is to demonstrate how to replicate the
analysis described in Chapter 3: Producing Data.

\section{Design of experiments}

\subsection{Randomizing subjects}

It's straightforward to randomly divide 40 students into two groups of 20 students
each (as described in Example 3.11 on page 185).  

<<>>=
students = 1:40    # equivalent to seq(from=1, to=40, by=1)
group1 = sample(students, size=20)
sort(group1)
group2 = students[-group1]    # all but those values are included
sort(group2)
@

\section{Sampling design}

\subsection{Simple random samples}

We reproduce a random sampling of resorts (from Figure 3.8, page 202).
<<>>=
resorts = c("Aloha Kai", "Captiva", "Palm Tree", "Sea Shell", "Anchor Down",
  "Casa del Mar", "Radisson", "Silver Beach")
# generate a SRS of size 3
sampled = sample(resorts, size=3)
sampled
@

\section{Toward statistical inference}

\subsection{Simulate a random sample}

It's straightforward to use R to generate simple random samples.  
Example 3.32 (page 214) describes how this is done by using a table 
of random digits.  It's more generalizable to do this with a set of possible
options each with specified probabilities (probability frustrated=0.6, probability
not-frustrated=0.4):
<<>>=
srs1 = sample(c("Frustrating", "Not-frustrating"), size=100, prob=c(0.6, 0.4), 
  replace=TRUE)
tally(srs1)
@
We can repeat the process, which will (generally) give different answers.
<<>>=
n = 100
n
tally(sample(c("Frustrating", "Not-frustrating"), size=n, prob=c(0.6, 0.4), 
  replace=TRUE))
tally(sample(c("Frustrating", "Not-frustrating"), size=n, prob=c(0.6, 0.4), 
  replace=TRUE))
tally(sample(c("Frustrating", "Not-frustrating"), size=n, prob=c(0.6, 0.4), 
  replace=TRUE))
@

We can repeat the process many times using the {\tt do()} function, which 
saves the results.
<<>>=
res = do(1000) * tally(sample(c("Frustrating", "Not-frustrating"), size=n, 
  prob=c(0.6, 0.4), replace=TRUE))
histogram(~ Frustrating, xlab="Number reporting shopping frustrating", data=res)
@
We see that the sampling distribution for the number reporting \emph{Frustrating} 
in m=1000 simple random samples each of size n=100 is centered at the value of around 60, which we
would expect since the true probability of being Frustrating is in fact $p=0.60$.

The results are equivalent if rescaled as a proportion (by dividing by the sample size).
<<>>=
sd(~ Frustrating/n, data=res)
histogram(~ Frustrating/n, xlab="Proportion reporting shopping frustrating", data=res)
@


What happens if we take samples of size n=2500 (as displayed in Example 3.33, on 
pages 214--215).   

<<>>=
n=2500
n
res = do(1000) * tally(sample(c("Frustrating", "Not-frustrating"), size=n, 
  prob=c(0.6, 0.4), replace=TRUE))
sd(~ Frustrating/n, data=res)
histogram(~ Frustrating/n, xlab="Proportion reporting shopping frustrating", 
  data=res)
@
The sampling distribution is much narrower, given the much larger sample size.


\subsection{Capture-recapture sampling}

R can be used as a calculator, as for the calculations in Example 3.34 (page 220).  
<<>>=
200*120/12
@

\end{document}