[IPSUR-commits] r147 - pkg/IPSUR/inst/doc
noreply at r-forge.r-project.org
noreply at r-forge.r-project.org
Mon Jan 18 21:01:31 CET 2010
Author: gkerns
Date: 2010-01-18 21:01:30 +0100 (Mon, 18 Jan 2010)
New Revision: 147
Modified:
pkg/IPSUR/inst/doc/IPSUR.Rnw
Log:
finished fixing BLANKs
Modified: pkg/IPSUR/inst/doc/IPSUR.Rnw
===================================================================
--- pkg/IPSUR/inst/doc/IPSUR.Rnw 2010-01-18 17:05:46 UTC (rev 146)
+++ pkg/IPSUR/inst/doc/IPSUR.Rnw 2010-01-18 20:01:30 UTC (rev 147)
@@ -2274,8 +2274,8 @@
peak and moderately sized tails. The standard example of a mesokurtic
distribution is the famous bell-shaped curve, also known as the Gaussian,
or normal, distribution, and the binomial distribution can be mesokurtic
-for specific choices of $p$. See Sections \ref{sec:The-Binomial-Distribution}
-and \ref{sec:The-Normal-Distribution}.
+for specific choices of $p$. See Sections \ref{sec:binom-dist} and
+\ref{sec:The-Normal-Distribution}.
\subsection{Clusters and Gaps\label{sub:Clusters-and-Gaps}}
@@ -5280,7 +5280,7 @@
@
Note that the exact value is $21/40$; we will learn a quick way to
-compute this in Section \ref{sec:Other-Discrete-Distributions}. What
+compute this in Section \ref{sec:other-discrete-distributions}. What
is the probability of observing \inputencoding{latin9}\lstinline[showstringspaces=false]!"red"!\inputencoding{utf8},
then \inputencoding{latin9}\lstinline[showstringspaces=false]!"green"!\inputencoding{utf8},
then \inputencoding{latin9}\lstinline[showstringspaces=false]!"red"!\inputencoding{utf8}?
@@ -6003,10 +6003,10 @@
\item how to make new discrete random variables from old ones
\end{itemize}
-\section{Discrete Random Variables\label{sec:Discrete-Random-Variables}}
+\section{Discrete Random Variables\label{sec:discrete-random-variables}}
-\subsection{Probability Mass Functions\label{sub:Probability-Mass-Functions}}
+\subsection{Probability Mass Functions\label{sub:probability-mass-functions}}
Discrete random variables are characterized by their supports which
take the form \begin{equation}
@@ -6047,7 +6047,7 @@
\end{example}
-\subsection{Mean, Variance, and Standard Deviation\label{sub:Mean,-Variance,-and}}
+\subsection{Mean, Variance, and Standard Deviation\label{sub:mean-variance-sd}}
There are numbers associated with PMFs. One important example is the
mean $\mu$, also known as $\E X$:\begin{equation}
@@ -6156,7 +6156,7 @@
@
-\section{The Discrete Uniform Distribution\label{sec:The-Discrete-Uniform}}
+\section{The Discrete Uniform Distribution\label{sec:disc-uniform-dist}}
We have seen the basic building blocks of discrete distributions and
we now study particular models that statisticians often encounter
@@ -6247,7 +6247,7 @@
The default name for the variable is \inputencoding{latin9}\lstinline[showstringspaces=false,tabsize=2]!disunif.sim1!\inputencoding{utf8}.
-\section{The Binomial Distribution\label{sec:The-Binomial-Distribution}}
+\section{The Binomial Distribution\label{sec:binom-dist}}
The binomial distribution is based on a \emph{Bernoulli trial}, which
is a random experiment in which there are only two possible outcomes:
@@ -6479,12 +6479,12 @@
-\section{Expectation and Moment Generating Functions\label{sec:Expectation-and-Moment}}
+\section{Expectation and Moment Generating Functions\label{sec:expectation-and-mgfs}}
-\subsection{The Expectation Operator\label{sub:The-Expectation-Operator}}
+\subsection{The Expectation Operator\label{sub:expectation-operator}}
-We next generalize some of the concepts from Section \ref{sub:Mean,-Variance,-and}.
+We next generalize some of the concepts from Section \ref{sub:mean-variance-sd}.
There we saw that every%
\footnote{Not every, only those PMFs for which the (potentially infinite) series
converges.%
@@ -6531,7 +6531,7 @@
\end{proof}
-\subsection{Moment Generating Functions\label{sub:Moment-Generating-Functions}}
+\subsection{Moment Generating Functions\label{sub:MGFs}}
\begin{defn}
Given a random variable $X$, its \emph{moment generating function}
(abbreviated MGF) is defined by the formula\begin{equation}
@@ -6618,9 +6618,9 @@
\begin{example}
Let $X\sim\mathsf{binom}(\mathtt{size}=n,\,\mathtt{prob}=p)\mbox{ with \ensuremath{M(t)=(q+p\me^{t})^{n}}}$.
We calculated the mean and variance of a binomial random variable
-in Section \ref{sec:The-Binomial-Distribution} by means of the binomial
-series. But look how quickly we find the mean and variance with the
-moment generating function.
+in Section \ref{sec:binom-dist} by means of the binomial series.
+But look how quickly we find the mean and variance with the moment
+generating function.
\begin{alignat*}{1}
M'(t)= & n(q+p\me^{t})^{n-1}p\me^{t}\left|_{t=0}\right.,\\
@@ -6682,7 +6682,7 @@
and \inputencoding{latin9}\lstinline[showstringspaces=false]!kurtosis!\inputencoding{utf8}.
-\section{The Empirical Distribution\label{sec:The-Empirical-Distribution}}
+\section{The Empirical Distribution\label{sec:empirical-distribution}}
Do an experiment $n$ times and observe $n$ values $x_{1}$, $x_{2}$,
\ldots{}, $x_{n}$ of a random variable $X$. For simplicity in most
@@ -6788,7 +6788,7 @@
Chapter \ref{cha:Resampling-Methods}.
-\section{Other Discrete Distributions\label{sec:Other-Discrete-Distributions}}
+\section{Other Discrete Distributions\label{sec:other-discrete-distributions}}
The binomial and discrete uniform distributions are popular, and rightly
so; they are simple and form the foundation for many other more complicated
@@ -6798,10 +6798,10 @@
offer.
-\subsection{Dependent Bernoulli Trials\label{sec:Non-Bernoulli-Trials}}
+\subsection{Dependent Bernoulli Trials\label{sec:non-bernoulli-trials}}
-\subsubsection*{The Hypergeometric Distribution\label{sub:Hypergeometric-Distribution}}
+\subsubsection*{The Hypergeometric Distribution\label{sub:hypergeometric-dist}}
Consider an urn with 7 white balls and 5 black balls. Let our random
experiment be to randomly select 4 balls, without replacement, from
@@ -7232,7 +7232,7 @@
the solution can be found by simple substitution.
\begin{example}
Let $X\sim\mathsf{nbinom}(\mathtt{size}=r,\,\mathtt{prob}=p)$. We
-saw in \ref{sec:Other-Discrete-Distributions} that $X$ represents
+saw in \ref{sec:other-discrete-distributions} that $X$ represents
the number of failures until $r$ successes in a sequence of Bernoulli
trials. Suppose now that instead we were interested in counting the
number of trials (successes and failures) until the $r$$^{\text{th}}$
@@ -8253,7 +8253,7 @@
The exponential distribution is closely related to the Poisson distribution.
If customers arrive at a store according to a Poisson process with
rate $\lambda$ and if $Y$ counts the number of customers that arrive
-in the time interval $[0,t)$, then we saw in Section \ref{sec:Other-Discrete-Distributions}
+in the time interval $[0,t)$, then we saw in Section \ref{sec:other-discrete-distributions}
that $Y\sim\mathsf{pois}(\mathtt{lambda}=\lambda t).$ Now consider
a different question: let us start our clock at time 0 and stop the
clock when the first customer arrives. Let $X$ be the length of this
@@ -8305,10 +8305,10 @@
$Y$ denotes this random time then $Y\sim\mathsf{gamma}(\mathtt{shape}=3,\,\mathtt{rate}=1/2)$.
\end{example}
-\subsection{The Chi Square, Student's $t$, and Snedecor's $F$ Distributions\label{sub:The-Chi-Square-t-F}}
+\subsection{The Chi square, Student's $t$, and Snedecor's $F$ Distributions\label{sub:The-Chi-Square-t-F}}
-\subsection*{The Chi Square Distribution\label{sub:The-Chi-Square}}
+\subsection*{The Chi square Distribution\label{sub:The-Chi-Square}}
A random variable $X$ with PDF\begin{equation}
f_{X}(x)=\frac{1}{\Gamma(p/2)2^{p/2}}x^{p/2-1}\me^{-x/2},\quad x>0,\end{equation}
@@ -8319,15 +8319,31 @@
\inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!qchisq!\inputencoding{utf8},
and \inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!rchisq!\inputencoding{utf8},
which give the PDF, CDF, quantile function, and simulate random variates,
-respectively. See Figure \ref{dchisq}. In an obvious notation we
-may define $\chi_{\alpha}^{2}(p)$ as the number on the $x$-axis
-such that there is exactly $\alpha$ area under the $\mathsf{chisq}(\mathtt{df}=p)$
+respectively. See Figure \ref{fig:chisq-dist-vary-df}. In an obvious
+notation we may define $\chi_{\alpha}^{2}(p)$ as the number on the
+$x$-axis such that there is exactly $\alpha$ area under the $\mathsf{chisq}(\mathtt{df}=p)$
curve to its right.
+The code to produce Figure \ref{fig:chisq-dist-vary-df} is
+
+<<eval = FALSE>>=
+curve(dchisq(x, df = 3), from = 0, to = 20, ylab = "y")
+ind <- c(4, 5, 10, 15)
+for (i in ind) curve(dchisq(x, df = i), 0, 20, add = TRUE)
+@
+
%
\begin{figure}
-\caption{Chi-Square densities with various df\label{dchisq}}
+\begin{centering}
+<<echo = FALSE, fig=true, height = 4.5, width = 6>>=
+curve(dchisq(x, df = 3), from = 0, to = 20, ylab = "y")
+ind <- c(4, 5, 10, 15)
+for (i in ind) curve(dchisq(x, df = i), 0, 20, add = TRUE)
+@
+\par\end{centering}
+\caption{Chi square distribution for various degrees of freedom\label{fig:chisq-dist-vary-df}}
+
\end{figure}
\begin{rem}
@@ -8354,45 +8370,17 @@
\begin{equation}
f_{X}(x)=\frac{\Gamma\left[(r+1)/2\right]}{\sqrt{r\pi}\,\Gamma(r/2)}\left(1+\frac{x^{2}}{r}\right)^{-(r+1)/2},\quad-\infty<x<\infty\end{equation}
is said to have \emph{Student's} $t$ distribution with $r$ \emph{degrees
-of freedom} ($\mathtt{df}$), and we write $X\sim\mathsf{t}(\mathtt{df}=r)$.
-The associated \textsf{R} functions are \inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!dt(x, df)!\inputencoding{utf8},
+of freedom}, and we write $X\sim\mathsf{t}(\mathtt{df}=r)$. The associated
+\textsf{R} functions are \inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!dt!\inputencoding{utf8},
\inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!pt!\inputencoding{utf8},
\inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!qt!\inputencoding{utf8},
and \inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!rt!\inputencoding{utf8},
which give the PDF, CDF, quantile function, and simulate random variates,
-respectively.
+respectively. See Section \ref{sec:sampling-from-normal-dist}.
-Similar to that done for the normal we may define $t_{\alpha}^{(\mathtt{df})}$
-as the number on the $x$-axis such that there is exactly $\alpha$
-area under the $\mathsf{t}(\mathtt{df}=r)$ curve to its right.
-\begin{example}
-We find $t_{0.01}^{(23)}$with the quantile function:
-<<>>=
-qt(0.01, df = 23, lower.tail = FALSE)
-@
+\subsection*{Snedecor's $F$ distribution\label{sub:snedecor-F-distribution}}
-\end{example}
-\begin{rem}
-We can say the following:
-\begin{enumerate}
-\item The $\mathsf{t}(\mathtt{df}=r)$ distribution looks just like a $\mathsf{norm}(\mathtt{mean}=0,\,\mathtt{sd}=1)$
-distribution except with heavier tails.
-\item The $\mathsf{t}(\mathtt{df}=1)$ distribution is also known as a standard
-{}``Cauchy distribution'', which is implemented in \textsf{R} with
-the \inputencoding{latin9}\lstinline[basicstyle={\ttfamily}]!dcauchy!\inputencoding{utf8}
-function and its relatives. The Cauchy distribution is quite pathological
-and is often a counterexample to many famous results.
-\item The standard deviation of $\mathsf{t}(\mathtt{df}=r)$ is undefined
-(that is, infinite) unless $r>2$. When $r$ is more than 2, the standard
-deviation is always bigger than one, but decreases to 1 as $r\to\infty$.
-\item The $\mathsf{t}(\mathtt{df}=r)$ distribution approaches a $\mathsf{norm}(\mathtt{mean}=0,\,\mathtt{sd}=1)$
-distribution as $r\to\infty$.
-\end{enumerate}
-\end{rem}
-
-\subsection*{Snedecor's $F$ distribution\label{sub:Fisher's-F-distribution}}
-
A random variable $X$ with p.d.f.
\begin{equation}
@@ -8404,19 +8392,21 @@
\inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!qf!\inputencoding{utf8},
and \inputencoding{latin9}\lstinline[breaklines=true,showstringspaces=false,tabsize=4]!rf!\inputencoding{utf8},
which give the PDF, CDF, quantile function, and simulate random variates,
-respectively. We define $F_{\alpha}^{(m,n)}$ as the number on the
-$x$-axis such that there is exactly $\alpha$ area under the $\mathsf{f}(\mathtt{df1}=m,\,\mathtt{df2}=n)$
+respectively. We define $F_{\alpha}(m,n)$ as the number on the $x$-axis
+such that there is exactly $\alpha$ area under the $\mathsf{f}(\mathtt{df1}=m,\,\mathtt{df2}=n)$
curve to its right.
\begin{rem}
-Here are some important notes about the $F$ distribution.
+Here are some notes about the $F$ distribution.
\begin{enumerate}
-\item If $X\sim\mathsf{f}(\mathtt{df1}=m,\,\mathtt{df2}=n)$, then $(1/X)\sim\mathsf{f}(\mathtt{df1}=n,\,\mathtt{df2}=m)$.
-Historically, this fact was especially convenient. In the old days,
-statisticians used printed tables for their statistical calculations.
-Since the $F$ tables were symmetric in $m$ and $n$, it meant that
-publishers could cut the size of their printed tables in half. It
-plays less of a role today, now that personal computers are widespread.
+\item If $X\sim\mathsf{f}(\mathtt{df1}=m,\,\mathtt{df2}=n)$ and $Y=1/X$,
+then $Y\sim\mathsf{f}(\mathtt{df1}=n,\,\mathtt{df2}=m)$. Historically,
+this fact was especially convenient. In the old days, statisticians
+used printed tables for their statistical calculations. Since the
+$F$ tables were symmetric in $m$ and $n$, it meant that publishers
+could cut the size of their printed tables in half. It plays less
+of a role today now that personal computers are widespread.
\item If $X\sim\mathsf{t}(\mathtt{df}=r)$, then $X^{2}\sim\mathsf{f}(\mathtt{df1}=1,\,\mathtt{df2}=r)$.
+We will see this again in Section \ref{sub:slr-overall-F-statistic}.
\end{enumerate}
\end{rem}
@@ -9800,7 +9790,7 @@
round(ProbTable, 3)
@
-Do some examples of \inputencoding{latin9}\lstinline[showstringspaces=false]!rmultinom!\inputencoding{utf8}
+Do some examples of \inputencoding{latin9}\lstinline[showstringspaces=false]!rmultinom!\inputencoding{utf8}.
Here is another way to do it%
\footnote{Another way to do the plot is with the \inputencoding{latin9}\lstinline[basicstyle={\ttfamily}]!scatterplot3d!\inputencoding{utf8}
@@ -9931,10 +9921,10 @@
as bias and variance
\end{itemize}
-\section{Simple Random Samples\label{sec:Simple-Random-Samples}}
+\section{Simple Random Samples\label{sec:simple-random-samples}}
-\subsection{Simple Random Samples\label{sub:Simple-Random-Samples}}
+\subsection{Simple Random Samples\label{sub:simple-random-samples}}
\begin{defn}
If $X_{1}$, $X_{2}$, \ldots{}, $X_{n}$ are independent with $X_{i}\sim f$
for $i=1,2,\ldots,n$, then we say that $X_{1}$, $X_{2}$, \ldots{},
@@ -9954,7 +9944,7 @@
The next fact will be useful to us when it comes time to prove the
-Central Limit Theorem in Section \ref{sec:The-Central-Limit}.
+Central Limit Theorem in Section \ref{sec:Central-Limit-Theorem}.
\begin{prop}
\label{pro:mgf-xbar}Let $X_{1}$, $X_{2}$, \ldots{}, $X_{n}$ be
a $SRS(n)$ from a population distribution with MGF $M(t)$. Then
@@ -9977,10 +9967,10 @@
-\section{Sampling from a Normal Distribution\label{sec:Sampling-from-Normal}}
+\section{Sampling from a Normal Distribution\label{sec:sampling-from-normal-dist}}
-\subsection{The Distribution of the Sample Mean\label{sub:Samp-Mean-Dist}}
+\subsection{The Distribution of the Sample Mean\label{sub:samp-mean-dist-of}}
\begin{prop}
Let $X_{1}$, $X_{2}$, \ldots{}, $X_{n}$ be a $SRS(n)$ from a
$\mathsf{norm}(\mathtt{mean}=\mu,\,\mathtt{sd}=\sigma)$ distribution.
@@ -10044,7 +10034,7 @@
T=\frac{Z}{\sqrt{V/r}},\end{equation}
where $r=n-1$.
-We know from Section \ref{sub:Samp-Mean-Dist} that $Z\sim\mathsf{norm}(\mathtt{mean}=0,\,\mathtt{sd}=1)$
+We know from Section \ref{sub:samp-mean-dist-of} that $Z\sim\mathsf{norm}(\mathtt{mean}=0,\,\mathtt{sd}=1)$
and we know from Section \ref{sub:Samp-Var-Dist} that $V\sim\mathsf{chisq}(\mathtt{df}=n-1)$.
Further, since we are sampling from a normal distribution, Theorem
\ref{thm:Xbar-andS} gives that $\Xbar$ and $S^{2}$ are independent
@@ -10063,51 +10053,75 @@
it takes the form
\begin{equation}
-f_{X}(x)=\frac{\Gamma[(r+1)/2]}{\sqrt{r\pi}\ \Gamma(r/2)}\left(1+\frac{x^{2}}{r}\right)^{-(r+1)/2},\quad\-\infty<x<\infty\end{equation}
+f_{X}(x)=\frac{\Gamma[(r+1)/2]}{\sqrt{r\pi}\ \Gamma(r/2)}\left(1+\frac{x^{2}}{r}\right)^{-(r+1)/2},\quad-\infty<x<\infty\end{equation}
-Any random variable $T$ with the preceding PDF is said to have Student's
-$t$ distribution with $r$ \emph{degrees of freedom} ($\mathtt{df}$),
-and we write $T\sim\mathsf{t}(\mathtt{df}=r)$. The shape of the PDF
-is similar to the normal, but the tails are considerably heavier.
-See Figure BLANK. As with the Normal distribution, there are four
-functions in \textsf{R} associated with the $t$ distribution, namely
-\texttt{dt()}, \texttt{pt()}, \texttt{qt()}, and \texttt{rt()}, which
-compute the p.d.f., c.d.f., quantiles, and generate random variates,
-respectively.
+Any random variable $X$ with the preceding PDF is said to have Student's
+$t$ distribution with $r$ \emph{degrees of freedom}, and we write
+$X\sim\mathsf{t}(\mathtt{df}=r)$. The shape of the PDF is similar
+to the normal, but the tails are considerably heavier. See Figure
+\ref{fig:Student's-t-dist-vary-df}. As with the normal distribution,
+there are four functions in \textsf{R} associated with the $t$ distribution,
+namely \inputencoding{latin9}\lstinline[showstringspaces=false]!dt!\inputencoding{utf8},
+\inputencoding{latin9}\lstinline[showstringspaces=false]!pt!\inputencoding{utf8},
+\inputencoding{latin9}\lstinline[showstringspaces=false]!qt!\inputencoding{utf8},
+and \inputencoding{latin9}\lstinline[showstringspaces=false]!rt!\inputencoding{utf8},
+which compute the PDF, CDF, quantile function, and generate random
+variates, respectively.
+The code to produce Figure \ref{fig:Student's-t-dist-vary-df} is
+
+<<eval = FALSE>>=
+curve(dt(x, df = 30), from = -3, to = 3, lwd = 3, ylab = "y")
+ind <- c(1, 2, 3, 5, 10)
+for (i in ind) curve(dt(x, df = i), -3, 3, add = TRUE)
+@
+
+%
+\begin{figure}
+\begin{centering}
+<<echo = FALSE, fig=true, height = 4.5, width = 6>>=
+curve(dt(x, df = 30), from = -3, to = 3, lwd = 3, ylab = "y")
+ind <- c(1, 2, 3, 5, 10)
+for (i in ind) curve(dt(x, df = i), -3, 3, add = TRUE)
+@
+\par\end{centering}
+
+\caption{Student's $t$ distribution for various degrees of freedom\label{fig:Student's-t-dist-vary-df}}
+
+\end{figure}
+
+
Similar to that done for the normal we may define $\mathsf{t}_{\alpha}(\mathtt{df}=n-1)$
as the number on the $x$-axis such that there is exactly $\alpha$
area under the $\mathsf{t}(\mathtt{df}=n-1)$ curve to its right.
\begin{example}
-Find $t_{0.01}^{(23)}$ with the quantile function:
+Find $\mathsf{t}{}_{0.01}(\mathtt{df}=23)$ with the quantile function.
\end{example}
-\texttt{\textcolor{red}{> qt(0.01, df=23, lower.tail=FALSE)}}\texttt{}~\\
-\texttt{\textcolor{blue}{{[}1{]} 2.499867}}
-
-Notice the \texttt{df} parameter.
+<<>>=
+qt(0.01, df = 23, lower.tail = FALSE)
+@
\begin{rem}
There are a few things to note about the $\mathtt{t}(\mathtt{df}=r)$
distribution.
\begin{enumerate}
-\item It looks a lot like a $\mathsf{norm}(\mathtt{mean}=0,\,\mathtt{sd}=1)$
-distribution, except with heavier tails.
-\item When $r=1$, the $\mathtt{t}(\mathtt{df}=r)$ distribution is the
-same as the $\mathtt{cauchy}(\mathtt{location}=0,\,\mathtt{scale}=1)$
-distribution.
-\item The standard deviation -- if it exists -- is always bigger than one,
-but decreases to one as $r\to\infty$.
+\item The $\mathtt{t}(\mathtt{df}=1)$ distribution is the same as the $\mathsf{cauchy}(\mathtt{location}=0,\,\mathtt{scale}=1)$
+distribution. The Cauchy distribution is rather pathological and is
+a counterexample to many famous results.
+\item The standard deviation of $\mathsf{t}(\mathtt{df}=r)$ is undefined
+(that is, infinite) unless $r>2$. When $r$ is more than 2, the standard
+deviation is always bigger than one, but decreases to 1 as $r\to\infty$.
\item As $r\to\infty$, the $\mathtt{t}(\mathtt{df}=r)$ distribution approaches
-the $\mathsf{norm}(\mathtt{mean}=0,\,\mathtt{sd}=1)$ distribution.
+the $\mathsf{norm}(\mathtt{mean}=0,\,\mathtt{sd}=1)$ distribution.
\end{enumerate}
\end{rem}
-\section{The Central Limit Theorem\label{sec:The-Central-Limit}}
+\section{The Central Limit Theorem\label{sec:Central-Limit-Theorem}}
In this section we study the distribution of the sample mean when
the underlying distribution is \emph{not} normal. We saw in Section
-\ref{sec:Sampling-from-Normal} that when $X_{1}$, $X_{2}$, \ldots{},
-$X_{n}$ is a $SRS(n)$ from a $\mathsf{norm}(\mathtt{mean}=\mu,\,\mathtt{sd}=\sigma)$
+\ref{sec:sampling-from-normal-dist} that when $X_{1}$, $X_{2}$,
+\ldots{}, $X_{n}$ is a $SRS(n)$ from a $\mathsf{norm}(\mathtt{mean}=\mu,\,\mathtt{sd}=\sigma)$
distribution then $\Xbar\sim\mathsf{norm}(\mathtt{mean}=\mu,\,\mathtt{sd}=\sigma/\sqrt{n})$.
In other words, we may say (owing to Fact \ref{fac:lin-trans-norm-is-norm})
when the underlying population is normal that the sampling distribution
@@ -10129,7 +10143,7 @@
as $n\to\infty$. \end{thm}
\begin{rem}
We suppose that $X_{1}$, $X_{2}$, \ldots{}, $X_{n}$ are i.i.d.,
-and we learned in Section \ref{sub:Simple-Random-Samples} that $\Xbar$
+and we learned in Section \ref{sub:simple-random-samples} that $\Xbar$
has mean $\mu$ and standard deviation $\sigma/\sqrt{n}$, so we already
knew that $Z$ has mean 0 and standard deviation 1. The beauty of
the CLT is that it addresses the \emph{shape} of $Z$'s distribution
@@ -10141,7 +10155,7 @@
is not mentioned in Theorem \ref{thm:central-limit-thrm}; indeed,
the result is true for any population that is well-behaved enough
to have a finite standard deviation. In particular, if the population
-is normally distributed then we know from Section \ref{sub:Samp-Mean-Dist}
+is normally distributed then we know from Section \ref{sub:samp-mean-dist-of}
that the distribution of $\Xbar$ (and $Z$ by extension) is \emph{exactly}
normal, for \emph{every} $n$.
\end{rem}
@@ -10614,7 +10628,7 @@
and we have observed $x=3$ of them to be white. What is the probability
of this?
-Looking back to Section \ref{sec:Other-Discrete-Distributions}, we
+Looking back to Section \ref{sec:other-discrete-distributions}, we
see that the random variable $X$ has a $\mathsf{hyper}(\mathtt{m}=M,\,\mathtt{n}=F-M,\,\mathtt{k}=K)$
distribution. Therefore, for an observed value $X=x$ the probability
would be\[
@@ -11294,10 +11308,10 @@
are unknown. This leads us to the following:
\begin{itemize}
\item If both sample sizes are large, then we may appeal to the CLT/SLLN
-(see \ref{sec:The-Central-Limit}) and substitute $S_{X}^{2}$ and
-$S_{Y}^{2}$ for $\sigma_{X}^{2}$ and $\sigma_{Y}^{2}$ in the interval
-\ref{eq:two-samp-mean-CI}. The resulting confidence interval will
-have approximately $100(1-\alpha)\%$ confidence.
+(see \ref{sec:Central-Limit-Theorem}) and substitute $S_{X}^{2}$
+and $S_{Y}^{2}$ for $\sigma_{X}^{2}$ and $\sigma_{Y}^{2}$ in the
+interval \ref{eq:two-samp-mean-CI}. The resulting confidence interval
+will have approximately $100(1-\alpha)\%$ confidence.
\item If one or more of the sample sizes is small then we are in trouble,
unless
@@ -11365,9 +11379,9 @@
\end{itemize}
We are given an $SRS(n)$ $X_{1}$, $X_{2}$, \ldots{}, $X_{n}$
distributed $\mathsf{binom}(\mathtt{size}=1,\,\mathtt{prob}=p)$.
-Recall from Section \ref{sec:The-Binomial-Distribution} that the
-common mean of these variables is $\E X=p$ and the variance is $\E(X-p)^{2}=p(1-p)$.
-If we let $Y=\sum X_{i}$, then from Section \ref{sec:The-Binomial-Distribution}
+Recall from Section \ref{sec:binom-dist} that the common mean of
+these variables is $\E X=p$ and the variance is $\E(X-p)^{2}=p(1-p)$.
+If we let $Y=\sum X_{i}$, then from Section \ref{sec:binom-dist}
we know that $Y\sim\mathsf{binom}(\mathtt{size}=n,\,\mathtt{prob}=p)$
and that \[
\Xbar=\frac{Y}{n}\mbox{ has }\E\Xbar=p\mbox{ and }\mathrm{Var}(\Xbar)=\frac{p(1-p)}{n}.\]
@@ -13235,7 +13249,7 @@
line is positive.
-\subsection{Overall \emph{F} statistic}
+\subsection{Overall \emph{F} statistic\label{sub:slr-overall-F-statistic}}
There is another way to test the significance of the linear regression
model. In SLR, the new way also tests the hypothesis $H_{0}:\beta_{1}=0$
@@ -15517,7 +15531,7 @@
is, so let us \emph{estimate} it, just like we would with any other
parameter. The statistic we use is the \emph{empirical CDF}, that
is, the function that places mass $1/n$ at each of the observed data
-points $x_{1},\ldots,x_{n}$ (see Section \ref{sec:The-Empirical-Distribution}).
+points $x_{1},\ldots,x_{n}$ (see Section \ref{sec:empirical-distribution}).
As the sample size increases, we would expect the approximation to
get better and better (with i.i.d.~observations, it does, and there
is a wonderful theorem by Glivenko and Cantelli that proves it). And
@@ -15594,7 +15608,7 @@
underlying population is $\mathsf{norm}(\mathtt{mean}=3,\,\mathtt{sd}=1)$.
Of course, we do not really need a bootstrap distribution here because
-from Section \ref{sec:Sampling-from-Normal} we know that $\Xbar\sim\mathsf{norm}(\mathtt{mean}=3,\,\mathtt{sd}=1/\sqrt{n})$,
+from Section \ref{sec:sampling-from-normal-dist} we know that $\Xbar\sim\mathsf{norm}(\mathtt{mean}=3,\,\mathtt{sd}=1/\sqrt{n})$,
but we will investigate how the bootstrap performs when we know what
the answer should be ahead of time.
@@ -15652,7 +15666,7 @@
\emph{bootstrap estimate of bias}. Since the estimate is so small
we would expect our original statistic ($\Xbar$) to have small bias,
but this is no surprise to us because we already knew from Section
-\ref{sub:Simple-Random-Samples} that $\Xbar$ is an unbiased estimator
+\ref{sub:simple-random-samples} that $\Xbar$ is an unbiased estimator
of the population mean.
Now back to our original problem, we would like to estimate the standard
@@ -17290,8 +17304,7 @@
\item Write your report as an \inputencoding{latin9}\lstinline[showstringspaces=false]!.odt!\inputencoding{utf8}
document in OO.o just as you would any other document. Call this document
\inputencoding{latin9}\lstinline[showstringspaces=false]!infile.odt!\inputencoding{utf8},
-and make sure that it is saved in your working directory (see Section
-BLANK).
+and make sure that it is saved in your working directory.
\item At the places you would like to insert \textsf{R} code in the document,
write the code chunks in the following format:
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