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\begin{verbatim}
Name: ___________________________________
\end{verbatim}
\begin{center}
{\bf Computing in Statistics}, STAT:5400\\
Midterm 2, Fall 2017 \\
\end{center}
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\section{Simulation study}

The {\tt prop.test} function, with the Yates correction argument set to
FALSE, computes large sample normal-theory confidence intervals.  With
appropriate adjustment to the to the numeric arguments, it can also compute
plus-four method intervals.

Conduct a simulation study to compare the true coverage probabilities of
large-sample normal theory intervals versus plus-four method intervals,
when the nominal coverage is 90\%, the sample size is 25 and the true success 
probability is 0.16.

Use a large enough number of replicate datasets so that the standard error of
your estimated coverage probabilities  will be no greater than 0.0025.

Time how long it takes to run your simulation study.

Submit:
\begin{itemize}
	\item Your R code.
	\item Your R output containing the results of the simulation study.
	\item R output showing how long it took to run the simulation study.
	\item A brief paragraph interpreting the results.
\end{itemize}


<<>>=

doit <- function( n, conf, se, truep, seed)
{
	getInt <- function(v) 
	{
		prop.test( v[1], v[2], conf.level = conf,
			  correct = FALSE)$conf.int
	}

	# how many replicate datasets, based on required stand error
	S <- 1/(4 * se^2)


	# generate data
	set.seed( seed )

	mydat <- rbinom( S , size = n, prob = truep )
	normal <- apply( cbind(mydat, rep(n,S) ), 1, getInt )
	plus4 <- apply( cbind(mydat+2, rep(n+4,S) ), 1, getInt )

	normalcov <- 0.16 > normal[1,] & 0.16 < normal[2,]
	plus4cov <- 0.16 > plus4[1,] & 0.16 < plus4[2,]


	coverage <- c(mean(normalcov), mean(plus4cov) )
        names(coverage) = c("normal","plus4")
	coverage
}

system.time(coverage <- doit( 25, .90, .0025, 0.16, 17 ))
print(coverage)

@

This is one of the fairly rare conditions under which the plus-four method
performs more poorly than the large-sample, normal-theory method.  The plus-four actually is anti-conservative here -- its actual coverage is lower than its
nominal coverage.


However, prop.test must not be doing exactly what is advertised.  If you code the computations yourself, you will see both methods coming in right around 
90\% coverage!

<<>>=
mydat <- rbinom(40000, 25, 0.16)
z <- qnorm(0.95)

computeInt <- function( y, n)
{
	phat <- y/n
	se <- sqrt( phat * (1-phat) / n )
	mult <- c(-1,1)
	interval <- phat + mult * z * se
}

normInts <- plus4Ints <- matrix(0, nrow=40000, ncol=2)

for(i in 1:40000)
{
normInts[i,] <- computeInt( mydat[i], 25 )

plus4Ints[i,] <- computeInt( mydat[i] + 2, 29)
}

normcov <- mean( .16 > normInts[,1] & .16 < normInts[,2])
plus4cov <- mean( .16 > plus4Ints[,1] & .16 < plus4Ints[,2])

print(c(normcov, plus4cov))

@
\section{Root-finding}

The function 
\[ f(x) \ = \ x^4 - 2 x^3 + 1 \]
has two real roots in the interval [0.5, 2.0] .

Use the {\tt uniroot} function in such a way that you are able to find both
roots.  

Submit:
\begin{itemize}
	\item Your R code.
	\item R output containing results for each root.
	\item Any other R output that helped you do this.
\end{itemize}

<<fig=TRUE>>=

myfunc <- function(x) {x^4 - 2 * x^3 + 1}

x <- seq(0.5,2.0, 0.01)

plot(x, myfunc(x), type="l")

result1 <- uniroot( myfunc, c(0.5, 1.5) )

result2 <- uniroot( myfunc, c(1.5, 2.0) )

result1
result2

@

\section{Relational database structures}

A journal editor wishes to store information about the papers published
in her journal and the authors of those papers.  Below are the attributes
that she wishes to store.  Develop a relational database structure, in
third normal form, for these data.  For each table, indicate any primary
and/or foreign keys.  Note that the same person could be an author of more
than one paper, and a paper may have more than one author.

This is a many-to-many relationship, so a linking table will be needed.

\begin{verbatim}
Paper table
----------

Paper ID      (primary key)
Paper title
Paper volume number
Paper issue number
Paper starting page number


Author table
------------
Author ID    (primary key)
Author name 
Author affiliation 
Author email 


Linking table
-------------

(Primary key is combination of the two fields)

Paper ID   (foreign key)
Author ID  (foreign key)

\end{verbatim}



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