\documentclass[11pt,titlepage]{article} \usepackage{amsmath} \usepackage{graphicx} \allowdisplaybreaks \jot=.2in \pagestyle{plain} \setlength{\topmargin}{-.4in} % \setlength{\footheight}{0 in} \setlength{\textheight}{9. in} \setlength{\oddsidemargin}{-0.2in} \setlength{\evensidemargin}{-0.1in} \setlength{\textwidth}{6.5in} \font\heada=cmbx10 scaled\magstep3 \font\headb=cmsl10 scaled\magstep1 \font\headc=cmr8 \pretolerance=10000 \setlength{\parindent}{2 em} %\input macros \newdimen\digitwidth \newdimen\minuswidth \setbox0=\hbox{\rm0} \digitwidth=\wd0 \setbox1=\hbox{$-$} \minuswidth=\wd1 \newdimen\starr \setbox2=\hbox{${}^*$} \starr=\wd2 {\catcode`?=\active \def?{\kern\digitwidth} \catcode`@=\active \def@{\kern\minuswidth} \catcode`|=\active \def|{\kern\starr}} \begin{document} \noindent {\heada Project 5}\\ \noindent {\headb Statistics 401: Spring 2007}\\ {\it Due: Friday, March 9} \bigskip \noindent Your write-up must be typed. Please number your answers as the questions were numbered. Your grade will be determined by how well you answer the questions, your justification for your answers, and by the professionalism and clarity of our write-up. \begin{enumerate} \item A random sample of 100 calls made to the customer service center of a small bank in a month is given in the file ServiceCalls.txt, which can be downloaded from the Stat 401 website (Moore and McCabe's \underline{Introduction to the Practice of Statistics}). \begin{enumerate} \item Construct a density plot, a boxplot, and a normal probability plot to display the distribution of the calls data. Include these plots in your report. \item Provide a {\bf summary table} displaying the sample mean, sample standard deviation, and and five number summary for the data. \item Compute the correlation between the normal scores and the calls data. Since Table 7.1 on page 320 of your textbook does not have a critical $r$ value for $n=100$, use critical $r$ of .98 to check for normality of the data. \item \label{normal} Does the distribution of calls data appear to depart from a normal distribution? If so, why? Use more than a single output from R to justify your answer. \item Regardless of your answer to problem \#\ref{normal}, use the boxcox function to estimate the optimal lambda value to use in the transformation. Include the plot in your report. Your chosen $\lambda$ value should be in the confidence interval, preferably near the center. It is best to choose a simple value such as $-1, -\frac{1}{4}, \frac{-1}{2}, 0, \frac{1}{4}, \frac{1}{2}, 1,$ or $2$ rather than a value such as $-0.47352$. \item Power transform the data using the $\lambda$ you just chose. For example, if you chose $\lambda=-\frac{1}{2}$, then in R execute: \begin{verbatim} > calls.transformed = calls^(-1/2) \end{verbatim} \item Construct a density plot, a boxplot, and a normal probability plot to display the distribution of the transformed calls data. Include these plots in your report. \item Provide a summary table displaying the sample mean, sample standard deviation, and five number summary for the transformed calls data. \item Compute the correlation between the normal scores and the transformed calls data. Use .98 as the critical $r$ value to check for normality. \item Does the distribution of the transformed calls data appear to depart from a normal distribution? If so, why or why not? Use more than a single output from R to justify your answer. Did the transformation work? \end{enumerate} \item Let $X$ be the number of SPAM emails received per day per employee at a large software engineering company. Suppose that the distribution of $X$ is: \begin{center} \begin{tabular}{||c|ccc|}\hline $X$ & $0$ & $1$ & $2$\\ \hline $P(X=x)$ & $0.60$ & $0.30$ & $0.10$\\ \hline \end{tabular} \end{center} Compute $\mu_x$ and $\sigma_x$. See the Chapter 8 Handout for definitions of $\mu$ and $\sigma^2$. \item Consider problem 8.10(a) (so don't do (b)) on page 340 of your textbook. \begin{enumerate} \item Compute $\mu$ and $\sigma$ for the population of four textbooks. Use the fact that $P(x)=\frac{1}{N}=\frac{1}{4}$ in your computations. Page 118 of your Course Notes show how the formulas for $\mu$ and $\sigma$ simplify. \item Compute the sampling distribution of $\overline X$ for all samples of size 2. Note that there are six ways to choose two books from four books. \item {\bf From the sampling distribution of $\overline X$}, compute $\mu_{\overline x}$, $\sigma^2_{\overline x}$, and $\sigma_{\overline x}$. \end{enumerate} Include the following two tables in your report which summarize your results from (b) and (c): \begin{center} {\catcode`?=\active \def?{\kern\digitwidth} \begin{tabular}{||l|c||}\hline \multicolumn{1}{||c}{Sample} & $\bar x$ \\ \hline $?1$. \underline{\hspace{.25in}} \underline{\hspace{.25in}} & \underline{\hspace{.5in}} \\[.05in] $?2$. \underline{\hspace{.25in}} \underline{\hspace{.25in}} & \underline{\hspace{.5in}} \\[.05in] $?3$. \underline{\hspace{.25in}} \underline{\hspace{.25in}} & \underline{\hspace{.5in}} \\[.05in] $?4$. \underline{\hspace{.25in}} \underline{\hspace{.25in}} & \underline{\hspace{.5in}} \\[.05in] $?5$. \underline{\hspace{.25in}} \underline{\hspace{.25in}} & \underline{\hspace{.5in}} \\[.05in] $?6$. \underline{\hspace{.25in}} \underline{\hspace{.25in}} & \underline{\hspace{.5in}} \\ \hline \hline \end{tabular} \\ \vspace{0.5in} \begin{tabular}{|c|c|} \multicolumn{2}{c}{Sampling Distribution of $\overline X$} \\[.1in] \hline Value of $\bar x$ & $P(\bar x)$ \\ \hline \underline{\hspace{.5in}} & \underline{\hspace{.5in}} \\[.05in] \underline{\hspace{.5in}} & \underline{\hspace{.5in}} \\[.05in] \underline{\hspace{.5in}} & \underline{\hspace{.5in}} \\[.05in] \underline{\hspace{.5in}} & \underline{\hspace{.5in}} \\[.05in] \underline{\hspace{.5in}} & \underline{\hspace{.5in}} \\[.05in] \underline{\hspace{.5in}} & \underline{\hspace{.5in}} \\ \hline $\mu_{\bar X}$: & \underline{\hspace{.5in}} \\[.05in] $\sigma^2_{\bar X}$: & \underline{\hspace{.5in}} \\[.05in] $\sigma_{\bar X}$ & \underline{\hspace{.5in}} \\\hline \end{tabular} \vspace{0.5in} } \end{center} \item Do problem 8.16 on page 350. \item Do problem 8.18 on page 350. In addition to (a) and (b) in the textbook, also answer the following: \begin{enumerate} \item[(c)] What is the approximate sampling distribution of $\overline X$ when the sample size is $n=50$? Explain why your answer is correct. \item[(d)] What is the probability that the average wait time for 50 individuals is longer than 25 seconds? \end{enumerate} \item Do problem 8.32 on page 357. \end{enumerate} \end{document} %\item \#8.20, page 350. %\item \#8.22, page 350.