\documentclass[11pt,titlepage]{article} \usepackage{amsmath} \usepackage{graphicx} \allowdisplaybreaks \jot=.2in \pagestyle{empty} \setlength{\topmargin}{-0.5in} % \setlength{\footheight}{0 in} \setlength{\textheight}{9.5in} \setlength{\oddsidemargin}{-0.2in} \setlength{\evensidemargin}{0in} \setlength{\textwidth}{6.7in} \font\heada=cmbx10 scaled\magstep3 \font\headb=cmsl10 scaled\magstep1 \font\headc=cmr8 \pretolerance=10000 \setlength{\parindent}{2 em} \begin{document} \begin{center} {\heada Project 9 - One-way ANOVA}\\ {\headb Statistics 401: Spring 2007}\\ {\it Due Wednesday, April 25} \end{center} \bigskip \noindent Turn in your answers in a type-written report. Number the problems appropriately. \vspace{0.1in} \begin{enumerate} \item Explain why it is not appropriate to perform an ANOVA for Problem 15.4 on page 676 of your textbook. %\item For Problem 15.4 on page 676 of your textbook, perform an ANOVA %for the first $k=3$ groups: ``beer only", ``wine only", ``spirits %only." Write out all 6 steps of the ANOVA hypothesis test for these %three groups. Include a one-way ANOVA table to summarize your %calculations. Assume that each group is normal, but confirm that %the constant variance assumption is violated. Perform the ANOVA %anyways. \item \label{jobsharing} Do problem 15.6 on page 677. \begin{enumerate} \item Show your work for the degrees of freedom calculations! \item Perform the test by answering the questions below, which you will need to do by hand since we do not have the data. Assume that the assumptions for the ANOVA are met. You will perform the follow-up test in \#\ref{followup}. \begin{enumerate} \item \underline{Hypotheses}: \item \underline{Test statistic value}: \item \underline{Distribution of the test statistic}: \item \underline{$p$-value}: \item \underline{Decision} at $\alpha=.05$: \item \underline{Conclusion}: \end{enumerate} \item When computing $MSE$, show your work! \item Summarize all of your previous calculations by filling in an ANOVA table as in the Chapter 15 handout. Label and reference this table from the body of your report. \item Fill in the following table, label it, and include it in your report. \begin{tabular}{||c|c|c||} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... parameter & estimate & Explanation of the parameter in English in terms of the problem\\\hline Estimate for $\mu_1$ & & \\\hline Estimate of $\sigma$ & &\\ \hline \end{tabular} \end{enumerate} \item \label{followup} Do problem 15.22 on page 685. \begin{enumerate} \item Why is it appropriate to perform a follow-up analysis to \#2: problem 15.6 on page 677? \item Perform a Tukey's follow-up test using a family $\alpha=.05$, which you'll need to do by hand since we do not have the data. Use R's qtukey(.95,nmeans = \underline{\hspace{.5in}},df=\underline{\hspace{.5in}}) to get the critical value for each of the CI's. Show your work and put the results in a table. \item Indicate all significant differences between the means. What is your conclusion in terms of the problem? \end{enumerate} \newpage \item A National Public Radio broadcast on February 19, 2007 described how researchers at the University of California at Davis conducted an experiment to test their hypothesis that gratitude is the key to happiness. Three groups kept daily journals. The first group kept a ``Gratitude Journal", where the subjects were instructed to write about something that they were grateful for in each entry. The second group kept a ``Stress Journal," where the subjects were instructed to focus on something that they were stressed about in each entry. The last group kept a ``Regular Journal" where the subjects were not instructed to focus on anything in particular. The average amount of sleep (in hours) for each group was compared. \bigskip The ``gratitude.txt" data file can be found on the STAT401 website. Use R to complete this problem. Attach all R commands and R output used in an appendix. Label all necessary Figures and Tables and refer to these figures and tables from the text of your report. \begin{enumerate} \item Assuming that the experiment was a completely randomized design, how must the experiment have been conducted? \item Give the value of $x_{2,13}$ and explain what this value is in terms of the problem. \item Construct side-by-side boxplots for a visual comparison of the groups. Include a figure of the plot in your report. \item Give the null and alternative hypotheses to be tested by a one-way ANOVA. \item Fit the ANOVA model. Include a table that displays the one-way ANOVA table as in the Chapter 15 notes. \item Check the assumptions. \begin{enumerate} \item Does the evidence suggest that the data for each group are not normal? Include the a normal probability plot, a smoothed histogram of the studentized residuals, and the correlation test of the studentized residuals to justify your answer. Include the plots in a figure in your report. \item Does the assumption of constant variance hold? Why or why not? \end{enumerate} \item Give the distribution of the $F$ statistic assuming the ${\rm H}_0$ is true. \item What is your decision regarding ${\rm H}_0$? \item Assume that the individuals in the experiment are not from a random sample. Second, assume that that the experiment was a CRD. Is it appropriate to make a cause and effect conclusion? Why? To what population can these results be inferred? Give a conclusion in terms of the problem, and mention cause-and-effect and the appropriate population. \end{enumerate} \item \begin{enumerate} \item Is it appropriate to conduct a follow-up test for the gratitude ANOVA? Why or why not? \item If it is appropriate, compute 95\% Tukey confidence intervals for the pairwise differences between means. Provide a table of the results of the simultaneous confidence intervals as in the Chapter 15 notes. \item Summarize the results of this study (i.e. give a conclusion). In particular, is there a treatment group which has \underline{significantly} the longest night's sleep on average? Justify your answer. \item Give estimates for $\mu_3$ and $\sigma$, and interpret each of these values in terms of the problem. Give your answers in a table as in \#\ref{jobsharing}(e). \end{enumerate} \vspace{0.1in} \end{enumerate} \end{document}