\documentclass[11pt,titlepage]{article} \usepackage{amsmath} \usepackage{graphicx} \allowdisplaybreaks \jot=.2in \pagestyle{plain} \setlength{\topmargin}{-.5in} % \setlength{\footheight}{0 in} \setlength{\textheight}{9.4in} \setlength{\oddsidemargin}{0.1in} \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} \begin{document} \begin{center} {\heada Project 7 - Hypothesis Testing}\\ {\headb Statistics 401: Spring 2007}\\ {\it Due Monday, April 2} \end{center} \noindent Turn in your solutions to this project in a typed report. When a problem requires calculations, do these calculations ``by hand" and include this work in your report. Use a significance level of $\alpha$=.05 if one is not given. \begin{enumerate} \item Do problem 10.4 on page 409. \item Read the January 2007 {\em Discover} article ``Gay Influence Found." \begin{enumerate} \item What is the ``base rate" for homosexuality in men with no older brothers? \item \label{gayhyp} Write out the appropriate null and alternative hypotheses that correspond to testing whether the proportion of men (with one older brother) who are gay is larger than the the base rate. \item With respect to the hypotheses in \eqref{gayhyp}, describe Type I and Type II errors in the context of this problem. \item Test the hypotheses you give in \eqref{gayhyp}. Of the 944 men in the Ontario study, let us assume that $n=283$ had exactly one older brother (Aside: $378$ of 944 had older brothers and $3/4(378)\approx 283$; source: {\small http://www.narth.com/docs/domothers.html}). From the article, assume that a sample proportion of $p=.052$ of men (who have one older brother) are gay. \begin{enumerate} \item Write out each of the six steps of this hypothesis test. You already gave the hypotheses in \eqref{gayhyp}. Show the calculation of the test statistic and p-value. Clearly indicate how you are checking assumptions, what decision you are making, and make clear your conclusion. \item What is the distribution of the test statistic in the test? \item Use R's prop.test function to confirm the ``by hand" calculations that you got and include the R code and R output in an Appendix in your report. \end{enumerate} \item Does the article describe an observational study or an experiment? Explain. \item Is the conclusion ``having an older brother causes men to be gay more often" valid? Explain. \end{enumerate} \item Read the article ``Sleep Deprived Teens," available at the STAT401 web site, which appeared in the {\em Bozeman Daily Chronicle} on March 29, 2006. This article refers to a study, released in 2006, by the National Sleep Foundation (if your interested, check out {\small http://www.sleepfoundation.org/site/c.huIXKjM0IxF/b.2417141/k.C60C/Welcome.htm}) \begin{enumerate} \item Who are the individuals in this study? \item What variable is being measured on each individual? \item Give the sample space for this variable. \item Is the variable discrete or continuous? Explain. \item According to the National Center on Sleep Disorders Research at the National Institutes of Health, what is the minimum amount of sleep needed by school-aged children and teens? \item \label{sixhyp} Write out the appropriate null and alternative hypotheses that correspond to testing whether sixth graders are getting less sleep than recommended by the National Center on Sleep Disorders Research. \item With respect to the hypotheses in \eqref{sixhyp}, describe Type I and Type II errors in the context of this problem. \item \label{sixmean} According to the Sleep Foundation's poll, how much sleep, on average, are sixth graders getting each night? \item \label{sixstd} To estimate the sample standard deviation of nightly sleep for the sixth grade respondents who took part in the study, consider the formula $s=\frac{\mbox{range}}{4}$ from Chapter 9. If the range in values for the sample is between 2 and 12, what is your estimated sample standard deviation? \item Assume that there are $n=228$ sixth graders who took part in the Sleep Foundation's study. (Aside: I took the 1602 adolescent respondents reported in the article, and divided by 7 since there are seven age groups from 11-17). Use the sample mean and sample standard deviation from \eqref{sixmean} and \eqref{sixstd} to test whether sixth graders are getting less sleep than recommended by the National Center on Sleep Disorders Research. Write out each of the six steps of this hypothesis test. You already gave the hypotheses in \eqref{sixhyp}. Show the calculation of the test statistic and $p$-value. Clearly indicate how you are checking assumptions, what decision you are making at $\alpha=.05$, and make clear your conclusion. \item What was the distribution of the test statistic that you used in the test? \end{enumerate} \item Do problem 10.58 on page 440. The data is on the CD which came with your textbook. Of course, you always type the data in like I did: {\small {\small{\small \begin{verbatim} nitrate=c(7251,6871,9632,6866,9094,5849,8957,7978,7064,7494,7883,8178, 7523,8724,7468) \end{verbatim} }}} \begin{enumerate} \item Even though we do not have a sample size over 30, we can still safely use a $t$-test. Use the techniques from Chapter 7 to explain why. \item Perform all six steps of the test ``by hand." \item Use R's t.test function to confirm the ``by hand" calculations that you got and include the R code and R output in an Appendix in your report. \end{enumerate} \item Do problem 10.60abcde on page 448 (i.e. skip (f) and (g)). You may make the sketch required in (d) by hand and turn it in with your report. For help with (e), see Example 10.17 on page 443. \item For problem 10.60 on page 448, calculate the power of the test when $\mu=153$. \item Do problem 10.62 on page 449. \begin{enumerate} \item Use R's prop.test to do the test and include the R code and R output in the Appendix. \item {\em Hint}: See Example 10.18 on page 445. \end{enumerate} \end{enumerate} \end{document}