\documentclass[11pt,titlepage]{article} \usepackage{amsmath} \usepackage{graphicx} \usepackage{verbatim} \allowdisplaybreaks \jot=.2in \pagestyle{plain} \setlength{\topmargin}{-0.5in} \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} %\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 4: Probability}\\ \vspace{0.05in} {\bf STAT 401: Spring 2007} \hspace{2in}{\it Due Friday, February 23} \vspace{.1in} \noindent For the probability calculations in this project, you must show your work. Your grade will be determined by how well you answer the questions and by the professionalism and clarity of our write-up. \begin{enumerate} \item \label{beautiful} Use the November 2006 {\em Discover} article ``Do Beautiful Parents have more daughters" from the STAT401 web-site to answer the following questions: \begin{enumerate} \item Is this an observational study or an experiment? Explain. \item Who are the individuals? \item The University of North Carolina web site http://www.cpc.unc.edu/projects/addhealth states that ``The National Longitudinal Study of Adolescent Health (Add Health) is a nationally representative study [of the US population]." Do you suspect that the individuals in this study are a random sample from the US population? Why or why not? \item What is the explanatory variable and what is the response variable? \item Give the sample size of the data for this study. \item This study is the work of evolutionary psychologist Satoshi Kanazawa of the London School of Economics. Comment on his comment that ``by marrying a beautiful spouse you are slightly increasing the chance that you'll have a daughter." Is this a valid conclusion? Why or why not? \item What type of inference can you make, if any, to the US population? Can a cause and effect conclusion be made? \item Indicate a confounding variable that may explain Kanazawa's conclusions. That is, what extraneous variable could account for a higher proportion of daughters among the beautiful people in this sample? \end{enumerate} \item \label{stratsamp} Suppose that we wish to repeat the ``Beautiful Parent" study on a sample from the US population. Regardless of whether this is feasible, assume that from the beautiful parents in the US, we take a SRS of size 4. From the ``non-beautiful" parents in the US, we take a SRS of size 4. Suppose that the percentages given in the article in problem \#\ref{beautiful} are truly representative of all U.S. adults, so that: \begin{center} \begin{tabular}{rcc} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $P($ daughter first $|$ beautiful parent ) &=& .56\\ $P($ daughter first $|$ non-beautiful parent ) &=& .48\\ \end{tabular} \end{center} \begin{enumerate} \item Give the name of the sampling design used to get the sample of 4 beautiful parents and 4 non-beautiful parents. \item What is the probability that none of the 4 beautiful parents has a daughter as their first child? \item What is the probability that at least one of the 4 non-beautiful parents has a daughter as their first child? \item What is the probability that at least one of the 4 of the beautiful parents has a daughter as their first child AND that at least one of the 4 non-beautiful parents does not have a daughter as their first child? \end{enumerate} \newpage \item \label{mosquito} Use the January 2006 {\em Discover} article ``Malaria Parasite Makes Humans Smell More Attractive to Mosquitoes" from the STAT401 web-site to answer the following questions: \begin{enumerate} \item \label{tentprob} What is the probability that out of the 100 mosquitos in the experiment, that a randomly selected mosquito flies towards kids carrying gametocytes? \item \label{cond} Sixty seven of the 100 mosquitos in the experiment fly towards kids with gametocytes. When randomly selecting ({\bf without replacement}) two mosquitos out of these 100 mosquitos, what is the probability that BOTH mosquitos go to the tent with kids carrying gametocytes? \item \label{ind} Suppose that the probability given in problem \#\ref{tentprob} is true for all mosquitos. Two friends are out camping in separate tents. One is carrying gametocytes and the other is not. Two mosquitos find the two friends in the middle of the night. Assuming independence, what is the probability that BOTH of the mosquitos fly towards the infected kid? \item Your answers for problems \#\ref{cond} and \ref{ind} should be almost identical. Why is this case? What property about simple random samples from a finite population support your answer? \end{enumerate} \item Do Exercise 7.28 on page 316 of your textbook. \item To determine if the results from the mosquito experiment in problem \#\ref{mosquito} are repeatable, many more experiments will be performed. For each experiment, the sample proportion of the 100 mosquitos who fly towards the tent which holds the kids infected with gametocytes will be calculated. So the sample proportion $p$ has a distribution, called a {\bf sampling distribution}, and which is approximately normal $$p~\dot \sim ~N(\mu=\frac{2}{3},\sigma=\frac{\sqrt{2}}{30})$$ when the data is from a large random sample. (This fact follows from the Central Limit Theorem). \begin{enumerate} \item What is the probability that in an experiment, the sample proportion of mosquitos who fly towards the infected tent is over 70\%? {\em Hint:} Take a square root in R by using {\bf sqrt( )}. \item What is the probability that in an experiment, the sample proportion of mosquitos who fly towards the infected tent is exactly $.6$? \end{enumerate} \end{enumerate} \end{document}