\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{graphicx} \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 \begin{document} \begin{center} {\heada PROJECT 2 SOLUTIONS}\\ {\headb Statistics 401: Spring 2007}\\ \end{center} \begin{enumerate} \item (5 pts) What's more important, the public's right to know the details of the slaying of Jason Wright or the fair-trial rights of two former Montana State University athletes accused of killing him? According to attorneys for the accused, ``a random phone survey of 401 self-described eligible jurors" from Gallatin County were asked whether they had heard of the homicide case (95\% said they had) and whether they thought the athletes were guilty (37\% said guilty) (Bozeman Daily Chronicle, October 27, 2006). \begin{enumerate} \item This survey amounts to an observational study since no treatment was applied to the respondents \item This ``random phone survey of 401" Gallatin county residents is fraught with potential selection and non-response bias. Selection bias is possible since citizens without telephones were not considered for the sample. Non-response bias is possible since respondents may have not answered their phone or hung up when contacted by the surveyors. \item \label{StratRS} To ensure that the sample is representative of both MSU-affiliates and those not affiliated with MSU, one may stratify the Gallatin population into two strata: ``MSU-affiliated" and ``not-MSU-affiliated." Now, select a SRS from each of these strata. \item The sample design described in (\ref{StratRS}) is a \underline{stratified random sample}. \end{enumerate} \item (10 pts) Much research has purported to prove that owning a pet makes for a healthier and happier human. Suppose you want see whether there is a ``pet effect" on the health of students at MSU. \begin{enumerate} \item A sampling design which will gather a sample representative of the MSU student body would be a SRS: Acquire a list of all students currently enrolled at MSU, then randomly select 100 students. \item One experimental design to test for the ``pet effect" is as follows. Take physiological measurements from all 100 students. Physiological measures of health are taken, for example: blood pressure, pulse, weight, blood work, urinalysis. After this initial measurement, randomly assign 50 of the students to the ``pet group." The other 50 students are in the control group. The students in the ``pet group" are required to spend at least 4 hours each week actively interacting with animals. For example, walking a dog, riding a horse, or playing with a cat. The students in the control group are forbidden from having interaction with animals. After a month, the physiological measures of health are taken again. \item To directly control for the effect of weather on health, all students go through the experiment for the same month. To directly control for the potential confounding effects of different types of pets, only dog, cat and horse interactions are acceptable for the ``pet group." We could be more restrictive and directly control for ``pet time" by requiring that all subjects in the pet group spend exactly 4 hours each week in pet interactions. \item Random assignment into each group will tend to yield subjects in each group who have the same variability of extraneous variables like age, socioeconomic status, and gender. Thus, we will be able to make cause-and-effect conclusions. \item Having fifty students in each group will allow the researchers to accurately estimate how the control and ``pet treatments" affect the physiological responses of the subjects. \end{enumerate} \item (1 pt, Exercise 2.10 on page 37) The ``danger" of using a sample of volunteers is that such a sample is not representative of the population from which these volunteers come. Instead, they are representative of people who tend to volunteer for experiments, which is likely a different population than the one which the researcher is interested in. \item (2 pts) Exercise 2.30 on page 42: \begin{enumerate} \item This is an example of an observational study. Thus, cause-and-effect conclusions are inappropriate.%% \item This survey was completed by volunteers who found out about the survey at DietSmart.com. Thus, this sample is not necessarily representative of any large US or TV watching population, except for perhaps the population DietSmart.com users. \end{enumerate} \item (8 pts) Exercise 2.48, page 53: \begin{enumerate} \item The population in this experiment is composed of families with elevated blood lead levels. \item Random assignment of the families in the sample to each of the two groups helps make sure that some value of some extraneous variable does occur more often in one group than the other. Thus, we will be able to make cause-and-effect conclusions. \item The control group in this study, the families who were merely given a pamphlet, allows the researchers to compare the effect of house cleaning to homes where cleaning was not performed. \item The treatment is housecleaning, with two levels: cleaning and not-cleaning. There are two response variables: lead levels in the children's blood and lead levels in the household dust. \item Possible extraneous variables are the types of cleaning supplies and how often the cleaning is performed. Direct control can be used to control for these: all families get the same supplies, and instructed in how often to clean. \item Another extraneous variable is age of the child, since younger children may crawl and chew on contaminated surfaces more often than older children. This extraneous variable can be controlled by blocking the children by age. The youngest children who do not crawl could be one block, young crawlers and walkers another block, and then six year olds and up a third block. Now, the treatments are randomly assigned within each of these blocks. \item The Experimental Design in (f) is called a Randomized Block Design. \end{enumerate} \item (1 pt) If one is interested in making an inference from a sample to a population, then random sampling is most important. \item (1 pt) If one wishes to make a causal inference then random assignment is most important. \item (1 pt) A cluster random sample is more efficient than a simple random sample when the population is ``naturally" divided into roughly equally sized groups, each of which is representative of the total population. \item (1 pt) A systematic random sample is more efficient than a simple random sample if the population list is either not related to the response variable, or if it is related, the relationship is not cyclic. \end{enumerate} \end{document}