The Completely Randomized Design with a Numerical ResponseComparative StudiesDawn's Study; Various ToolsA Connection Between Pictures and NumbersThe Standard DeviationComputingSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 1The Completely Randomized Design with aNumerical ResponseA Completely Randomized Design (CRD) is a particular type of comparative study. The worddesign means that the researcher has a very specific protocol to follow in con ducting the study.The word randomized refers to the fact that the p rocess of randomization is part of the design.The word completely tells us that complete randomization is required, in contrast to some form ofincomplete randomizati on, such as the rando mized pairs design we will study later in these notes.What is a numerical response? See t he following section.1.1 Comparative StudiesSo, what is a comparative study? Let’s look at its two words, beginning with the word study.According to dictionary.com (http://dictionary.reference.com) the fifth definit ionof study is:Research or a detailed ex amination and analysis of a subject, ph enomenon, etc.This reasonably well fits what I mean by a study. Next, again according to dictionary.com, the firstdefinition of compare (the root word of comparative) i s:To examine (two or more objects, ideas, people, et c.) in order to no te similarities anddifferences.Because of time limitations , for the most part in these notes we will restrict attention to exactlytwo, as opposed to two or more, things being compared.In the examples of the first two chapters, Dawn wants to compare two flavors of cat treats;Kymn wants to compare two settings on an exercise machine; Sara wants to compare two golfclubs; and Cathy wants to compare two routes for jogging. In the practice and homework problemsof these first two chapters you will be introduced to several other comparative studies. Indeed, alarge majority of the chapters in this book are devoted to comparative studies. Why? Two reasons:31. Comparative s tudies are extremely important in science.2. The discipline of Statistics includes several good ideas and methods that help scientists per-form and analyze comparative studies.Next, some terminology: the two things being compared are called the two levels of the studyfactor. For our examples we have the following study factors and levels.• Dawn’s study factor is the flavor of the cat treat, wit h levels equal to chicken-flavored andtuna-flavored.• For Kymn ’s study, her exercise apparatus is called an ergometer which requires two choicesby its operator. Kymn’s study factor is the machine setting with first level defined as smallgear with the vent closed; her second level is large gear with the vent open.• Sara’s st udy factor is the golf club she used with levels 3-Wood and 3-Iron,• Cathy’s study factor is the route for her on e mile run with levels at her local high school andthrough the park.The remaining components of a comparative study are:• The units that provide the researcher with information.• The response wh ich is the particular information from the unit of interest to the researcher.• One of the following methods:– The researcher identifies each unit wit h its level of the study factor, or– The researcher assigns each unit to a level of the study factor.I cho ose to introduce you to the units and the response for each o f our studies in the vario u s sectionsbelow. I do want to s ay a bit about the method in the last bullet of the above list.Examples of identifying, sometimes called classifying, are: comparing men and women; com-paring old people with young people; comparing residents of Wyoming with residents of Wiscon-sin. Our development of randomization-based inference—beginning wi th Chapter 3 —in Part I ofthese notes, will not consider any studies that involve identifying units wit h levels.As the last sentence implies, randomization-based inference is rest ricted to studies in whichthe researcher has the o ption of assigning uni ts to levels. In fact, as the name sug gests, we attendonly t o those stu dies in whi ch the researcher exercised the option of assignment by using a methodcalled randomization. You will learn about randomization in Chapter 3.41.2 Dawn’s Study; Various ToolsDawn completed my class several years ago. In this section you will be introduced to Dawn’sproject.The cho ice of a project topic, or, indeed, any research, s hould begin with a curiosity abou t howthe world operates. Here is Dawn’s motivation for her study, as she wrote in her report.I decided to study my cat’s preference to two different flavors of the same brand-namecat treats. I was interested in this s tudy because I figured that Bob, my cat, wouldprefer the tuna-flavored treats over the chicken-flavored because Bob absolutely lovescanned tuna with a p assion. My interest came when I looked at the ingredients on thetwo labels. I noticed that the percentage of real chicken in the chicken-flavored treatswas larger than the percentage of real tuna in the tu na-flavored t reats.Thus, Dawn had a p retty good idea of what she wanted to study. Her next step was to operationalizethe above notions in to the s tandard language of a comparative study. We know her study factorand its levels from the previous section. Now, we need to specify: the definition of the units andthe response.A unit consisted of presenting Bob w ith a pile of ten treats, all of th e sam e flavor. The flavor ofthe treats in the pile determined the level of the study factor for that unit: either chicken (level 1)or t una (level 2). The response is the number of treats that Bob consumed in five min utes.The technical term unit is not very descriptive. In thi s course there will be two types o f units:trials and subjects. Dawn’s units are trials. Essentially, we have trials if data collection involvesdoing something repeatedly. In Dawn’s case this something is setting out a pile of ten treats. Andthen doing it again the next day. By contrast, in many st udies the different u nits are differentpeople. When the u nits are people, o r other distinct objects, we call th em subjects. As we willsee later in these notes, sometimes the distincti on between trials and subjects is b lurry; fortunately,this doesn’t matt er.Dawn decided to collect data for 20 days, with one trial per day. Dawn further decided tohave 10 days assig ned to each level of her study factor. (Sometim es, as
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