Chapter 16 One Population with Two Dichotomous Responses This chapter focuses on a new idea Thus far in these notes a unit subject trial has yielded one response In this chapter we consider situations in which each unit yields two responses both dichotomies Later in these Course Notes we will examine situations in which both responses are numbers and the mixed situation of one response being a number and the other a dichotomy Multi category responses can be added to the mix but with one exception we won t have time for that topic Sometimes the examples of this chapter will look very much like our examples of Chapter 15 Other times it will be natural to view our two responses as paired data As a result you need to be extra careful as you read through this material 16 1 Populations Structure Notation and Results A population model for two dichotomous responses can arise for a collection of individuals a finite population or as a mathematical model for a process that generates two dichotomous responses per trial Here are two examples 1 Consider the population of students at a small college The two responses are sex with possible values female and male and the answer to the following question with possible values yes and no Do you usually wear corrective lenses when you attend lectures 2 Recall the data on Larry Bird in Chapter 15 presented in Table 15 16 on page 374 We view his shooting a pair of free throws as a trial with two responses the outcome of the first shot and the outcome of the second shot Recall that I treated the Larry Bird data as Chapter 15 data i e independent random samples from two Bernoulli trials processes Later in this chapter we will view his results as paired data Both 387 perspectives are valid but it will require some care for you to be comfortable with such moving between models Also my example of sex and lenses can be viewed as Chapter 15 data but I would find it awkward to refer to it as paired data We begin with some notation With two responses per unit sometimes it would be confusing to speak of successes and failures Instead we proceed as follows The first response has possible values A and Ac Note that Ac is read A complement or not A The second response has possible values B and B c Note that B c is read B complement or not B In the above example of a finite population A could denote female Ac could denote male B could denote the answer yes and B c could denote the answer no In the above example of trials A could denote that the first shot is made Ac could denote that the first shot is missed B could denote that the second shot is made and B c could denote that the second shot is missed In fact with data naturally viewed as paired such as Larry Bird s shots it is natural to view A B as a success on the first second response and Ac B c as a failure on the first second response It will be easier if we consider finite populations and trials separately We will begin with finite populations 16 1 1 Finite Populations Table 16 1 presents our notation for population counts for a finite population Remember that in practice only Nature would know these numbers This notation is fairly simple to remember all counts are represented by N with or without subscripts The symbol N without subscripts represents the total number of members of the population An N with subscripts counts the number of members of the population with the feature s given by the subscripts For example NAB is the number of population members with response values A and B NAc is the number of population members with value Ac on the first response i e for this we don t care about the second response Note also that these guys sum in the obvious way NA NAB NABc In words if you take the number of population members whose response values are A and B and add to it the number of population members whose response values are A and B c then you get the number of population members whose value on the first response is A It might help if we have some hypothetical values for the population counts I put some in Table 16 2 If we take the table of population counts and divide each entry by N we get the table of population proportions or probabilities see the discussion in the next paragraph I do this in Tables 16 3 and 16 4 for the general notation and our particular hypothetical data Now we must face a notational annoyance Consider the number 0 36 in Table 16 4 derived from our hypothetical population counts for the sex and lenses study There are two ways to interpret this number First it is the proportion of the population who have value A female on the 388 Table 16 1 The table of population counts B A NAB c A NAc B Total NB Bc NABc NAc Bc NBc Total NA NAc N Table 16 2 Hypothetical population counts for the study of sex and corrective lenses Female A Male Ac Total Yes B 360 140 500 No B c 240 260 500 Total 600 400 1000 Table 16 3 The table of population proportions lower case p s with subscripts or probabilities upper case P s followed by parentheses B A pAB P AB Ac pAc B P Ac B Total pB P B Bc Total c pA P A pABc P AB pAc Bc P Ac B c pAc P Ac 1 pBc P B c Table 16 4 Hypothetical population proportions or probabilities for the study of sex and corrective lenses Female A Male Ac Total Yes B 0 36 0 14 0 50 389 No B c 0 24 0 26 0 50 Total 0 60 0 40 1 00 first response and value B yes on the second response From this perspective it is natural to view 0 36 as pAB because we use lower case p s for population proportions with a subscript if needed to clarify which one But consider our most commonly used chance mechanism when studying a finite population Select a member of the population at random For this chance mechanism it is natural to view 0 36 as the probability of selecting a person who is female and would answer yes We use upper case P to denote the word probability Hence it is also natural to write P AB 0 36 The point of all this is Well in this chapter pAB P AB and pA P A and so on the one we use will depend on whether we feel it is more natural to talk about proportions or probabilities 16 1 2 Conditional Probability Conditional probability allows us to investigate one of the most basic questions in science How do we make use of partial information Consider …
View Full Document
Unlocking...