Chapter 13Two Dichotomous Variables13.1 Populations and SamplingA population model for two dichotomous variables can arise for a collection of individuals—a finite population—or as a mathematical model for a process that generates two dichotomousvariables per trial.Here are two examples.1. Consider the population of students at a small college. The two variables are sex with possi-ble values female and male; and the answer to the following question, with possible valuesyes and no.Do you usually wear corrective lenses when you attend lectures?2. Recall the homework data on Larry Bird shooting pairs of free throws. If we view each pairas a trial, then the two variables are: the outcome of the first shot; and the outcome of thesecond shot.We begin with terminology and notation. With two responses per subject/trial, it sometimeswill be too confusing to speak of successes and failures. Instead, we proceed as follows.• The first variable has possible values A and Ac.• The second variable has possible values B and Bc.In the above example of a finite population, A could denote female; Accould denote male; B coulddenote the answer ‘yes;’ and Bccould denote the answer ‘no.’ In the above example of trials, Acould denote that the first shot is made; Accould denote that the first shot is missed; B coulddenote that the second shot is made; and Bccould denote that the second shot is missed.It is important that we now consider finite populations and trials separately. We begin withfinite populations.149Table 13.1: The Table of Population Counts.B BcTotalA NABNABcNAAcNAcBNAcBcNAcTotal NBNBcNTable 13.2: Hypothetical Population Counts for Study of Sex and Corrective Lenses.Yes (B) No (Bc) TotalFemale (A) 360 240 600Male (Ac) 140 260 400Total 500 500 100013.1.1 Finite PopulationsTable 13.1 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 subscriptsrepresents the total number of subjects in the population. An N with subscripts counts the numberin the population with the feature(s) given by the subscripts. For example, NABis the number ofpopulation members with variable values A and B; NAcis the number of population members withvalue Acon the first variable; i.e. for this, we don’t care about the second variable.Note also that these guys sum in the obvious way:NA= NAB+ NABc.In words, if you take the number of subjects whose variable values are A and B; and add to it thenumber of subjects whose variable values are A and Bcthen you get the number of subjects whosevalue on the first variable is A.It might help if we have some hypothetical values for the population counts. I put some inTable 13.2.If we take the table of population counts and divide each entry by N, we get the table ofpopulation proportions. I do this in Tables 13.3 and 13.4, for the general notation and our particularhypothetical data.Now we must face a notational annoyance. Consider the symbol pABwhich equals 0.36 forour hypothetical population. This says, literally, that the proportion of the population memberswhose variable values are A and B is 0.36; the proportion of our hypothetical students who areboth female and would answer ‘yes’ is 0.36. We use the lower case ‘p’ for this notion, because weuse lower case p’s to represent population proportions.But consider our most commonly used Chance Mechanism when studying a finite population:Select a member of the population at random. For this CM it is natural to view pAB= 0.36 as the150Table 13.3: The Table of Population Proportions.B BcTotalA pABpABcpAAcpAcBpAcBcpAcTotal pBpBc1Table 13.4: Hypothetical Population Proportions for Study of Sex and Corrective Lenses.Yes (B) No (Bc) TotalFemale (A) 0.36 0.24 0.60Male (Ac) 0.14 0.26 0.40Total 0.50 0.50 1.00probability of selecting a person who is female and would answer ‘yes.’ But we tend to use uppercase ‘P’ to denote the word probability. Hence, it is more natural to write this as P (AB) = 0.36.The point of all this is ...? Well, in this chapter pAB= P (AB) and the one we use will dependon whether we feel it is more natural to talk about proportions or probabilities.13.1.2 Conditional ProbabilityConditional probability allows us to investigate one of the most basic questions in science: Howdo we make use of partial information?Consider again the hypothetical population presented in Table 13.2 and 13.4. Consider the CMof selecting one person at random from this population. We see that P (A) = 0.60. In words, theprobability is 60% that we will select a female. But suppose we are given the partial informationthat the person selected answered ‘yes’ to the question. Given this information, what is the proba-bility the person selected is a female? We write this symbolically as P (A|B). The literal readingof this is: The probability that A occurs given that B occurs. How do we compute this?We reason as follows. Given that B occurs, we know that the selected person is among the500 in column B of Table 13.2. Of these 500 persons, reading up the column we see that 360 ofthem are female. Thus, by direct reasoning P (A|B) = 360/500 = 0.72, which is different thanP (A) = 0.60. In words, knowledge that the person usually wears corrective lenses in lecture,increases the probability that the person is female.We now repeat the above reasoning, but using symbols instead of numbers. Refer to Table 13.1.Given that B occurs, we know that the selected subject is among the NBsubjects column B. Ofthese NBsubjects, reading up the column we see that NABof them have property A. Thus, bydirect reasoning we obtain the following equation.P (A|B) = NAB/NB. (13.1)151Table 13.5: Conditional Probabilities of the B’s Given the A’s in the Hypothetical Study of Sexand Lenses.Yes (B) No (Bc) TotalFemale (A) 0.60 0.40 1.00Male (Ac) 0.35 0.65 1.00Unconditional 0.50 0.50 1.00Now, this is a perfectly good equation, relating the conditional probability of A given B to popu-lation counts. Most statisticians, however, prefer a modification of this equation. On the right sideof the equation divide both the numerator and denominator by N. This, of course, does not changethe value of the right side and has the effect of converting counts to probabilities. The result isbelow, the equation which is usually referred to as the definition of
View Full Document
Unlocking...