Comparing Two Numerical Response Populations: Independent SamplesNotation and AssumptionsCase 1: The Slutsky (Large Sample) Approximate MethodCase 2: Congruent Normal PopulationsCase 3: Normal Populations with Different SpreadMiscellaneous ResultsAccuracy of Case 2 Confidence LevelsSlutsky; SkewnessComputingComparison of MeansSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 19Comparing Two Numerical ResponsePopulations: Independent SamplesChapter 19 is very much like Chapter 15. The major—and obvious—difference is that in theearlier chapter the response was a dichotom y, but in this chapter the respon se is a number. Ifyou revisit the material on the fo ur types of studies in Section 15.2, you can see that the fact thatthe respon se was a dichotomy is i rrelevant. In oth er words, everything you learned earlier abouthow the interpretation of an analysis depends on the type of study remains true in this chapter.In particular, for an observational study, if you conclude that numerical populations di ffer, thenyou don’t know—based on the statistical analysis—why they differ. On the other hand, for anexperimental study, if you conclude th at numerical populations differ, then you may conclu de t hatthere is a causal link between the treatment and response.In addition, the meaning of independent random sa m ples for the different types of studiesremains the same in the current chapter. There is even an extension o f Simpson’s Paradox for anumerical respon se, but t ime limitations will prevent me from covering this t opic.It is also true that Chapter 19 builds on the work of Chapters 1 7 and 18. In particul ar, recallthat in Chapter 17 you learned that the population for a numerical response is a picture and thekind of picture depends on w hether the response is a count or a measurement.19.1 Notation and Assump tionsThe researcher has two populati o ns of interest. The methods of Chapters 17 and 18 may be usedto study the population s separately. In this chapter, you will learn how to compare the populations.• Population 1 has mean µ1, variance σ21and standard deviation σ1.• Population 2 has mean µ2, variance σ22and standard deviation σ2.I realize that specifying both the variance and standard deviation is redundant, but it will proveuseful to have bot h for some of the formulas we develop.We will consider procedures that compare the populations b y comparing their means.495• We assume that we will observe n1i.i.d. random variables from population 1, denoted by:X1, X2, X3, . . . , Xn1.These will be summarized by their mean¯X, variance S21and standard deviation S1. Theobserved values of these various random variables are denoted by:x1, x2, x3, . . . , xn1, ¯x, s21and s1, respectively.• We assume that we will observe n2i.i.d. random variables from population 2, denoted by:Y1, Y2, Y3, . . . , Yn2.These will be summarized by their mean¯Y , variance S22and standard deviation S2. T heobserved values of these various random variables are denoted by:y1, y2, y3, . . . , yn2, ¯y, s22and s2, respectively.• We assume that the two samples are independent .I apologize for t he cumbersome and confusing notation. In particular, in my µ’s, σ2’s, n’s S2’s, andso on, I use a sub script to d eno te the population, either 1 or 2; this is very user-friendly. You needto rememb er, however, that the random variables, data and s ome summaries from population 1 aredenoted by x’s and the corresponding not ions from population 2 are denoted by y’s. There is a longtradition of doing things this way in introductory Statistics. While i t is confus ing, its one virtue isthat it allows you to avoid double subscripts until you take a more advanced Statistics class.(Enrichment: Here is the problem with double subscripts—well, o ther than th e o bvious prob-lem that they s ound, and are, complicated. If I write x123does it mean:• Observation number 123 from one so urce of data?• Observation 23 from populati on 1? or• Observation 3 from population 12?This could be made clear with commas; use x1,23for the second answ er above and x12,3for thethird answer. The only problem is: In my experience, statisticians and mathematicians don’t wantto be bothered with commas!)The methods introduced in this chapter involve comparing the populations by comp arin g theirmeans. For tests of hypotheses, this translates to t he null h ypothesis being µ1= µ2, or, equiva-lently, µ1− µ2= 0. For estimation, µ1− µ2is the feature that will be estimated with confidence.Our point estimator of µ1− µ2is¯X −¯Y . There is a Central Limit Theorem for this problem,just as there was in Chapter 17. First, it shows us how to standardi ze our est imator:W =(¯X −¯Y ) − (µ1− µ2)q(σ21/n1) + (σ22/n2). (19.1)496Second, it states that we can app roximate probabilities for W by using the N(0,1 ) curve and that inthe limit, as both sample sizes become larger and larger, the approximations are accurate.In order to obtain formul as for estimation and testing, we need to eliminate the unknown pa-rameters in the denominator of W, σ21and σ22. We also will need to decide what to use for ourreference curve: the N(0,1) curve of the Central Lim it Theorem and Slu tsky or one of the t-curvesof Gosset.Statisticians suggest three meth ods for handl ing these two is sues, which I refer to as Cases 1,2 and 3. I won’t actuall y show you Case 3 because I believe that it n early worthless to a s ci entist;I will explain why I feel this way.We will begin with Case 1; I will follow th e popular terminology and call this the large sampleapproximate method.19.2 Case 1: The Slu t s ky (Large Sample) Approximate MethodThis method comes from Slutsky’s Theorem. In Equatio n 19.1 for W , replace each populationvariance by its sample variance. The resultant rando m variable is:W1=(¯X −¯Y ) − (µ1− µ2)q(S21/n1) + (S22/n2). (19.2)(Note that I have placed the subs crip t of ‘1’ on W to remind you that this is for Case 1.) It canbe shown that in the limit, as both sample sizes grow witho ut bound, the N(0,1) pdf providesaccurate probabilities for W1. Thus, for finite values of n1and n2, the N(0,1) pdf will be used t oobtain approximate probabilities for W1. As a general guideline, I recomm end using Case 1 onlyif n1≥ 30 and n2≥ 30.The usual algebraic manipulation o f the ratio that is W1yields the following result.Result 1 9.1 (Slutsky’s approximate
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