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UW-Madison STAT 371 - Ch. 17

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Inference for One Numerical PopulationResponses Obtained by CountingFinite Populations for CountsA Population of TrialsResponses Obtained by MeasuringThe General Definition of a Probability Density FunctionFamilies of Probability Density FunctionsEstimation of The AssumptionsGosset or Slutsky?Population is a Normal CurveLies, Damned Lies and Statistics TextsMore on SkewnessComputing SummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 17Inference for On e Numerical PopulationIn Chapter 10 you learned about finite popul ations. You learned about smart and dumb randomsamples from a finite population. You learned that i.i.d. trial s can be viewed as the outcomes ofa dumb random sample from a finite population . Chapter 11 developed these ideas in the specialcase of a dichotomous response. This was a very fruitful development, leading to all the results notnamed Poisso n in Chapters 12–16. And, of course, our results for the Poisson are related to ourresults for the binomial.In Chapters 17–20 we mimic the work of Chapters 11–16, but for a numerical response ratherthan a dichotomy. First, you will see the familiar distinction between a finite population and amathematical model for the process that generates the out comes of trials. Second, you will see thatresponses that are counts must be studied differently than responses that are measurements . Webegin by studying responses that are obtained by counting.Before we get to count responses, let me lay out some notation for this chapter. Recall thateither dumb random sampling from a finite populat ion or the assump tion that trials are i.i.d., resultin o ur observing n i.i.d. random variables:X1, X2, X3, . . . , Xn.The prob ability/sampling distribution for each of these random variables is determined by thepopulation. Recall that for a dichotomous response the popu lation is quite simple; it is determinedby the single number p. For a numerical response, as you will soon see, the population is morecomplex—it is a picture, not a single number. Finally, when I want to talk about a generic randomvariable—one observation of a trial or o ne populat ion member selected at random—I will use t hesymbol X, without a subscript.You may need—on occasion—to refer back to the preceding paragraph as you work throug hthis chapter.17.1 Response s Obtained by Coun tingI will begin with finite populations.425Table 17.1: The po pulation distribution for the cat population.x0 1 2 3 TotalP (X = x) 0.10 0.50 0.30 0.10 1.00Figure 17.1: The prob ability his togram for the cat population.01 230.100.500.300.1017.1.1 Finite Populations for CountsPlease remember that the two examples in this subsection are both hypothetical. In particular, Iclaim no knowledge of cat ownership or hous eho ld size in our society.Example 17.1 (The cat population.) A city consists of exactly 100,000 households. Nature knowsthat 10,000 of these household s have no cats; 50,000 of these househo lds have exactly one cat;30,000 of these households have exactly two cats; and the remaining 10,0 00 households haveexactly th ree cats.We can visualize the cat population as a pop ulation box that contains 100,000 cards, one foreach household. On a household’s card is its numb er of cats: 0, 1, 2 or 3. Consider the chancemechanism of selecting o ne card at random from the population box. (Equivalently, selecting onehousehold at random from the cit y.) Let X be the number on the card t hat will be selected. It iseasy to determine the sampling distribution of X and it is given in Table 17.1. For example, 50,000of the 100,000 hou seholds have exactly one cat; thus P (X = 1) = 50,000/100,000 = 0.50. It willbe useful to draw the probability histogram of the random variable X; it is presented in Figure 17.1.To this end, note that consecutive possible values of X differ by 1; thus, δ = 1 and the height ofeach rectangle in Figu re 17.1 equals the probability of its center value. For example, the rectanglecentered at 1 has a height of 0.50 because P (X = 1) = 0.50. Either the distribution i n Table 17.1 orits probability hist ogram in Figure 17.1 can p lay the role of the population. In the next section, wewill see that for a measurement response th e population is a pi ct u re, called the probability densityfunction. (Indeed, the populat ion must be a picture for mathematical reasons—trust me on th is.)426Because we have no choice with a m easurement—the population is a picture—for consistency, Iwill refer t o the probability histogram of a count response as the population. Except when I don’t;occasionally, it will be convenient for me to view the probability distribution—such as th e one inTable 17.1—as being th e population. As Oscar Wilde reportedl y said,Consistency is the last refuge of the unimaginative.It can be shown that the mean, µ, of the cat pop u lation equals 1.40 cats per household and itsstandard deviation, σ, equals 0.80 cats per ho usehold. I suggest you trust me on the accuracy ofthese values. Certainly, if one imagines a fulcrum placed at 1.40 in Figure 17.1, it appears that thepicture will balance. If you really enjoy hand computations, you can use Equatio ns 7.1 and 7.3on pages 147 and 148 to obtain µ = 1.40 and σ2= 0.64. Finally, if you refer to my originaldescription of the cat population in Example 17.1, you can easily verify that the median of the100,000 population values is 1. (In the sorted lis t, posit ions 10,001 through 60,0 00 are all hometo t he respo nse value 1. Thus, the two center positions, 50,000 and 50,001 both house 1’s; hence,the median is 1.) For future use it is convenient to have a Greek letter to represent the median of apopulation; we will use ν, pronounced as new.You have now seen the veracity of my comment in the first paragraph of this chapter; thepopulation for a count response— a probability histog ram—is m uch more complicated than thepopulation for a dichotomy—the number p.Thus far with the cat popu lation, I have focused exclusively on Nature’s perspective. We nowturn to the view of a researcher.Imagine that you are a researcher who is interested in the cat population. All you would know isthat the respons e is a count; thus, the population is a probability histogram. But which probabilityhistogram? It is natural to begin with the idea of using data to estimate the population’s probabilityhistogram. How shou ld you do that?Mathematically, the answer is simple: Select a random


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UW-Madison STAT 371 - Ch. 17

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