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chap5 - Chapter 5 One-Population Models 5.1 Study...

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Chapter 5 One-Population Models 5.1 Study Suggestions In the first four chapters of the text, the results of a study literally were restricted to the subjects in the study. Chapter 5 begins the investigation of methods for extending what has been learned in a study. The notion of a population is the fundamental concept in- volved in these extensions. Throughout Chapter 5 and the remainder of the text, I stress the existence of two types of popula- tions that correspond to the two types of subjects in- troduced in Chapter 1, distinct individuals or trials. At first my approach may appear to be a wasteful ex- travagance because the mathematical techniques that work for one type of population also work for the other type. There are a number of advantages, how- ever, in having two types of populations; some of these advantages are discussed below. A finite population is a tangible entity. The pop- ulation of students registered at my university this semester is very concrete. By contrast, an infinite population is a bit more slippery for a nonmathematician. The notion of “a mathemat- ical model for the process that generates the out- comes of my shooting free throws,” is a bit over- whelming for many of my students. A census is possible for a finite population, but not for an infinite population. Obtaining a random sample presents a different challenge for the two types of populations. For a finite population the challenge lies in obtain- ing a listing of the population members, select- ing a sample of subjects at random, finding the selected subjects, and convincing them to par- ticipate in the study. For trials, the challenge lies in verifying (see below) the assumptions of Bernoulli trials. The goals of inference can depend on the type of population. For example, in many applica- tions prediction is the most important method of inference for trials, but it is rarely important for a finite population. For the material in Chapter 15, it is important to realize that the population in Chapter 5 is a single number p . Pages 153–55 of the text provide a brute-force jus- tification of the multiplication rule. This argument does not work well for my students and I no longer use it in my class. Instead, I appeal to the long- run relative frequency interpretation of probability to motivate the multiplication rule. More precisely, suppose p = 0 . 6 , and suppose interest lies in the event ( X 1 = 1 , X 2 = 0) . The condition for invoking the long-run relative fre- quency interpretation of probability is met: the ex- periment of selecting two cards at random with re- placement from the population box can certainly be repeated a large number of times under identical con- ditions. Suppose that the experiment is, in fact, to be repeated a large number of times. Since p = 0 . 6 , about 60 percent of the first selections will yield a card marked 1. Regardless of what happens on the first selection, because q = 0 . 4 , about 40 percent of the second selections will yield a card marked 0. Hence, of the 60 percent of first selections that 39
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