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www.downloadslide.com 2.12 Random Sampling 77 EXAMPLE 2.25 Compute the probabilities for each value of Y in Example 2.24. Solution The event {Y = 0} results only from sample point E 4 . If the coins are balanced, the sample points are equally likely; hence, P(Y = 0) = P(E 4 ) = 1/4. Similarly, P(Y = 1) = P(E 2 ) + P(E 3 ) = 1/2 and P(Y = 2) = P(E 1 ) = 1/4. A more detailed examination of random variables will be undertaken in the next two chapters. Exercises 2.139 Refer to Exercise 2.112. Let the random variable Y represent the number of radar sets that detect a particular aircraft. Compute the probabilities associated **with** each value of Y . 2.140 Refer to Exercise 2.120. Let the random variable Y represent the number of defective refrigerators found after three refrigerators have been tested. Compute the probabilities for each value of Y . 2.141 Refer again to Exercise 2.120. Let the random variable Y represent the number of the test in which the last defective refrigerator is identified. Compute the probabilities for each value of Y . 2.142 A spinner can land in any of four positions, A, B, C, and D, **with** equal probability. The spinner is used twice, and the position is noted each time. Let the random variable Y denote the number of positions on which the spinner did not land. Compute the probabilities for each value of Y . 2.12 Random Sampling As our final topic in this chapter, we move from theory to application and examine the nature of experiments conducted in statistics. A statistical experiment involves the observation of a sample selected from a larger body of data, existing or conceptual, called a population. The measurements in the sample, viewed as observations of the values of one or more random variables, are then employed to make an inference about the characteristics of the target population. How are these inferences made? An exact answer to this question is deferred until later, but a general observation follows from our discussion in Section 2.2. There we learned that the probability of the observed sample plays a major role in making an inference and evaluating the credibility of the inference. Without belaboring the point, it is clear that the method of sampling will affect the probability of a particular sample outcome. For example, suppose that a fictitious

www.downloadslide.com 78 Chapter 2 Probability population contains only N = 5 elements, from which we plan to take a sample of size n = 2. You could mix the elements thoroughly and select two in such a way that all pairs of elements possess an equal probability of selection. A second sampling procedure might require selecting a single element, replacing it in the population, and then drawing a single element again. The two methods of sample selection are called sampling **with**out and **with** replacement, respectively. If all the N = 5 population elements are distinctly different, the probability of drawing a specific pair, when sampling **with**out replacement, is 1/10. The probability of drawing the same specific pair, when sampling **with** replacement, is 2/25. You can easily verify these results. The point that we make is that the method of sampling, known as the design of an experiment, affects both the quantity of information in a sample and the probability of observing a specific sample result. Hence, every sampling procedure must be clearly described if we wish to make valid inferences from sample to population. The study of the design of experiments, the various types of designs along **with** their properties, is a course in itself. Hence, at this early stage of study we introduce only the simplest sampling procedure, simple random sampling. The notion of simple random sampling will be needed in subsequent discussions of the probabilities associated **with** random variables, and it will inject some realism into our discussion of statistics. This is because simple random sampling is often employed in practice. Now let us define the term random sample. DEFINITION 2.13 Let N and n represent the numbers of elements in the population and sample, respectively. If the sampling is conducted in such a way that each of the ( ) N n samples has an equal probability of being selected, the sampling is said to be random, and the result is said to be a random sample. Perfect random sampling is difficult to achieve in practice. If the population is not too large, we might write each of the N numbers on a poker chip, mix all the chips, and select a sample of n chips. The numbers on the poker chips would specify the measurements to appear in the sample. Tables of random numbers have been formed by computer to expedite the selection of random samples. An example of such a table is Table 12, Appendix 3. A random number table is a set of integers (0, 1, ..., 9) generated so that, in the long run, the table will contain all ten integers in approximately equal proportions, **with** no trends in the patterns in which the digits were generated. Thus, if one digit is selected from a random point on the table, it is equally likely to be any of the digits 0 through 9. Choosing numbers from the table is analogous to drawing numbered poker chips from the mixed pile, as mentioned earlier. Suppose we want a random sample of three persons to be selected from a population of seven persons. We could number the people from 1 to 7, put the numbers on chips, thoroughly mix the chips, and then draw three out. Analogously, we could drop a pencil point on a random starting point in Table 12, Appendix 3. Suppose the point falls on the 15th line of column 9 and we decide to use the rightmost digit of the group of five, which is a5inthis case. This process is like drawing the chip numbered 5. We may now proceed in any direction to

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