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Unformatted text preview: Chapter 3 Approximating a Sampling Distribution Table 3.1: Heights of the rectangles in the probabil ity histogram of the sampling distribution of the test statistic for Fisher’s test for the Ballerina study. Height of rectangle x P ( X = x ) P ( X = x ) / . 08 . 40 . 0009 . 01125 . 32 . 0081 . 10125 . 24 . 0387 . 48375 . 16 . 1127 1 . 40875 . 08 . 2104 2 . 63000 . 00 . 2584 3 . 23000 . 08 . 2104 2 . 63000 . 16 . 1127 1 . 40875 . 24 . 0387 . 48375 . 32 . 0081 . 10125 . 40 . 0009 . 01125 Total 1 . 0000 3.1 Study Suggestions Chapter 1 introduced the CRD as a device for conducting a comparative study of two treatments. Chapter 2 introduced hypothesis testing as a tech nique for deciding whether the treatments have an identical effect. A hypothesis test yields a number, the Pvalue, which quantifies the debate between the Skeptic and the Advocate. A practical problem has arisen, however; the Pvalue can be difficult to com pute. In fact, if a study has a large number of sub jects, the Pvalue can be impossible to compute, even with an electronic computer equipped with any of the popular existing statistical software packages. Thus, Chapter 3 addresses the problem of finding an easy way of obtaining an approximate Pvalue. The two approximation methods introduced in Chapter 3 are much easier to understand if a person has learned to represent a sampling distribution with a picture—the probability histogram. Given a sam pling distribution, make sure you can draw its prob ability histogram by following the three steps in the key extract on page 80 of the text. In particular, prac tice obtaining the height of a value’s rectangle by di viding the probability of the value by δ . (Remember, after sorting, or ordering, the possible values of the test statistic from smallest to largest, δ equals the dif ference between any two successive values.) In addition, given a probability histogram, be cer tain that you can create its sampling distribution. The centers of the bases of the rectangles of the proba bility histogram correspond to the possible values of the test statistic, and the area of a rectangle equals the probability of its value (center). Some examples of these techniques follow. Consider the Ballerina study introduced in Chap ter 1 of the text. The sampling distribution for the Ballerina study is presented in Table 3.1. In order to obtain its probability histogram, first we must de termine the value of δ , the (constant) difference be tween any two of the ordered possible values of the test statistic. From Table 3.1, clearly δ = 0 . 08 . (Al ternatively, you can remember that δ = n/ ( n 1 n 2 ) . For the current study, this formula gives δ = 50 / [25(25)] = 0 . 08 .) Second, we determine the height of each rectan gle. This is tedious, but basically simple. Each rect angle is centered at a possible value of x ; the height of the rectangle is the probability of that value di vided by δ . The heights are given in Table 3.1.The heights are given in Table 3....
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This note was uploaded on 10/23/2009 for the course STAT STATS 371 taught by Professor Professorwardrop during the Fall '09 term at University of Wisconsin.
 Fall '09
 ProfessorWardrop
 Statistics, Probability

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