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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 P-value, which quantifies the debate between the Skeptic and the Advocate. A practical problem has arisen, however; the P-value can be difficult to com- pute. In fact, if a study has a large number of sub- jects, the P-value 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 P-value. 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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