371chapter5 - Chapter 5 The Goodness of Fit Test 5.1 Dice,...

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Chapter 5 The Goodness of Fit Test 5.1 Dice, Computers and Genetics The CM of casting a die was introduced in Chapter 1. We assumed that the six possible outcomes of this CM are equally likely; i.e. we assumed the ELC. Later I mentioned that I own two round- cornered dice and I suspect that the ELC is not reasonable for either of them. How can we decide whether to believe in the ELC for a die? In this chapter we will learn about the (Chi-squared) Goodness of Fit Test . This test was developed circa 1900 by Karl Pearson (1857–1936), in part to investigate theories of genetic in- heritance. While I cannot give you an exact reference, sometime in the 1990’s Scientific American (or a similarly themed journal—sorry) published an issue devoted to ‘The 20 greatest scientific discoveries of the 20th Century.’ Right along side such obvious entries as the jet engine and the splitting of the atom was . . . the test of this chapter! This was a curious inclusion for at least two reasons. 1. Whereas it is true that modern statisticians do not condemn this test, they don’t use it that much. 2. With all the wondrous discoveries of those one hundred years, I can’t imagine putting any statistical method on the list! Many of my more zealous colleagues might disagree with my last statement, but I would be truly amazed if any of them selected the Goodness of Fit Test as our main contribution. When I read this issue of the journal, my sense was that there were two reasons for including our test. First, the test is important historically b/c it provided a confirmation that Mendel’s ‘genes’ made sense. This was important b/c genes provided a mechanism for Darwin’s work. (I am not a biologist and indeed understand the subject poorly, but as I understand things, Darwin provided no mechanism for all these wonderful things he described. It reminds me a bit of all the physicists running around trying to get experimental evidence to support Einstein’s Relativity.) Second, I believe that the editors wanted to take the most inclusive view of science for the issue. Hence, even Statistics gets some attention. 43
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In any event, unless you work for a casino or are interested in gambling (sorry, I refuse to call it ‘gaming’), you might think that the study of dice is bit frivolous. Well, as mentioned above, there are applications to genetics. But why do I mention computers in this section title? Well, we hear all the time about computer models that help us learn about the world. There are computer models for the climate, the mutation of species or viruses, and so on. These com- puter models typically include CMs and at some point in the analysis the computer programmer will simulate the operations of these various CMs by using a program called a random number generator . For example, a random number generator might promise to select a digit at random (this implies ELC in this setting) from 0, 1, 2, . . . , 9. But how does the programmer know that the program works as advertised? The test of this chapter can be used to investigate this issue.
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371chapter5 - Chapter 5 The Goodness of Fit Test 5.1 Dice,...

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