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371chapter11f2011 - Chapter 11 Inference for One Numerical...

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Chapter 11 Inference for One Numerical Population 11.1 Counting Suppose that we can observe i.i.d. random variables, X 1 , X 2 , X 3 , . . . , X n that are count variables; i.e. the possible values of each random variable are the integers . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . or some subset of the integers. We have studied this problem: for BTs (giving the binomial); and for the Poisson. In this section we consider the general problem. For example, consider the population of students at UW-Madison this semester with a response equal to the total number of credits that will be completed. Personally, I would not be willing to study this as either the binomial or the Poisson. The general problem is as follows. The probability distribution of X 1 is given by a collection of equations: P ( X 1 = j ) = p j , for j = . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . .. The ideal situation would be when we know all of the p j ’s, for then we could compute the proba- bility of any event. But the ideal is not realistic in science. The next best would be to have a parametric family such as the Poisson of binomial. In these cases all we need to do is estimate one parameter (or more, sometimes a family has more than one parameter) and then we would have estimates of all the p j ’s. This is a fruitful area that we cannot pursue in this course because of time limitations. In addition to the binomial and Poisson, parametric families include: the geometric , the hypergeometric and the negative binomial distributions. Instead, we opt for a much more modest goal. We will use our data to draw inferences about the mean μ of the probability distribution. Just as for the binomial and Poisson, you can visualize μ as follows: Given the p j ’s we can draw a probability histogram; the center of gravity of the probability histogram is the mean of the population. 127
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We will compute the mean and standard deviation of our data, denoted by ¯ x and s , as in Chapter 10. We will begin with estimation. 11.1.1 Estimation of μ Our point estimate of μ , the mean of the population, is simply ¯ x , the mean of our data. But, of course, we want to have a confidence interval estimate too. It turns out that without a parametric family to guide us, an exact answer is impossible; we have no choice but to use an approximate method. In order to obtain an approximate CI, we need to be able to compute approximate probabilities for ¯ X . Fortunately, there is a wonderful result in probability theory that can help us. It is called the Central Limit Theorem (CLT). Let’s examine this three word name. Theorem, of course, means it is an important mathematical fact. Limit means that the truth of the theorem is achieved only as n grows without bound. In other words, for any finite value of n the result of the theorem is only an approximation. (Here is an example that you might have seen in calculus. As n tends to infinity, the value 1 /n converges to 0 in the limit. For any finite n the limit, 0, can be viewed as an approximation to 1 /n .) The quality of the approximation we obtain from the CLT is an important and vexing issue that we will deal with below. Finally, it is called Central because it is viewed as very important, i.e. central, to all of
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