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Unformatted text preview: + + Chapter 6: Inference for a population Consider the CM of selecting n cards at ran dom w/replacement from Box( N ; p ). Before we operate it, we can consider calculating probabilities. But suppose that p is unknown, the usual sit uation in science. This means that we cannot actually calculate probabilities. Thus, lets forget about calculating probabili ties and fast forward to after we operate the CM. We now know two numbers: n and the total number of successes in the sample, x . What do we do with these? + 149 + + We calculate p = x/n , the proportion of suc cesses in the sample. Our interest is in p . We can hope that we have a representative sample, which would mean that p = p . We provide an alternative to just hoping. First, we need a term. We call p the point estimate of p . I approve of onehalf of this name: point. Recall that every number is a point on the number line and every point on the number line is a number. Thus, p is definitely a point. I am not so happy with the word estimate. Here is why. Discuss. Well, I am not tsar of the world, so we must use the word estimate. + 150 + + Here is how I remember this: The word es timate signifies that the statistician is taking data from a sample to make a statement (wild guess?) about a feature of a population. Well, it is trivially easy to calculate p , so we turn our attention to studying its properties. To this end, it is convenient to introduce the term nature to represent an allknowing en tity. In particular, nature knows the value of p . In order to gain insight into what we should do as researchers, we (temporarily) pretend we are nature. For example, suppose that nature knows that p = 0 . 40 for a population. There are 10 researchers and each researcher (independently) selects a sample of size n = 100 ARWR (at random w/replacement) from the population. + 151 + + Here is what happened: Res. x p Correct? 1 47 0.470 No 2 31 0.310 No 3 36 0.360 No 4 41 0.410 No 5 35 0.350 No 6 46 0.460 No 7 40 0.400 Yes 8 41 0.410 No 9 42 0.420 No 10 36 0.360 No BTW, we say that a point estimate is correct if, and only if, p = p . There are two features in this table: Being correct is rare. Only nature knows who is correct. Below is our method for dealing with the first of these features. First note that res. 4, 8 and 9, while not correct, are very close. + 152 + + Close counts in horseshoes, hand grenades and estimation. We expand the idea of a point estimate to an interval estimate . For example, in the table above, if we estimate p by p . 05 then res. 35 and 710 are correct. If we estimate it by p . 10 then all researchers are correct....
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This note was uploaded on 10/23/2009 for the course STAT STATS 301 taught by Professor Professorwardrop during the Fall '08 term at Wisconsin.
 Fall '08
 ProfessorWardrop
 Statistics

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