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lecturenotesch6a

# lecturenotesch6a - Chapter 6 Inference for a population...

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+ + 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, let’s 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

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+ + We calculate ˆ 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 = ˆ p . We provide an alternative to ‘just hoping.’ First, we need a term. We call ˆ p the point estimate of p . I approve of one-half 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, ˆ 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 ˆ p , so we turn our attention to studying its properties. To this end, it is convenient to introduce the term nature to represent an all-knowing 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

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+ + Here is what happened: Res. x ˆ 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, ˆ 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 ˆ p ± 0 . 05 then res. 3–5 and 7–10 are correct. If we estimate it by ˆ p ± 0 . 10 then all researchers are correct. Define: h = 1 . 282 radicalBigg ˆ p ˆ q n . I suggest we estimate p by ˆ p ± h . This is called the 80% confidence interval estimate of p , or, more simply, the 80% con- fidence interval (CI) for p . + 153

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+ + In the table below, let l = ˆ p h and u = ˆ p + h , the endpoints of the CI. For example, for re- searcher 1, h = 1 . 282 radicalBig 0 . 47(0 . 53) / 100 = 1 . 282(0 . 050) = 0 . 064 .
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lecturenotesch6a - Chapter 6 Inference for a population...

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