Chapter 19 Notes.doc - Chapter 19 Confidence Intervals for...

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Chapter 19: Confidence Intervals for Proportions A Confidence Interval What happens when you take a sample and come up with your estimate, ˆ p , but you are actually trying to find the true proportion, p? What if you are trying to find the true standard deviation? Since we don’t know p, we can’t find the true standard deviation of the sampling distribution model. Since you do know the observed proportion, ˆ p , we’ll just use what we know, and we estimate. That may not seem like a big deal, but it gets a special name. Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error (SE). For a sample proportion ˆ p the standard error is: SE( ˆ p ) = ˆ p ˆ q n . Now we know that the sampling model for ˆ p should look like this: Great…what does that tell us? Well, because it’s Normal, it says that 68% of all sample of size n will have ˆ p within 1 SE of p. And about 95% of all these samples will be within p 2 SE’s. But where is our sample proportion in this picture? And what value does p have? We still don’t know!!! We do know that for 95% of our random samples, ˆ p will be no more than 2 SEs away from p. So let’s look at this from ˆ p ’s point of view. If I’m ˆ p , there’s a 95% chance that p is no more than 2 SEs away from me. If I reach out 2 SEs away from me on both sides, I’m 95% sure that p will be within my grasp. Now I’ve got him! Probably . Of course, ˆ p 3 SE ( ˆ p ) ˆ p 2 SE ( ˆ p ) ˆ p 1 SE ( ˆ p ) ˆ p ˆ p 1 SE ( ˆ p ) ˆ p 2 SE ( ˆ p ) ˆ p 3 SE ( ˆ p )
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even if my interval does catch p, I still don’t know its true value. The best I can do is an interval, and even then I can’t be positive it contains p. So what can we really say about p? Here’s a list of things we’d like to be able to say, in order of strongest to weakest and the reasons we can’t say most of them: 1. We can’t say that our estimate ˆ p is equal to the true proportion p. We don’t have enough information to do that. There’s no way to be sure that the population proportion is the same as the sample proportion; in fact, it almost certainly isn’t. Observations vary. Another sample would yield a different sample proportion. 2. We can’t say that it is probably true that our estimate ˆ p is equal to the true proportion p. Again, we can be pretty sure that whatever the true proportion is, it’s not exactly ˆ p . 3. We don’t know exactly what the true proportion is, but we KNOW that it’s within the interval from ___% to ___% ( ˆ p + 2SEs and ˆ p - 2SEs). This is getting closer, but we still can’t be certain. We can’t know for sure that the true proportion is in this interval. Here’s the best way to say what we do know: We are 95% confident that between ___% and ____% of ___________________ (fill in the blank here with whatever the context of your problem is) Statements like these are called confidence intervals . They’re the best we can do.
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