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# 371chapter3 - Chapter 3 Estimation of p 3.1 Point and...

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Chapter 3 Estimation of p 3.1 Point and Interval Estimates of p Suppose that we have BT. So far, in every example I have told you the (numerical) value of p . In science, usually the value of p is unknown to the researcher. In such cases, scientists and statisticians use data from BT to estimate the value of p . Note that the word estimate is a technical term that has a precise definition in this course. I don’t particularly like the choice of the word estimate for what we do, but I am not the tsar of the Statistics world! It will be very convenient for your learning if we distinguish between two creatures. First, is Nature , who knows everything and in particular knows the value of p . Second is the researcher who is ignorant of the value of p . Here is the idea. A researcher plans to observe n BT, but does not know the value of p . After the BT have been observed the researcher will use the information obtained to make a statement about what p might be. After observing the BT, the researcher counts the number of successes, x , in the n BT. We define ˆ p = x/n , the proportion of successes in the sample, to be the point estimate of p . For example, if I observe n = 20 BT and count x = 13 successes, then my point estimate of p is ˆ p = 13 / 20 = 0 . 65 . It is trivially easy to calculate ˆ p = x/n ; thus, based on your experiences in previous math courses, you might expect that we will move along to the next topic. But we won’t. What we do in a Statistics course is evaluate the behavior of our procedure. What does that mean? Well, recall that in first lecture I stated that statisticians evaluate procedures by seeing how they perform in the long run . We say that the point estimate ˆ p is correct if, and only if, ˆ p = p . Obviously, any honest researcher wants the point estimate to be correct. Let’s go back to the example of a researcher who observes 13 successes in 20 BT and calculates ˆ p = 13 / 20 = 0 . 65 . The researcher schedules a press conference and the following exchange is recorded. Researcher: I know that all Americans are curious about the value of p . I am here today to announce that based on my incredible effort, wisdom and brilliance, I estimate p to be 0.65. 23

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Reporter: Great, but what is the actual value of p ? Are you saying that p = 0 . 65 ? Researcher: Well, I don’t actually know what p is, but I certainly hope that it equals 0.65. As I have stated many times, nobody is better than me at obtaining correct point estimates. Reporter: Granted, but is anybody worse than you at obtaining correct point estimates? Researcher: (Mumbling) Well, no. You see, the problem is that only Nature knows the actual value of p . No mere researcher will ever know it. Reporter: Then why are we here? Before we follow the reporter’s suggestion and give up, let’s see what we can learn.
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371chapter3 - Chapter 3 Estimation of p 3.1 Point and...

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