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UW-Madison STAT 371 - Chapter 3 - Estimation of p

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Chapter 3 Estimation of p 3 1 Point and Interval Estimates of p Suppose that we have Bernoulli Trials BT So far in every example I have told you the numerical value of p In science however the value of p typically is unknown to the researcher In such cases scientists and statisticians use data from the 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 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 p 13 20 0 65 It is trivially easy to calculate 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 this mean Statisticians evaluate procedures by seeing how they perform in the long run We say that the point estimate p is correct if and only if 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 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 Reporter Great but what is the actual value of p Are you saying that p 0 65 33 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 I 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 Let s bring Nature into the analysis Suppose that Nature knows that p 0 75 Well Nature knows that the researcher in the above press conference has an incorrect point estimate But let s proceed beyond that one example Consider a researcher who decides to observe n 20 BT and use them to estimate p What will happen Well we don t know what will happen The researcher might observe x 15 successes giving p 15 20 0 75 which would be a correct point estimate Sadly of course the researcher would not know it is correct only Nature would Given what we were doing in Chapters 1 and 2 it occurs to us to calculate a probability After all we use probabilities to quantify uncertainty about the future operation of a CM in the current case the CM is the process of collecting the data So before the researcher observes the 20 BT Nature decides to calculate the probability that the point estimate will be correct This probability is of course P X 15 20 0 75 15 0 25 5 15 5 which I find with the help of the binomial website to be 0 2023 There are two rather obvious undesirable features to this answer 1 Only Nature knows whether the point estimate is correct indeed before the data are collected only Nature can calculate the probability the point estimate will be correct 2 The probability that the point estimate will be correct is disappointingly small And note that for most values of p it is impossible for the point estimate to be correct For one of countless possible examples suppose that n 20 as in the current discussion and p 0 43 It is impossible to obtain p 0 43 As we shall see repeatedly in this course what often happens is that by collecting more data our procedure becomes better in some way Thus suppose that the researcher plans to observe n 100 BT with p still equal to 0 75 The probability that the point estimate will be correct is P X 75 100 0 75 75 0 25 25 75 25 which I find with the help of the website to be 0 0918 This is very upsetting More data makes the probability of a correct point estimate smaller not larger 34 The difficulty lies in our desire to have p be exactly correct Close is good too In fact statisticians like to say Close counts in horse shoes hand grenades and estimation But what do I mean by close Well for an example to move us along suppose we decide that if p is within 0 05 of p then it is close enough for us to be happy Revisiting the two computations above we see that for n 20 and p 0 75 close enough means 14 X 16 The probability of this happening again with the help of the website is 0 5606 For n 100 close enough means 70 X 80 The probability of this happening is 0 7967 As a final example for n 1000 close enough means 700 X 800 The probability of this happening is 0 9998 a virtual certainty to a statistician Here is another way to view my close enough argument above Instead of estimating p by the single number point p we use an interval estimate in this example the closed interval is p 0 05 As you may have learned in a math class a closed interval is an interval the includes its endpoints In this class all interval estimates are closed intervals Analogous to our earlier definition we say that the interval estimate is correct if and only if the interval contains p Thus saying that p is within 0 05 of p my working definition of close enough in the example above is equivalent to saying that p is in the interval estimate i e the interval estimate is correct Henceforth I will not talk about p being close enough to p I will talk about whether an …


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UW-Madison STAT 371 - Chapter 3 - Estimation of p

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