A Statistical Test of HypothesesStep 1: Choice of HypothesesAn Artificial StudyWhat if the Skeptic is Correct?Four Examples of the Skeptic Being IncorrectFinally! The Alternative Hypothesis is SpecifiedStep 2: The Test Statistic and Its Sampling DistributionStep 3: Calculating the P-valueThe P-value for the alternative >The P-value for the alternative <The P-value for the alternative =Some Relationships Between the Three P-valuesComputingA Final CommentSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 12Inference for a Binomial pIn Part 1 of these Course Notes you learned a great deal about statistical tests of hypoth eses .These tests explore the unknowable; in particular, whether or not the Skeptic’s Argument is true.In Section 11.4, I briefly introduced you to three tests that explore whether or not a sequence ofdichotomous trials are Bernoull i trials. In this chapter, we will assume that we have Bernoullitrials and t urn our attention to th e value of the parameter p. Later i n this chapter we will explorea statistical test of hy potheses concerning the value of p. First, h owever, I will i ntroduce you tothe inference procedure called estimation. I will point out that for Bernoulli trials, estimation isinherently much more interesting than testing.The estimation methods in this chapter are relatively straightforward. This does not mean,however, that th e material will be easy; you will be exposed to several new ways of th inking aboutthings and this will prove challenging.After you complete this chapter, however, you wil l have a solid understanding of the two typesof inference that are used by scientists and statisticians: testing and estimation. Most of the remain-der of the material in these Course Notes will focus on introducing you to new scientific s cenarios,and then learning how to test and estimate in these scenarios. (In some scenarios you also willlearn about the closely related topic o f predi ct ion.) Thus, for the most part, after this chapt er youwill have been exposed to the major ideas of this course, and your remaining work, being familiar,should be easier to master.12.1 Point and Interval Estimates of pSuppose that we pl an to observe n Bernoulli Trials. More accurately, we plan to observe n di-chotomous trials and we are willing to assume—for the moment, at l east —that the assumptions ofBernoulli trials are met. Throughout these Course Notes, un less I state otherwise, we always willassume that the researcher knows the value of n.Before we observe the n Bernoulli trials, if we know the numerical value of p, then w e cancompute probabilities for X, the total n umber of successes that wi ll be observed. If we do notknow that numerical value of p, then we cannot compute probabilities for X. I would argue—no teveryone agrees with me—that there is a gray area between these extremes; refer to my example283concerning basketball player Kobe Bryant on page 264 of Chapter 11; i.e., if I have a massiveamount of previous data from the process that generates my future Bernoulli trials, then I might bewilling to use the proportion of successes in the massive data s et as an approximate value of p.Still assuming that the numerical value of p is unknown to the researcher, after n Bernoullitrials are observed, if one is willing to condition on the total number o f successes, then one cancritically examine the assump tion of Bernoulli trials using t he method s presented in Section 11.4.Alternatively, we can use the data we collect—the obs erved value x of X—to make an inferenceabout the unknown numerical value o f p. Such inferences will always involve some uncertainty. Tosummarize, if the value of p is unkn own a researcher wil l attemp t to infer its value by loo king at thedata. It is convenient to create Nature—introduced in Chapter 8 in th e di scussion of Table 8.8—who kn ows the value of p.The sim plest i nference possible involves the idea of a point estimate/estimator, as defined be-low.Definition 12.1 (Point estimate/estimator.) A researcher observes n B ernoulli tri als, counts thenumber of successes, x and calculates ˆp = x/n. This proportion, ˆp, is called the point estima te o fp. It is the observed value of the random variableˆP = X/n, which is called the point estim a to r ofp. For convenience, we write ˆq = 1 − ˆp, for the proportion of failures in the data; ˆq is the observedvalue of the random variableˆQ = 1 −ˆP .Before we collect data, we focus on the random variable, t he point estimator. After we collectdata, we compute th e value of the point estimate, which is, of course, the ob served value of t hepoint estimator.I don’t l ike the t echnical term, point estimate/estimator. More precisely, I don’t like half of it.I like the word point because we are talking about a single number. (I recall the lesson I learnedin math years ago, “Every number is a point on the number line and every point on the numberline is a number.”) I don’t particularl y like t he use of the word estimate/estimator. If I becom etsar of the Statistics world, I might change the term inology. I say might instead of will because,frankly, I can’t actually suggest an im provement on estimate/ estimator. I recommend that yousimply remember th at estimate/estim at or is a word statisticians use whenever they take observeddata and try to infer a feature of a populati on.It is trivially easy to calculate ˆp = x/n; thus, based on experiences in previous math courses,you might expect t hat we will move along to the next topic. But we won’t. In a Statist ics coursewe evaluate the behavior of a procedure. What does this mean? Statisticians evaluate proceduresby seeing how they perform in the long run.We say that the point est imate ˆp is correct if, and onl y if, ˆp = p. Obviously, any honestresearcher wants the point estimate to be correct. As we will see now, whereas having a correctpoint estimate is desirable, the concept has som e serious difficulties.Let’s suppose that a researcher observes n = 100 Bernoulli trials and counts a total of x = 55successes. Thus, ˆp = 55/100 = 0.55 and this point estimate is correct if, and only if, p = 0.55.This leads us to the first difficulty with the concept of being correct.• Nat ure k nows whether ˆp is correct; th e researcher never knows.284The above example takes place after the data have been collected. We can see this because weare told that a total of x = 55
View Full Document
Unlocking...