Lecture13_570统计笔记.doc

Review of Confidence Intervals for Population Mean and ProportionMotivation & simple random sampleEg) We wish to estimate the average height of adult US males Take a random sample.- “Simple” random sample: every subject in the population has the same chance to be selected.Introduction to statistical inference on one population meanFor a “random sample” of size n: Point estimatior sample mean ()Other estimators: median, mode, trimmed mean, Confidence Interval (C.I.)Eg) 95% C.I. for 99.9999% C.I. (6-9 in the manufacture industry) Hypothesis TestEg) : : Point Estimator, C.I., Test Statistical Inference- Draw some conclusion on the population (parameters of interest) based on a random sample.1. The Exact Confidence Interval for when the population is normal & 2 is known Point estimator and confidence interval for - When the population is normal and the population variance is known.- Let be a random sample for a normal population with mean and variance . That is, . - For now, we assume that is known. Point Estimator for : is an unbiased estimator of - Intuitively, this means if you take “many” samples of size n from the population, then the mean of these samples means would be equal to if you take a large enough # of samples.- is also a maximum likelihood estimator (MLE) of .- is also a method of moment estimator (MOME) of .- Other good properties too. Confidence Interval for - Intuitive approach (backwards derivation for the CI boundaries C1and C2): PC1C2=0.95P-C1-C2=0.95PX-C1X-X-C2=0.95PX-C1/nX-/nX-C2/n=0.95Since we knowZ=X-/nN(0,1)We can compute the expressions for C1and C2.However, one question is that there are MANY ways to choose the Cs.Later you will see that for pivotal quantity with symmetric pdfs, the symmetric CIs are the optimal in that they have the shortest lengths for the given confidence level 100(1-a)%.Now we present a general approach to derive the CIs. General approach for deriving CIs : the Pivotal Quantity (P.Q.) approach*Definition: A pivotal quantity is a function of the sample and the parameter of interest. Furthermore, its distribution is entirely known.1. We start by looking at the point estimator of . * Is a pivotal quantity for ? is not because is unknown.* function of and : Yes, it is pivotal quantity.* Another function of and : Yes, it is pivotal quantity.So, Pivotal Quantity is not unique.2. Now that we have found the pivotal quantity Z, we shall start the derivation for the symmetrical CIs for from the PDF of the pivotal quantity Z100(1-a)% CI for m, 0a1(e.g. a=0.05 95% C.I.) the 100(1-)% C.I. for is *Note, some special values for and the corresponding Z/2values are:1. The 95% CI, where =0.05 and the corresponding Z2=Z0.025=1.962. The 90% CI, where =0.1 and the corresponding Z2=Z0.05=1.6453. The 99% CI, where =0.01 and the corresponding Z2=Z0.005=2.575Example 1. A random sample of 400 adult US male was taken and the sample mean was found to be X=57”=67 inches . Based on past studies, it is believed that the population distribution of all adult US male is normal and the standard deviation is 30 inches. Please construct a 95% confidence interval for the average height of all adult US male based on this sample. Solution: The 95% CI for is X-1.96n, X+1.96n =67-1.9630400, 67+1.9630400 64, 70That is, the estimated 95% confidence interval for the average height of all adult US male is 54”, 510”. This means that we are 95% sure the population mean would lie between 54” and 510”.Recall the 100(1-)% symmetric C.I. for is *Please note that this CI is symmetric around XThe length of this CI is: Now we derive a non-symmetrical CI: 100(1-)% C.I. for Compare the lengths of the C.I.s, one can prove theoretically that: You can try a few numerical values for , and see for yourself. For example,=0.05HW: Please derive the 100(1-)% symmetric C.I. for based on a random sample from a normal population with unknown variance2. (Large Sample) Confidence interval for a population mean (*any population) and a population proportion p Central Limit TheoremZ=X-/nnN(0,1)When n is large enough, we haveZ=X-/nN(0,1)That means Z follows approximately the normal (0,1) distribution.Application #1. Inference on when the population distribution is unknown but the sample size is largeBy Slutskys Theorem We can also obtain another pivotal quantity when is unknown by plugging the sample standard deviation S as follows:We subsequently obtain the C.I. using the second P.Q. for : Application #2. Inference on one population proportion p when the population is Bernoulli(p) * Let , please find the 100(1-)% CI for p.Point estimator : (ex. , )Our goal: derive a 100(1-)% C.I. for pThus for the Bernoulli population, we have:=EX=p2=VarX=p(1-p)Thus by the CLT we have:Furthermore, we have for this situation: X=pTherefore we obtain the following pivotal quantity Z for p:By Slustkys theorem, we can replace the population proportion in the denominator with the sample proportion and obtain another pivotal quantity for p:Z*=p-pp1-pnN(0,1)# Thus the (approximate, or large sample) C.I. for p based on the second pivotal quantity Z* is:= The large sample C.I. for p is .# CLT = n large usually means # special case for the inference on p based on a Bernoulli population. The sample size n is large meansLet , large sample means: (*Here X= total # of S), and (*Here n-X= total # of F)Example 2. During one of the “beer wars” in the early 1980s, a taste test between Schlitz and Budweiser was the focus of