Social Statistics 社会統計学社会統計学 K112 Lecture 6Tu Bao Ho School of Knowledge ScienceSampling and Estimation 標本抽出推定標本抽出推定K112 - Lecture 62 21. Introduction 2. The Sampling Distribution of a Statistic 統計量標本分布3. Distribution of the Sample Mean and the Central Limit Theorem標本平均値分布中心極限定理4. Point Estimation of a Population Mean母平均値点推定5. Confidence Interval for a Population Mean母平均値信頼区間Sampling and estimation抽出予測K112 - Lecture 63 3IntroductionThe generalizations in statistics are founded on the understanding of the manner in which variation in the population is transmitted, by sampling, to variation in statistics like the sample mean. We are interested in learning some numerical feature of the population.A numerical feature of a population is called a parameter.The true value of a population parameter is an unknown constant.A statistic（統計量）（統計量）is a numerical valued function of the sample observations.Example: The sample mean (標本平均値)is a statistic because its numerical value can be computed once the sample data, consisting of the values of X1, , Xn, are variable.nXXXn+=.1K112 - Lecture 64 41. Introduction 2. The Sampling Distribution of a Statistic 統計量標本分布統計量標本分布3. Distribution of the Sample Mean and the Central Limit Theorem標本平均値分布中心極限定理4. Point Estimation of a Population Mean母平均値点推定5. Confidence Interval for a Population Mean母平均値信頼区間Sampling and estimation抽出予測K112 - Lecture 65 5The sampling distribution of a statistic 統計量標本分布統計量標本分布Three crucial points:A sample is only a part of the population, the numerical value of a statistic cannot be expected to give us the exact value of the parameter.The observed value of a statistic depends on the particular sample that happens to be selected.There will be some variability in the value of statistic over different occasions (場合) of sampling.As any statistic varies from sample to sample, it is a random variable. The probability distribution of a statisticis called its sampling distribution (標本分布標本分布)K112 - Lecture 66 6Illustration of a sampling distributionThere are 3 housing units, Xis # of rooms in each unit as follows:Consider drawing a random sample of size 2 with replacement, say, X1 and X2. Find the sampling distribution of SOLUTION:2/ )(21XXX+=The population distributionX,xx of valuesingcorrespond theand )( sample possible The21K112 - Lecture 67 7Illustration of a sampling distribution (cond)There are 9 equally possible samples with probability 1/9 of each. But there are only 5 different values of as shown in the table:Suppose the population consists of 200 houses of which 100 have 2 rooms, 100 have 3 rooms, and 100 have 4 rooms. It would make little difference whether or not we replace the unit after the first selection. We have PX=2 = PX=3 =PX=4 = 2/3 which characterizes the population. XK112 - Lecture 68 8What are the key conditions required for a sample to be random?The observations X1, X2, Xnare a random sample of size n from the population distribution if they result from independent selections (独立選択独立選択), and each has the same distribution as the population.Random sample確率標本K112 - Lecture 69 91. Introduction 2. The Sampling Distribution of a Statistic 統計量標本分布3. Distribution of the Sample Mean and the Central Limit Theorem標本平均値分布中心極限定理標本平均値分布中心極限定理4. Point Estimation of a Population Mean母平均値点推定5. Confidence Interval for a Population Mean母平均値信頼区間Sampling and estimation抽出予測K112 - Lecture 61010Statistical inference about the population mean from the sample mean is of prime practical importance. We have the basic properties of the sampling distribution of sample mean in relation with the population.Distribution of the sample mean and the central limit theorem標本平均値分布中心極限定理K112 - Lecture 61111Mean and variance of (X1+X2)/2 ? (Tables in pages 6 and 7)K112 - Lecture 61212deviation standard andmean on with distributi normal thehas Xmean sample the,deviation standard and meanwith population afromsampling randomIn nnormalRandom sampling from a normal distributionK112 - Lecture 61313In practice the normal approximation is usually adequate when nis greater than 30.Central limit theorem 中心極限定理) 1 , 0(ely approximat is ly,Consequent.ndeviation standard and mean with normalely approximat isis X ofon distributi thelarge, is when ,deviation standard andmean with populationarbitrary an from sampling randomIn NnXZn=large. is n when normal telyapproprimais X of ondistributi the ,population theWhatever K112 - Lecture 61414Consider a population with mean = 82 and std. deviation =12. (1) A random sample selected with n = 64, P80.8 sample mean (61%)S = sqrt(2.32 + 0.32 + 5.32 + 2.72 + 1.32 + 5.72 + 0.72)/6 = sqrt(91.2/6) = sqrt(15.2) = 2.3 ? sample standard deviation (7.6%)S.E. = 2.3/sqrt(30) = 0.41 (4.1%)95% error margin is 1.96 x 0.41 = 0.8, if d = 0.5 ? n = 1.96 x 2.3/0.52 = 8295% confidence interval for is