1STAT 13, UCLA, Ivo Dinov Slide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health ScienceszInstructor: Ivo Dinov, Asst. Prof. In Statistics and NeurologyzTeaching Assistants: Tom Daula and Kaiding Zhu,UCLA StatisticsUniversity of California, Los Angeles, Fall 2002http://www.stat.ucla.edu/~dinov/STAT 13, UCLA, Ivo Dinov Slide 2Chapter 7: Sampling DistributionszParameters and EstimateszSampling distributions of the sample meanzCentral Limit Theorem (CLT)zEstimates that are approximately NormalzStandard errors of differenceszStudent’s t-distributionSTAT 13, UCLA, Ivo DinovSlide 3Parameters and estimatesz A parameter is a numerical characteristic of a population or distributionz An estimate is a quantity calculated from the data to approximatean unknown parameterz NotationCapital letters refer to random variablesSmall letters refer to observed valuesSTAT 13, UCLA, Ivo DinovSlide 4Questionsz What are two ways in which random observations arise and give examples. (random sampling from finite population –randomized scientific experiment; random process producing data.)z What is a parameter? Give two examples of parameters. (characteristic of the data – mean, 1stquartile, std.dev.)z What is an estimate? How would you estimate the parameters you described in the previous question?z What is the distinction between an estimate (p^ value calculated form obs’d data to approx. a parameter)and an estimator (P^ abstraction the the properties of the ransom process and the sample that produced the estimate) ? Why is this distinction necessary? (effects of sampling variation in P^)STAT 13, UCLA, Ivo DinovSlide 5The sample mean has a sampling distributionSampling batches of Scottish soldiers and taking chest measurements. Population µ = 39.8 in, and σ = 2.05 in.12345678910121134 36 38 40 42 44 46(a) 12 samples of size n = 6mplemberSamplenumberChestmeasurements12 samples of size 6STAT 13, UCLA, Ivo DinovSlide 6Twelve samples of size 2434 36 38 40 42 44 46123456789101211Samplenumber12 samples of size 24Chestmeasurements2STAT 13, UCLA, Ivo DinovSlide 7Histograms from 100,000 samples, n=6, 24, 100393837 40 41420.00.5(a) n = 6(b) n = 24393837 40 41420.00.51.0(c) n = 100393837 40 41 420.00.51.01.5Sample mean of chest measurements (in.)What do we see?!?1.Random nature of the means:individual sample meansvary significantly2. Increase of sample-sizedecreases the variability ofthe sample means!STAT 13, UCLA, Ivo DinovSlide 8E(sample mean) = Population meansize SamplePopulation = )SD(SDnsample meaMean and SD of the sampling distributionnnXXXXσσσσµµµµ================)(SD)(SD ,)(E)(ESTAT 13, UCLA, Ivo DinovSlide 9z We use both and to refer to a sample mean. For what purposes do we use the former and for what purposes do we use the latter?z What is meant by “the sampling distribution of ”?(sampling variation – the observed variability in the process of taking random samples; sampling distribution – the real probability distribution of the random sampling process)z How is the population mean of the sample averagerelated to the population mean of individual observations? (E( ) = Population mean)x X X X ReviewX STAT 13, UCLA, Ivo DinovSlide 10z How is the population standard deviation of related to the population standard deviation of individual observations? ( SD( ) = (Population SD)/sqrt(sample_size) )z What happens to the sampling distribution of if the sample size is increased? ( variability decreases ) z What does it mean when is said to be an “unbiased estimate” of µ ? (E( ) = µ.Are Y^= ¼ Sum, or Z^ = ¾ Sum unbiased?)z If you sample from a Normal distribution, what can you say about the distribution of ? ( Also Normal ) XX x X ReviewX x STAT 13, UCLA, Ivo DinovSlide 11z Increasing the precision of as an estimator of µis equivalent to doing what to SD( )? (decreasing)z For the sample mean calculated from a random sample, SD( ) = . This implies that the variability from sample to sample in the sample-means is given by the variability of the individual observations divided by the square root of the sample-size. In a way, averaging decreases variability.X X ReviewX nσσσσSTAT 13, UCLA, Ivo DinovSlide 12Central Limit Effect –Histograms of sample means0.0 0.2 0.40.6 0.81.00120.0 0.2 0.4 0.6 0.8 1.0012n = 10.0 0.2 0.4 0.6 0.8 1.00123n = 2TriangularDistributionSample means from sample sizen=1, n=2, 500 samplesArea = 1210210210Y=2 X3STAT 13, UCLA, Ivo DinovSlide 13Central Limit Effect -- Histograms of sample means0.0 0.2 0.4 0.6 0.8 1.0n = 40.00.2 0.4 0.6 0.8 1.0n = 10Triangular DistributionSample sizes n=4, n=10STAT 13, UCLA, Ivo DinovSlide 14Central Limit Effect –Histograms of sample means0.0 0.2 0.4 0.6 0.8 1.0012n = 10.0 0.2 0.4 0.6 0.8 1.0012n = 200.0 0.2 0.4 0.6 0.8 1.0123Area = 1Uniform DistributionSample means from sample sizen=1, n=2, 500 samplesY = XSTAT 13, UCLA, Ivo DinovSlide 15Central Limit Effect -- Histograms of sample meansn = 40.0 0.2 0.4 0.6 0.8 1.00123n = 100.0 0.2 0.4 0.6 0.8 1.001234Uniform DistributionSample sizes n=4, n=10STAT 13, UCLA, Ivo DinovSlide 16Central Limit Effect –Histograms of sample meansSample means from sample sizen=1, n=2, 500 samples01234560.00.20.40.60.81.0n = 101234560.00.20.40.60.81.0n = 2012340.00.20.40.60.8Area = 1Exponential Distribution),0[ , ∞∞∞∞∈∈∈∈−−−−xxeSTAT 13, UCLA, Ivo DinovSlide 17Central Limit Effect -- Histograms of sample meansn = 401230.00.20.40.60.81.0n = 100120.00.40.81.2Exponential DistributionSample sizes n=4, n=10STAT 13, UCLA, Ivo DinovSlide 18Central Limit Effect –Histograms of sample meansSample means from sample sizen=1, n=2, 500 samples0.00.20.4 0.60.8 1.00123n = 10.0 0.2 0.4 0.6 0.8 1.00123n = 20.00.20.40.60.81.00123Quadratic U DistributionArea = 1(((())))]1,0[ , 12221∈∈∈∈−−−−==== xxY4STAT 13, UCLA, Ivo DinovSlide 19Central Limit Effect -- Histograms of sample meansn = 40.00.2 0.40.6 0.8 1.00123n = 100.0 0.2 0.4 0.6 0.8 1.00123Quadratic U DistributionSample sizes n=4, n=10STAT 13, UCLA, Ivo DinovSlide 20Central Limit Theorem:When sampling from almost any distribution,is approximately Normally distributed in large samples.X Central Limit Theorem – heuristic formulationSTAT 13, UCLA, Ivo DinovSlide 21Let be a sequence of independentobservations from one specific random process. Let and and and both arefinite ( ). If
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