1STAT 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health Sciences!Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology!Teaching Assistants:Ming Zheng, Sovia Lau, Tianwei YuUCLA StatisticsUniversity of California, Los Angeles, Fall 2003http://www.stat.ucla.edu/~dinov/courses_students.htmlSTAT 13, UCLA, Ivo DinovSlide 2Statistics Online Compute Resources!http://socr.stat.ucla.edu/SOCR.html!Interactive Normal Curve!Online Calculators for Binomial, Normal, Chi-Square, F and T distributions!Galton's Board or QuincunxSTAT 13, UCLA, Ivo DinovSlide 3Chapter 7: Sampling Distributions!Parameters and Estimates!Sampling distributions of the sample mean!Central Limit Theorem (CLT)!Estimates that are approximately Normal!Standard errors of differences!Student’s t-distributionSTAT 13, UCLA, Ivo DinovSlide 4Parameters and estimates! A parameter is a numerical characteristic of a population or distribution! An estimate is a quantity calculated from the data to approximatean unknown parameter! Notation"Capital letters refer to random variables"Small letters refer to observed valuesSTAT 13, UCLA, Ivo DinovSlide 5Questions! What are two ways in which random observations arise and give examples. (random sampling from finite population –randomized scientific experiment; random process producing data.)! What is a parameter? Give two examples of parameters. (characteristic of the data – mean, 1stquartile, std.dev.)! What is an estimate? How would you estimate the parameters you described in the previous question?! 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 6The 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 62STAT 13, UCLA, Ivo DinovSlide 7Twelve samples of size 2434 36 38 40 42 44 46123456789101211Samplenumber12 samples of size 24ChestmeasurementsSTAT 13, UCLA, Ivo DinovSlide 8Histograms 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 41420.00.51.01.5Sample 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 9E(sample mean) = Population meansize Sample Population = )SD(SDnsample meaMean and SD of the sampling distributionnnXXXXσµ====)(SD)(SD ,)(E)(ESTAT 13, UCLA, Ivo DinovSlide 10! 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?! 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)! 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 11! How is the population standard deviation of related to the population standard deviation of individual observations? ( SD( ) = (Population SD)/sqrt(sample_size) )! What happens to the sampling distribution of if the sample size is increased? ( variability decreases ) ! 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?)! 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 12! Increasing the precision of as an estimator of µis equivalent to doing what to SD( )? (decreasing)! 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 nσ3STAT 13, UCLA, Ivo DinovSlide 13Central 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 XSTAT 13, UCLA, Ivo DinovSlide 14Central 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 15Central Limit Effect –Histograms of sample means0.0 0.2 0.4 0.6 0.8 1.0012n = 10.0 0.20.40.6 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 16Central 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 0.81.001234Uniform DistributionSample sizes n=4, n=10STAT 13, UCLA, Ivo DinovSlide 17Central 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 18Central Limit Effect -- Histograms of sample meansn = 401230.00.20.40.60.81.0n = 100120.00.40.81.2Exponential DistributionSample sizes n=4, n=104STAT 13, UCLA, Ivo DinovSlide 19Central 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 = 20.0 0.2 0.4 0.6 0.8 1.00123Quadratic U DistributionArea = 1()]1,0[ , 12221∈−= xxYSTAT 13, UCLA, Ivo DinovSlide 20Central 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 1.00123Quadratic U DistributionSample sizes n=4, n=10STAT 13, UCLA, Ivo DinovSlide 21Central Limit Theorem:When sampling from almost any distribution,is approximately Normally distributed in large samples.CLT Applet DemoApplets.dir/SamplingDistributionApplet.htmlX Central Limit Theorem –
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