1Stat 10, UCLA, Ivo DinovSlide 1UCLA STAT 10Introduction toStatistical ReasoningInstructor: Ivo Dinov, Asst. Prof. In Statistics and NeurologyTeaching Assistants: Yan Xiong and Will AndersonUCLA StatisticsUniversity of California, Los Angeles, Winter 2002http://www.stat.ucla.edu/~dinov/Stat 10, UCLA, Ivo DinovSlide 2Chapter 6 –Measurement Error, Chance and UncertaintyStat 10, UCLA, Ivo DinovSlide 3Newtonial science vs. chaotic scienceArticle by Robert May, Nature, vol. 411, June 21, 2001Science we encounter at schools deals with crisp certainties(e.g., prediction of planetary orbits, the periodic table as a descriptor of all elements, equations describing area, volume, velocity, position, etc.)As soon as uncertainty comes in the picture it shakes the foundation of the deterministic science, because only probabilistic statements can be made in describing a phenomenon (e.g., roulette wheels, chaotic dynamic weather predictions, Geiger counter, earthquakes, Others?)What is then science all about – describing absolutely certain events and laws alone, or describing more general phenomena in terms of their behavior and chance of occurring? Or may be both!Stat 10, UCLA, Ivo DinovSlide 4Variation in sample percentagesPoll: Do you consider yourselfoverweight? 1010We are getting closer toThe population mean, asis this a coincidence?∞→n50 60 70 80 90Samples of 20 peopleSamples of 500 peopleSample percentageTarget: True populationpercentage = 69%Figure 1.1.1Comparing percentages from 10 different surveys each of20 people with those from 10 surveys each of500 people (all surveys from same population).From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.Stat 10, UCLA, Ivo DinovSlide 5Measurement Error No matter how carefully a measurement of a single unit is made it oftern comes out a bit different. Do repeated measurements to find out by how much different each observation is! The SD of a series of repeated measurements estimates the likely size of the chance error in a single measurement of the process being observed. Examples?True value=67Observed values53Stat 10, UCLA, Ivo DinovSlide 6Random or chance error … Random or chance error is the difference between the sample-valueand the true population-value (e.g., 53 vs. 67, in the above example).2Stat 10, UCLA, Ivo DinovSlide 7The Subject of StatisticsStatistics is concerned with the process of finding out about the world and how it operates - in the face of variation and uncertainty by collecting and then making sense (interpreting, summarizing) of data.Stat 10, UCLA, Ivo DinovSlide 8Realproblems,curiosityQuestionsaboutworldDesignmethodof datacollectionCollectdataSummaryandanalysisof dataAnswerstooriginalquestions(a) (b) (c) (d) (e) (f)The investigative processStat 10, UCLA, Ivo DinovSlide 9Questions What are two ways in which random observations arise and give examples. (random sampling from finite population –randomized scientific experiment; random process producing data, observational data, surveys.) 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 10, UCLA, Ivo DinovSlide 10Review Let {x1,1, x1,2, x1,3,, …, x1,N,}{x2,1, x2,2, x2,3,, …, x2,N,} …{xK,1, xK,2, xK,3,, …, xK,N,}  As the number of samples and the number of observations within each sample increase we get a better estimate of the true population parameter (say the mean). Scottish soldiers chest measurements example … K samples ofsize N.Data comes froma distr’n withµ, σ, but we’reinterested inmean/std-devof sample averageStat 10, UCLA, Ivo DinovSlide 11The sample mean has a sampling distributionSampling batches of 6 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 10, UCLA, Ivo DinovSlide 12Twelve samples of size 2434 36 38 40 42 44 46123456789101211Samplenumber12 samples of size 24ChestmeasurementsDifference with the 6-unit samples?3Stat 10, UCLA, Ivo DinovSlide 13Histograms 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 10, UCLA, Ivo DinovSlide 14E(sample mean) = Population meansize SamplePopulation = )SD(SDnsample meaMean and SD of the sampling distributionnnXXXXσσσσµµµµ================)(SD)(SD ,)(E)(EStat 10, UCLA, Ivo DinovSlide 15 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? (sample mean estimate vs. estimator) 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 10, UCLA, Ivo DinovSlide 16Bias and Precision The bias in an estimator is the distance between between the center of the sampling distribution of the estimator and the true value of the parameter being estimated. In math terms, bias = , where theta, , is the estimator, as a RV, of the true (unknown) parameter . Example, Why is the sample mean an unbiasedestimate for the population mean? How about ¾ of the sample mean? θθθθ−−−−)ˆ(ΘEΘˆθθθθ011)ˆ( ====−−−−∑∑∑∑========−−−−µµµµµµµµnkkXnEE Θgeneral.in