Stat 321 – Day 24Lab 6 commentsPopulation size correction factorSlide 4Example 3: Estimating s2HW 7 CommentsLast Time – Point EstimatorsLab 8 PreviewStat 321 – Day 24Point Estimation (cont.)Lab 6 commentsParameterPopulationSampleProbabilityStatisticsxx xxxx xx x x x xxx x xx xx x x xxxx xx xxxx x xx xxxxx xxA statistic is an unbiased estimator for a parameter if its sampling distribution centers at the parameter value1. 2. Mean of empirical sampling distribution)(XE1. Humans not very good at selecting “random” samplesGettysburg, Literary Digest2. Sample size affects variability, not biasLarger probability of being close to 3. Population size doesn’t really matter!“bias” = systematic tendency to error in same directionNotes:- Sampling distribution vs. sample- Shape vs. spreadWith 664 randomly selected people, would have been within 5%Population size correction factorIf you do have a finite population, can apply a correction factor to the standard deviationBinomialHypergeometricN=303,572,923; n = 650 (+ 5%)N=303,572,923; n = 1500 (+ 2.5%))1()(,)( pnpXVnpXE NMnMnNnNXVNMnXE 11)(,)(Lab 6 commentsParameterPopulationSampleProbabilityStatisticsxx xxxx xx x x x xxx x xx xx x x xxxx xx xxxx x xx xxxxx xxA statistic is an unbiased estimator for a parameter if its sampling distribution centers at the parameter value1. 2. Mean of empirical sampling distribution)(XE ˆˆ?1)(θˆ211nXXniiMean = 71412Unbiased!Example 3: Estimating 2S2 is an unbiased estimator for 2 (p. 233)Although S is a biased estimator for 1)()(1)()(11)(1)()(1)()θˆE(212212212211nXEXVnXEXVnXEnXEnXnEXEnXXEiiniiiinininiiiiHW 7 CommentsMake sure you show lots details in deriving expected values and variancesInterpreting interval (# 3) vs. level (lab 7)Make sure have something random before making probability statementsSample size calculationsAlways round up to integer valueE(Y) = P(430.5 << 446.1) = .95? 1)//(2/2/nzXnzXPLast Time – Point EstimatorsFor parameters other than a mean or a proportion, need to think about best choice of estimatorMathematical formula for what you will do with your sample data to calculate an estimate of the parameter for a particular sampleGood properties to have:Unbiased: Sampling distribution of estimator centers at parameter, E(estimator) = Small variance/high precisionLab 8 PreviewDuring World War II, wanted to estimate the number of German tanksTurns out, the Mark V tanks were produced with sequential serial numbers, 1,…, NCan we use the numbers from n captured tanks to estimate N?Example: 170, 101, 5, 202, 43With one partner, suggest 3 estimators for NE.g., mean(Xi)Turn in with
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