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1Albert R Meyer, May 10, 2010 lec 14M.1 Mathematics for Computer ScienceMIT 6.042J/18.062JSampling & ConfidenceAlbert R Meyer, May 10, 2010 lec 14M.2 Estimate % contaminated fish in Charles River?Procedure: catch n fish, test each, use %contaminated in catch as estimate of %contaminated in whole river SamplingAlbert R Meyer, May 10, 2010 lec 14M.3 Sampling Questions Catch 500 fish; what isprobability that estimateis within 0.1 of the actualfraction?Albert R Meyer, May 10, 2010 Model as Coin Tosses lec 14M.4 p ::= fraction contaminated in river test a fish  toss bias p coin catch n fish  toss n coins An::= fraction contaminated in the sample of nAlbert R Meyer, May 10, 2010 Pr An- μ > {} 1n2Pr A500- p > 0.1{}1500 0.12Pairwise Independent Sampling 1/2n = 500,μ= p,= 0.1worst =12Albert R Meyer, May 10, 2010 Pr An- μ > {} 1n2Pr A500- p > 0.1{}1500 0.12Pairwise Independent Sampling 1/2Pr A500-p  0.1{}> 0.95n = 500,μ= p,= 0.12Albert R Meyer, May 10, 2010 lec 14M.7 With probability 0.95 ourestimated fraction will be within 0.1 of the actualfraction of contaminated fish in the whole river. Confidence in our estimateAlbert R Meyer, May 10, 2010 Better estimate: lec 14M.8 Sampling using Binomial PDF AnisBn,pnPr An p {}= Pr Bn,p np n{}Albert R Meyer, May 10, 2010 Pr Bn,p np n{}Pr B500,p500p 0.06()500{} 30Better estimate: lec 14M.10 Sampling using Binomial PDF n = 500,= 0.06Albert R Meyer, May 10, 2010 lec 14M.11 How to bound this probability when we don’t know p?Lemma: is min when p = 1/2Sampling using Binomial PDF Pr Bn,p np n{}Albert R Meyer, May 10, 2010 Pr B500,p500p 0.06()500{}lec 14M.12 Sampling using Binomial PDF 301/2250Pr 220  B500,1/2 280{}Albert R Meyer, May 10, 2010 Pr 220  B500,1/2 280{}lec 14M.13 Sampling using Binomial PDF =500ii=220280 2500 0.993Albert R Meyer, May 10, 2010 lec 14M.14 We can actually be 99%confident that our estimated fraction is with 0.06 of the true fraction of contaminated fish in the whole river. Confidence in our estimateAlbert R Meyer, May 10, 2010 lec 14M.15 ConfidenceNow suppose we sample 500 fish and discover 230 are contaminated. So we estimate p is 230/500 = 0.46It’s tempting to say “the probability thatp = 0.46± 0.06is at least 0.99”--technically wrong! not Probable Reality Albert R Meyer, May 10, 2010 lec 14M.16 p is the actual fraction of bad fish in the river. p is unknown,but not a random variable! ConfidenceAlbert R Meyer, May 10, 2010 lec 14M.17 The possible outcomes of oursampling procedure is a random variable. We can say that the “probabilitythat our sampling process will yield a fraction that is ± 0.06 of the true fraction at least 0.99”ConfidenceAlbert R Meyer, May 10, 2010 lec 14M.20 for simplicity we say that p = 0.46 ±0.06 at the 99% confidence level ConfidenceAlbert R Meyer, May 10, 2010 Moral: when you are told that some fact holds at a highconfidence level, rememberthat a random experiment lies behind this claim. lec 14M.21 ConfidenceAskyourself “what experiment?”4Albert R Meyer, May 10, 2010 Team Problems Problems14lec 14M.22MIT OpenCourseWarehttp://ocw.mit.edu6.042J / 18.062J Mathematics for Computer ScienceSpring 2010For information about citing these materials or our Terms of Use, visit:


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MIT 6 042J - Sampling & Confidence

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