ECE 510 Lecture 6 Confidence Liits Scott Johnson Glenn Shirley
Concepts 28 Jan 2013 S.C.Johnson, C.G.Shirley 2
Statistical Inference Population True ( population ) value = paraeter Saple Saple value = statistic Use a saple statistic to estiate a population paraeter 28 Jan 2013 S.C.Johnson, C.G.Shirley 3
Statistical Inference (Continuous) Population Mean=2.98 Stdev=0.50 Saple 1.5 2 2.5 3 3.5 4 4.5 1.5 2 2.5 3 3.5 4 4.5 paraeters Mean=3.00 Stdev=0.48 statistics 2.0W 2.5W 3.0W 3.5W 4.0W Exaple of continuous case: Use saple to estiate population ean and standard deviation 28 Jan 2013 S.C.Johnson, C.G.Shirley 4
Population Statistical Inference (Discrete) Saple 25,000 DPM paraeter 40,000 DPM statistic Good Unit Bad Unit Exaple of discrete case: Use saple to estiate population defect DPM (DPM=Defects Per Million) 28 Jan 2013 S.C.Johnson, C.G.Shirley 5
Note: Saples Must Be Rando! Population Not rando Population = 55,000 DPM Saple = 204,000 DPM Saple Good Unit Bad Unit Saples ust be representative of the entire population! Best to select saples truly randoly Not the first lot available or other partly-rando ethods No statistical analysis can correct for non-rando saples 28 Jan 2013 S.C.Johnson, C.G.Shirley 6
Probability Population Distributions of Statistics True ( population ) value = paraeter Saple Saple value = statistic Distribution of statistic: 0.2 0.15 Probability of Finding V when True Value = 200 Measured statistic is not enough Need to add either Confidence interval or liits Answer to a statistically-well-posed question ( hypothesis test ) Calculated fro distributions of statistics If we looked at any saples fro any identical populations, what values of the statistics ight we get? 0.1 0.05 0 0 200 400 600 Measured Value V 28 Jan 2013 S.C.Johnson, C.G.Shirley 7
Probability Probability Population has one true distribution: Distributions of Statistics (Continuous) Different saples have different distributions: σ (pop stdev) 2.0 3.0 4.0 μ (population ean) S 2.0 3.0 4.0 x S 2.0 3.0 4.0 x S 2.0 3.0 4.0 x Properties of saple distributions are statistics. We can calculate distributions of these statistics: 0.3 0.25 0.2 0.15 0.1 0.05 0 Distribution of Means (Noral) 2 2.5 3 3.5 4 Measured ean Distribution of Standard Deviations 0.2 0.15 0.1 0.05 0 (Chi-square) 0 0.5 1 1.5 Measured standard deviation We get one value for each fro our one saple. 28 Jan 2013 S.C.Johnson, C.G.Shirley 8
Probability Distributions of Statistics (Discrete) Population has one true DPM: 25,000 DPM Different saples have different DPMs: 20,000 DPM (1 fail) 0 DPM (0 fail) 40,000 DPM (2 fail) 60,000 DPM (3 fail) 20,000 DPM (1 fail) 40,000 DPM (2 fail) The easured saple DPM is a statistic. We can calculate the distribution of this statistic: PDF - Probability of Seeing x Fails 0.4 0.3 (Binoial) 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Nuber of failures in saple We get one value fro our one saple. 28 Jan 2013 S.C.Johnson, C.G.Shirley 9
DPM Siulation 28 Jan 2013 S.C.Johnson, C.G.Shirley 10
Population Window Shows 10,000 units, ost good, a few bad 28 Jan 2013 S.C.Johnson, C.G.Shirley 11
The Saple You can ove the saple box 28 Jan 2013 S.C.Johnson, C.G.Shirley 12
DPM Indicator on DPM Histogra 6000 DPM 2000 DPM 28 Jan 2013 S.C.Johnson, C.G.Shirley 13
Binoial Histogra Gives probability of getting each easureent given the true DPM 28 Jan 2013 S.C.Johnson, C.G.Shirley 14
Binoial Distribution =binodist (6, 1000, 0.005, false) 6 fails 1000 saples 5000 DPM Not cuulative binodist N f N f f, N, p, false p 1 p f 28 Jan 2013 S.C.Johnson, C.G.Shirley 15
True DPM True DPM is adjustable Not in the real world, only the siulation! 28 Jan 2013 S.C.Johnson, C.G.Shirley 16
Low DPM 28 Jan 2013 S.C.Johnson, C.G.Shirley 17
High DPM 28 Jan 2013 S.C.Johnson, C.G.Shirley 18
Please put True DPM back to 5,000 28 Jan 2013 S.C.Johnson, C.G.Shirley 19
Sall Saple Size 28 Jan 2013 S.C.Johnson, C.G.Shirley 20
Large Saple Size 28 Jan 2013 S.C.Johnson, C.G.Shirley 21
Statistical Measureent Uncertainty Sall saple (400) = wide range 0 12,500 Large saple (2000) = narrow range 1500 8500 28 Jan 2013 S.C.Johnson, C.G.Shirley 22
(A) Set saple size = 1000 Exercise 6.1 (B) Set True DPM = 1100 DPM and look for a saple with 3 fails what DPM does that represent? (C) Set True DPM = 6700 DPM and look for a saple with 3 fails what DPM does that represent? 28 Jan 2013 S.C.Johnson, C.G.Shirley 23
Why We Need Confidence Liits Did you get a bad saple fro a good population? or a good saple fro a bad population? 28 Jan 2013 S.C.Johnson, C.G.Shirley 24
Confidence Liits 28 Jan 2013 S.C.Johnson, C.G.Shirley 25
Confidence Interval Meaning True DPM Confidence level = 90% = 0.9 Risk of being wrong = 1 confidence level = α = 10% = 0.1 90% of rando saple eans with this confidence interval include the true population ean 28 Jan 2013 S.C.Johnson, C.G.Shirley 26
1-Sided vs. 2-Sided 2-sided α/2 α/2 Upper α 1-sided Lower α 28 Jan 2013 S.C.Johnson, C.G.Shirley 27
1-Sided UCL Meaning True DPM 90% of rando saple eans with this confidence interval include the true population ean 28 Jan 2013 S.C.Johnson, C.G.Shirley 28
Calculating Confidence Liits 28 Jan 2013 S.C.Johnson, C.G.Shirley 29
Exercise 6.2 Monte Carlo deterination of binoial CL: In each row, siulate 10 pass/fail saples and count the nuber of fails Make a histogra of the count of runs that got each fail% Add the binoial prediction for each fail% Plot both as a bar chart Calculate cuulative values for your MC and calculated distributions Plot those with a line plot Use the cu plots to find the UCL and LCL for 3 fails / 10 units Copare to the analytic expressions (T&T section 11.3): LCL = BETAINV(5%, fails, saples fails+1) UCL = BETAINV(95%, fails+1, saples fails) 28 Jan 2013 S.C.Johnson, C.G.Shirley 30
Monte Carlo Exponential CL λ=0.022 MTTF = 45 hr λ=0.031 MTTF = 32 hr λ=0.042 MTTF = 24 hr 95% 5% 0.022 0.042 28 Jan 2013 S.C.Johnson, C.G.Shirley 31
Exercise 6.3 Monte Carlo deterination of exponential CL: In each row, siulate 50 exponentially distributed saples Deterine the best labda (exponential paraeter) for each row Make a CDF plot of the labda values Find the UCL and LCL for n=50 saples that found a labda of 3 Copare to the analytic expressions (T&T table 3.5): LCL = CHIINV(5%, 2*n) / (2*n) UCL = CHIINV(95%, 2*n+1) / (2*n) 28 Jan 2013 S.C.Johnson, C.G.Shirley 32
Analytic Exponential CL Answer: a gaa or a chi-square distribution Confidence intervals taken fro that 30 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 33
The End 28 Jan 2013 S.C.Johnson, C.G.Shirley 34