ECE 510 Lecture 6 Confidence Limits. Scott Johnson Glenn Shirley
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1 ECE 510 Lecture 6 Confidence Limits Scott Johnson Glenn Shirley
2 Concepts 28 Jan 2013 S.C.Johnson, C.G.Shirley 2
3 Statistical Inference Population True ( population ) value = parameter Sample Sample value = statistic Use a sample statistic to estimate a population parameter 28 Jan 2013 S.C.Johnson, C.G.Shirley 3
4 Statistical Inference (Continuous) Population Mean=2.98 Stdev=0.50 Sample parameters Mean=3.00 Stdev=0.48 statistics 2.0W 2.5W 3.0W 3.5W 4.0W Example of continuous case: Use sample to estimate population mean and standard deviation 28 Jan 2013 S.C.Johnson, C.G.Shirley 4
5 Population Statistical Inference (Discrete) Sample 25,000 DPM parameter 40,000 DPM statistic Good Unit Bad Unit Example of discrete case: Use sample to estimate population defect DPM (DPM=Defects Per Million) 28 Jan 2013 S.C.Johnson, C.G.Shirley 5
6 Note: Samples Must Be Random! Population Not random Population = 55,000 DPM Sample = 204,000 DPM Sample Good Unit Bad Unit Samples must be representative of the entire population! Best to select samples truly randomly Not the first lot available or other partly-random methods No statistical analysis can correct for non-random samples 28 Jan 2013 S.C.Johnson, C.G.Shirley 6
7 Probability Distributions of Statistics Population Sample True ( population ) value = parameter Sample value = statistic Distribution of statistic: 0.2 Probability of Finding V when True Value = 200 Measured statistic is not enough Need to add either Confidence interval or limits Answer to a statistically-well-posed question ( hypothesis test ) Calculated from distributions of statistics If we looked at many samples from many identical populations, what values of the statistics might we get? Measured Value V 28 Jan 2013 S.C.Johnson, C.G.Shirley 7
8 Probability Probability Population has one true distribution: Distributions of Statistics (Continuous) Different samples have different distributions: σ (pop stdev) μ (population mean) S x S x S x Properties of sample distributions are statistics. We can calculate distributions of these statistics: Distribution of Means (Normal) Measured mean Distribution of Standard Deviations (Chi-square) Measured standard deviation We get one value for each from our one sample. 28 Jan 2013 S.C.Johnson, C.G.Shirley 8
9 Probability Distributions of Statistics (Discrete) Population has one true DPM: 25,000 DPM Different samples 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 measured sample DPM is a statistic. We can calculate the distribution of this statistic: PDF - Probability of Seeing x Fails (Binomial) Number of failures in sample We get one value from our one sample. 28 Jan 2013 S.C.Johnson, C.G.Shirley 9
10 DPM Simulation 28 Jan 2013 S.C.Johnson, C.G.Shirley 10
11 Population Window Shows 10,000 units, most good, a few bad 28 Jan 2013 S.C.Johnson, C.G.Shirley 11
12 The Sample You can move the sample box 28 Jan 2013 S.C.Johnson, C.G.Shirley 12
13 DPM Indicator on DPM Histogram 6000 DPM 2000 DPM 28 Jan 2013 S.C.Johnson, C.G.Shirley 13
14 Binomial Histogram Gives probability of getting each measurement given the true DPM 28 Jan 2013 S.C.Johnson, C.G.Shirley 14
15 Binomial Distribution =binomdist (6, 1000, 0.005, false) 6 fails 1000 samples 5000 DPM Not cumulative binomdist N f N f f, N, p, false p 1 p f 28 Jan 2013 S.C.Johnson, C.G.Shirley 15
16 True DPM True DPM is adjustable Not in the real world, only the simulation! 28 Jan 2013 S.C.Johnson, C.G.Shirley 16
17 Low DPM 28 Jan 2013 S.C.Johnson, C.G.Shirley 17
18 High DPM 28 Jan 2013 S.C.Johnson, C.G.Shirley 18
19 Please put True DPM back to 5, Jan 2013 S.C.Johnson, C.G.Shirley 19
20 Small Sample Size 28 Jan 2013 S.C.Johnson, C.G.Shirley 20
21 Large Sample Size 28 Jan 2013 S.C.Johnson, C.G.Shirley 21
22 Statistical Measurement Uncertainty Small sample (400) = wide range 0 12,500 Large sample (2000) = narrow range Jan 2013 S.C.Johnson, C.G.Shirley 22
23 Exercise 6.1 (A) Set sample size = 1000 (B) Set True DPM = 1100 DPM and look for a sample with 3 fails what DPM does that represent? (C) Set True DPM = 6700 DPM and look for a sample with 3 fails what DPM does that represent? 28 Jan 2013 S.C.Johnson, C.G.Shirley 23
24 Why We Need Confidence Limits Did you get a bad sample from a good population? or a good sample from a bad population? 28 Jan 2013 S.C.Johnson, C.G.Shirley 24
25 Confidence Limits 28 Jan 2013 S.C.Johnson, C.G.Shirley 25
26 Confidence Interval Meaning True DPM m m m m m m m m m m Confidence level = 90% = 0.9 Risk of being wrong = 1 confidence level = α = 10% = % of random sample means with this confidence interval include the true population mean 28 Jan 2013 S.C.Johnson, C.G.Shirley 26
27 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
28 1-Sided UCL Meaning m True DPM m m m m m m m m 90% of random sample means with this confidence interval include the true population mean m 28 Jan 2013 S.C.Johnson, C.G.Shirley 28
29 Calculating Confidence Limits 28 Jan 2013 S.C.Johnson, C.G.Shirley 29
30 Exercise 6.2a Monte Carlo determination of binomial CL 28 Jan 2013 S.C.Johnson, C.G.Shirley 30
31 Exercise 6.2b Compare Monte Carlo CL to analytic 28 Jan 2013 S.C.Johnson, C.G.Shirley 31
32 Monte Carlo Exponential CL λ=0.022 MTTF = 45 hr λ=0.031 MTTF = 32 hr λ=0.042 MTTF = 24 hr 95% 5% Jan 2013 S.C.Johnson, C.G.Shirley 32
33 Exercise 6.3a Monte Carlo determination of exponential CL 28 Jan 2013 S.C.Johnson, C.G.Shirley 33
34 Analytic Exponential CL Answer: a gamma or a chi-square distribution Confidence intervals taken from that 30 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 34
35 Exercise 6.3b Compare analytic to Monte Carlo values 28 Jan 2013 S.C.Johnson, C.G.Shirley 35
36 28 Jan 2013 S.C.Johnson, C.G.Shirley 36
37 28 Jan 2013 S.C.Johnson, C.G.Shirley 37
38 28 Jan 2013 S.C.Johnson, C.G.Shirley 38
39 28 Jan 2013 S.C.Johnson, C.G.Shirley 39
40 28 Jan 2013 S.C.Johnson, C.G.Shirley 40
41 The End 28 Jan 2013 S.C.Johnson, C.G.Shirley 41
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