MAE430 Reliability Engineering in ME Term Project I Jae Hyung Cho Andreas Beckmann

Contents Jae Hyung’s Data Set (n = 63) Andreas’s Data Set (n = 59) Combined Data Set (n = 122) Summary 2

Contents Jae Hyung’s Data Set (n = 63) Andreas’s Data Set (n = 59) Combined Data Set (n = 122) Summary 3

Normal Lognormal WeibullBiexponential Linearity Check: Symmetric Simple Cumulative Distribution 4

Linearity Check: Mean Rank 5 Normal Lognormal WeibullBiexponential

Normal Lognormal WeibullBiexponential Linearity Check: Median Rank 6

Normal Lognormal WeibullBiexponential Linearity Check: The Other Method 7

Linearity Check Results 8 S.S.C.D.MeanMedianOther NormalOOOO LognormalXXXX WeibullOOOO BiexponentialXXXX S.S.C.D.MeanMedianOther Normal Lognormal Weibull Biexponential Graph Linearity Adjusted R-squared of Linear Fit

Normal Lognormal WeibullBiexponential K-S Test: Symmetric Simple Cumulative Distribution 9 Clearly Failing Low significance levels (α = 0.01, and 0.05) are used throughout our project

Normal WeibullBiexponential K-S Test: Symmetric Simple Cumulative Distribution (Zoomed in) 10 α = 0.01 α = 0.05 α = α = 0.01α = 0.025α = 0.05 Normal003 Weibull003 BiExp001 Number of Outliers

Normal Lognormal WeibullBiexponential K-S Test: Mean Rank 11 Clearly Failing

Normal WeibullBiexponential K-S Test: Mean Rank (Zoomed in) 12 α = 0.01 α = 0.05 α = 0.01α = 0.05 Normal02 Weibull00 BiExp00 Number of Outliers

Normal Lognormal WeibullBiexponential Clearly Failing K-S Test: Median Rank 13

Normal WeibullBiexponential K-S Test: Median Rank (Zoomed in) 14 α = 0.01 α = 0.05 α = 0.01α = 0.05 Normal02 Weibull02 BiExp00 Number of Outliers

Normal Lognormal WeibullBiexponential Clearly Failing K-S Test: The Other Method 15

Normal WeibullBiexponential K-S Test: The Other Method (Zoomed in) 16 α = 0.01 α = 0.05 α = 0.01α = 0.05 Normal02 Weibull03 BiExp01 Number of Outliers

Median Rank Biexponential K-S Test Results 17 S.S.C.D.MeanMedianOther NormalXXXX LognormalXXXX WeibullXOXX BiexponentialXOOX K-S Test Result for α = 0.05 (O: Accepted, X: Rejected) Adj. R-squared = Comparison with Linearity Test No indication that biexp. distribution fits well A couple of outliers But Linear in General Sufficiently high Adj. R-squared value

Contents Jae Hyung’s Data Set (n = 63) Andreas’s Data Set (n = 59) Combined Data Set (n = 122) Summary 18

Normal Lognormal WeibullBiexponential Linearity Check: Symmetric Simple Cumulative Distribution 19

Linearity Check: Mean Rank 20 Normal Lognormal WeibullBiexponential

Normal Lognormal WeibullBiexponential Linearity Check: Median Rank 21

Normal Lognormal WeibullBiexponential Linearity Check: The Other Method 22

Linearity Check Results 23 S.S.C.D.MeanMedianOther NormalOOOO LognormalXXXX WeibullOOOO BiexponentialXXXX S.S.C.D.MeanMedianOther Normal Lognormal Weibull Biexponential Graph Linearity Adjusted R-squared of Linear Fit

Normal Lognormal WeibullBiexponential K-S Test: Symmetric Simple Cumulative Distribution 24 Clearly Failing

Normal Weibull K-S Test: Symmetric Simple Cumulative Distribution (Zoomed in) 25 α = 0.01 α = 0.20 α = 0.05 α = 0.01α = 0.05α = 0.20 Normal002 Weibull000 Number of Outliers

Normal Lognormal WeibullBiexponential K-S Test: Mean Rank 26 Clearly Failing

Normal Weibull K-S Test: Mean Rank (Zoomed in) 27 α = 0.01 α = 0.20 α = 0.05 α = 0.01α = 0.05α = 0.20 Normal000 Weibull000 Number of Outliers

Normal Lognormal WeibullBiexponential Clearly Failing K-S Test: Median Rank 28 Clearly Failing

Normal Weibull K-S Test: Median Rank (Zoomed in) 29 α = 0.01α = 0.05α = 0.20 Normal000 Weibull000 Number of Outliers α = 0.01 α = 0.20 α = 0.05

Normal Lognormal WeibullBiexponential Clearly Failing K-S Test: The Other Method 30 Clearly Failing

Normal Weibull K-S Test: The Other Method (Zoomed in) 31 α = 0.01α = 0.05α = 0.20 Normal000 Weibull000 Number of Outliers α = 0.01 α = 0.20 α = 0.05

S.S.C.D. Weibull K-S Test Results 32 S.S.C.D.MeanMedianOther NormalXOOO LognormalXXXX WeibullOOOO BiexponentialXXXX K-S Test Result for α = 0.20 (O: Accepted, X: Rejected) Adj. R-squared = Comparison with Linearity Test Best result with Weibull distribution Nice Linearity High Adj. R-squared value

Contents Jae Hyung’s Data Set (n = 63) Andreas’s Data Set (n = 59) Combined Data Set (n = 122) Summary 33

Normal Lognormal WeibullBiexponential Linearity Check: Symmetric Simple Cumulative Distribution 34

Linearity Check: Mean Rank 35 Normal Lognormal WeibullBiexponential

Normal Lognormal WeibullBiexponential Linearity Check: Median Rank 36

Normal Lognormal WeibullBiexponential Linearity Check: The Other Method 37

Linearity Check Results 38 S.S.C.D.MeanMedianOther NormalOOOO LognormalXXXX WeibullOOOO BiexponentialXXXX S.S.C.D.MeanMedianOther Normal Lognormal Weibull Biexponential Graph Linearity Adjusted R-squared of Linear Fit

Normal Lognormal WeibullBiexponential K-S Test: Symmetric Simple Cumulative Distribution 39 Clearly Failing

Normal Weibull K-S Test: Symmetric Simple Cumulative Distribution (Zoomed in) 40 α = 0.01 α = 0.05 α = 0.01α = 0.05 Normal59 Weibull59 Number of Outliers

Normal Lognormal WeibullBiexponential K-S Test: Mean Rank 41 Clearly Failing

Normal Weibull K-S Test: Mean Rank (Zoomed in) 42 α = 0.01 α = 0.05 α = 0.01α = 0.05 Normal28 Weibull28 Number of Outliers

Normal Lognormal WeibullBiexponential Clearly Failing K-S Test: Median Rank 43 Clearly Failing

Normal Weibull K-S Test: Median Rank (Zoomed in) 44 α = 0.01 α = 0.05 α = 0.01α = 0.05 Normal28 Weibull28 Number of Outliers

Normal Lognormal WeibullBiexponential Clearly Failing K-S Test: The Other Method 45 Clearly Failing

Normal Weibull K-S Test: The Other Method (Zoomed in) 46 α = 0.01 α = 0.05 α = 0.01α = 0.05 Normal38 Weibull38 Number of Outliers

K-S Test Results 47 S.S.C.D.MeanMedianOther NormalXXXX LognormalXXXX WeibullXXXX BiexponentialXXXX S.S.C.D.MeanMedianOther Normal Weibull Adjusted R-squared of Linear Fit K-S Test Result for α = 0.01 (O: Accepted, X: Rejected) Every distribution is rejected with the smallest α value Normal and Weibull distributions fit better than the other two Fewest outliers with mean and median rank methods

Contents Jae Hyung’s Data Set (n = 63) Andreas’s Data Set (n = 59) Combined Data Set (n = 122) Summary 48

Results Summary Jae Hyung’s Data Set (n = 63) –Best fitting distribution: Biexponential Distribution –Best CDF estimation method: Median Rank Andreas’s Data Set (n = 59) –Best fitting distribution: Weibull Distribution –Best CDF estimation method: Symmetric S. C. D. Combined Data Set (n = 122) –Best fitting distribution: Weibull Distribution –Best CDF estimation method: Median Rank 49