1. MAE430 Reliability Engineering in ME Term Project I Jae Hyung Cho 20101103 Andreas Beckmann 20156476

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

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

4. Normal Lognormal WeibullBiexponential Linearity Check: Symmetric Simple Cumulative Distribution 4

5. Linearity Check: Mean Rank 5 Normal Lognormal WeibullBiexponential

6. Normal Lognormal WeibullBiexponential Linearity Check: Median Rank 6

7. Normal Lognormal WeibullBiexponential Linearity Check: The Other Method 7

8. Linearity Check Results 8 S.S.C.D.MeanMedianOther NormalOOOO LognormalXXXX WeibullOOOO BiexponentialXXXX S.S.C.D.MeanMedianOther Normal 0.967400.973660.970570.96953 Lognormal 0.892260.890780.891770.89198 Weibull 0.979660.976800.978960.97934 Biexponential 0.923760.943490.933670.93040 Graph Linearity Adjusted R-squared of Linear Fit

9. 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

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

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

12. 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

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

14. 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

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

16. 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

17. 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 = 0.93367 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

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

19. Normal Lognormal WeibullBiexponential Linearity Check: Symmetric Simple Cumulative Distribution 19

20. Linearity Check: Mean Rank 20 Normal Lognormal WeibullBiexponential

21. Normal Lognormal WeibullBiexponential Linearity Check: Median Rank 21

22. Normal Lognormal WeibullBiexponential Linearity Check: The Other Method 22

23. Linearity Check Results 23 S.S.C.D.MeanMedianOther NormalOOOO LognormalXXXX WeibullOOOO BiexponentialXXXX S.S.C.D.MeanMedianOther Normal 0.952930.962840.957850.95621 Lognormal 0.898180.900600.899590.89916 Weibull 0.972930.977250.975760.97496 Biexponential 0.861190.887950.874360.86996 Graph Linearity Adjusted R-squared of Linear Fit

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

25. 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

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

27. 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

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

29. 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

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

31. 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

32. 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 = 0.97293 Comparison with Linearity Test Best result with Weibull distribution Nice Linearity High Adj. R-squared value

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

34. Normal Lognormal WeibullBiexponential Linearity Check: Symmetric Simple Cumulative Distribution 34

35. Linearity Check: Mean Rank 35 Normal Lognormal WeibullBiexponential

36. Normal Lognormal WeibullBiexponential Linearity Check: Median Rank 36

37. Normal Lognormal WeibullBiexponential Linearity Check: The Other Method 37

38. Linearity Check Results 38 S.S.C.D.MeanMedianOther NormalOOOO LognormalXXXX WeibullOOOO BiexponentialXXXX S.S.C.D.MeanMedianOther Normal 0.964020.970470.967160.96610 Lognormal 0.875280.874910.875260.8753 Weibull 0.974800.974450.975150.97514 Biexponential 0.898070.916510.907050.90403 Graph Linearity Adjusted R-squared of Linear Fit

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

40. 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

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

42. 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

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

44. 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

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

46. 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

47. K-S Test Results 47 S.S.C.D.MeanMedianOther NormalXXXX LognormalXXXX WeibullXXXX BiexponentialXXXX S.S.C.D.MeanMedianOther Normal 0.964020.970470.967160.96610 Weibull 0.974800.974450.975150.97514 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

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

49. 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