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Mark Andersen Reliability Engineering HW 1 Februrary 4, 2008 Problem 3 a) Descriptive Statistics: Fail Time (Hz) Variable Fail Time (Hz) N 22 N* 0 Variable Fail Time (Hz) Maximum 173.40 Mean 72.39 SE Mean 8.18 StDev 38.36 Minimum 17.88 Q1 44.73 Median 61.68 b) Histogram of Hertz 7 6 Frequency 5 4 3 2 1 0 40 80 120 Fail Time (Hertz) c) Distribution ID Plot for Fail Time (Hz) 160 Q3 100.26 Probability Plot for Fail Time (Hz) LSXY Estimates-Complete Data Weibull C orrelation C oefficient Weibull 0.984 Lognormal 0.990 Exponential * Loglogistic 0.991 Lognormal 99 90 P er cent P er cent 90 50 10 50 10 1 10 1 10 100 Fail T ime (H z) E xponential 100 Fail T ime (H z) Loglogistic 99 90 50 P er cent P er cent 90 10 50 10 1 1 10 100 Fail T ime (H z) 1 10 1000 100 Fail T ime (H z) Out of these graphs, the exponential is not a good fit, and the Lognormal and Loglogistic are the best fits. Distribution ID Plot for Fail Time (Hz) Probability Plot for Fail Time (Hz) LSXY Estimates-Complete Data 3-Parameter Weibull C orrelation C oefficient 3-P arameter Weibull 0.989 3-P arameter Lognormal 0.993 2-P arameter E xponential * 3-P arameter Loglogistic 0.992 3-Parameter Lognormal 99 90 90 Percent Percent 50 10 50 10 1 1 10 100 Fail Time (Hz) - Threshold 50 100 Fail Time (Hz) - Threshold 2-Parameter Exponential 3-Parameter Loglogistic 99 90 50 Percent Percent 90 10 50 10 1 1 00 0. 0 01 0. 0 10 0. 0 00 1. 00 .0 10 0 00 00 .0 0. 00 0 10 1 Fail Time (Hz) - Threshold 1 10 100 Fail Time (Hz) - Threshold 200 2-parameter exponential is not a good fit; 3-parameter Lognormal provides the best fit. Distribution ID Plot for Fail Time (Hz) Probability Plot for Fail Time (Hz) LSXY Estimates-Complete Data Smallest Extreme Value C orrelation C oefficient S mallest Extreme V alue 0.898 N ormal 0.961 Logistic 0.962 Normal 99 90 90 Percent Percent 50 10 50 10 1 1 0 100 Fail Time (Hz) 200 0 50 100 Fail Time (Hz) 150 Logistic 99 Percent 90 50 10 1 0 50 100 Fail Time (Hz) 150 Normal provides best fit. **However, from Histogram we can see that this does not follow a normal distribution. 3-Parameter Loglogistic Goodness-of-Fit Distribution Weibull Lognormal Exponential Loglogistic 3-Parameter Weibull 3-Parameter Lognormal 2-Parameter Exponential 3-Parameter Loglogistic Smallest Extreme Value Normal Logistic Anderson-Darling (adj) 1.021 0.744 4.616 0.755 0.846 0.747 2.192 0.763 3.311 1.160 1.118 Correlation Coefficient 0.984 0.990 * 0.991 0.989 0.993 * 0.992 0.898 0.961 0.962 Table of MTTF Distribution Weibull Lognormal Mean 71.8534 74.3210 Standard Error 7.4325 9.8527 95% Normal CI Lower Upper 58.6677 88.0027 57.3150 96.3729 Exponential Loglogistic 3-Parameter Weibull 3-Parameter Lognormal 2-Parameter Exponential 3-Parameter Loglogistic Smallest Extreme Value Normal Logistic 60.5627 75.7660 72.4721 73.5634 68.2119 74.6895 71.4949 72.3873 72.3873 11.8104 10.4729 7.9522 9.0339 10.3120 9.6469 8.0717 8.2714 8.4251 41.3248 57.7851 58.4481 55.8572 50.7200 55.7819 55.6746 56.1755 55.8744 88.7563 99.3421 89.8611 91.2696 91.7362 93.5972 87.3152 88.5990 88.9002 e) The Weibull distribution provides a good fit and the Histogram from the data appears to follow a Weibull distribution Distribution Overview Plot for Fail Time (Hz) Distribution Overview Plot for Fail Time (Hz) LSXY Estimates-Complete Data P robability D ensity F unction Table of S tatistics S hape 2.19377 S cale 81.1337 M ean 71.8534 S tD ev 34.5652 M edian 68.6505 IQ R 48.1811 F ailure 22 C ensor 0 A D* 1.021 C orrelation 0.984 Weibull 0.012 90 P DF P er cent 0.008 0.004 0.000 0 50 100 Fail T ime (H z) 50 10 1 150 10 S urv iv al F unction Hazard F unction 100 0.06 Rate P er cent 100 Fail T ime (H z) 50 0.04 0.02 0 0.00 0 50 100 Fail T ime (H z) 150 0 50 100 Fail T ime (H z) 150 NOTE: The parameter values for the Weibull distribution are: Shape = 2.19377 a.k.a. ά Scale = 81.1337 a.k.a. λ See HW1_Problem3e.pdf for derived expressions of F, R, f, z, MTTF, and MRL