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| =Hypothesis Tests=
| | #REDIRECT [[RGA_Appendix_B]] |
| {{common beta hypothesis test rsa}}
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| ==Laplace Trend Test==
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| <br>
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| The Laplace Trend Test tests the hypothesis that a trend does not exist within the data. The Laplace Trend test is applicable to the following data types: Multiple Systems-Concurrent Operating Times, Repairable and Fleet. The Laplace Trend Test can determine whether the system is deteriorating, improving, or if there is no trend at all. Calculate the test statistic, <math>U</math> , using the following equation:
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| ::<math>U=\frac{\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{X}_{i}}}{N}-\tfrac{T}{2}}{T\sqrt{\tfrac{1}{12N}}}</math>
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| where:
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| <br>
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| :• <math>T</math> = total operating time (termination time)
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| :• <math>{{X}_{i}}</math> = age of the system at the <math>{{i}^{th}}</math> successive failure
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| :• <math>N</math> = total number of failures
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| <br>
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| The test statistic <math>U</math> is approximately a standard normal random variable. The critical value is read from the Standard Normal tables with a given significance level, <math>\alpha </math> .
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| <br>
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| <br>
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| '''Example'''
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| <br>
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| Consider once again the data in Table B.1. Check for a trend within System 1 assuming a significance level of 0.10. Calculate the test statistic <math>U</math> for System 1 using Eqn. (Utatistic).
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| ::<math>U=-2.6121</math>
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| From the Standard Normal tables with a significance level of 0.10, the critical value is equal to 1.645. If <math>-1.645<U<1.645</math> then we would fail to reject the hypothesis of no trend. However, since <math>U<-1.645</math> then an improving trend exists within System 1. <br>
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| If <math>U>1.645</math> then a deteriorating trend would exist.
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| <br>
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| ==Critical Values for Cramér-von Mises Test==
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| <br>
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| Table B.2 displays the critical values for the Cramér-von Mises goodness-of-fit test given the sample size, <math>M</math> , and the significance level, <math>\alpha </math> .
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| <br>
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| <br>
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| {|style= align="center" border="1"
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| |colspan="6" style="text-align:center"|Table B.2 - Critical values for Cramér-von Mises test
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| | ||colspan="5" style="text-align:center;"|<math>\alpha </math>
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| |<math>M</math>|| 0.20|| 0.15|| 0.10|| 0.05|| 0.01
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| |2|| 0.138|| 0.149|| 0.162|| 0.175|| 0.186
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| |-
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| |3|| 0.121|| 0.135|| 0.154|| 0.184||0.23
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| |-
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| |4|| 0.121|| 0.134|| 0.155|| 0.191||0.28
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| |-
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| |5|| 0.121|| 0.137|| 0.160|| 0.199||0.30
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| |6|| 0.123|| 0.139|| 0.162|| 0.204||0.31
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| |7|| 0.124|| 0.140|| 0.165|| 0.208||0.32
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| |8|| 0.124|| 0.141|| 0.165|| 0.210||0.32
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| |9|| 0.125|| 0.142|| 0.167|| 0.212||0.32
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| |10|| 0.125|| 0.142|| 0.167|| 0.212||0.32
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| |11|| 0.126|| 0.143|| 0.169|| 0.214||0.32
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| |12|| 0.126|| 0.144|| 0.169|| 0.214||0.32
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| |13|| 0.126|| 0.144|| 0.169|| 0.214||0.33
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| |14|| 0.126|| 0.144|| 0.169|| 0.214||0.33
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| |15|| 0.126|| 0.144|| 0.169|| 0.215||0.33
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| |16|| 0.127|| 0.145|| 0.171|| 0.216|| 0.33
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| |17|| 0.127|| 0.145|| 0.171|| 0.217|| 0.33
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| |18|| 0.127|| 0.146|| 0.171|| 0.217|| 0.33
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| |19|| 0.127|| 0.146|| 0.171|| 0.217|| 0.33
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| |20|| 0.128|| 0.146|| 0.172|| 0.217|| 0.33
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| |30|| 0.128|| 0.146|| 0.172|| 0.218|| 0.33
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| |60|| 0.128|| 0.147|| 0.173|| 0.220|| 0.33
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| |100|| 0.129|| 0.147|| 0.173|| 0.220|| 0.34
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| |}
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| For application of the Cramér-von Mises critical values, refer to Sections 5.5.1 and 10.1.6.1.
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