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=Hypothesis Tests=
#REDIRECT [[RGA_Appendix_B]]
{{common beta hypothesis test rsa}}
 
==Laplace Trend Test==
<br>
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:
 
::<math>U=\frac{\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{X}_{i}}}{N}-\tfrac{T}{2}}{T\sqrt{\tfrac{1}{12N}}}</math>
 
 
where:
<br>
:• <math>T</math>  = total operating time (termination time)
:• <math>{{X}_{i}}</math>  = age of the system at the  <math>{{i}^{th}}</math>  successive failure
:• <math>N</math>  = total number of failures
<br>
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> .
<br>
<br>
'''Example'''
<br>
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).
 
 
::<math>U=-2.6121</math>
 
 
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>
If  <math>U>1.645</math>  then a deteriorating trend would exist.
 
<br>
 
==Critical Values for Cramér-von Mises Test==
<br>
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> .
 
<br>
<br>
 
{|style= align="center" border="1"
|-
|colspan="6" style="text-align:center"|Table B.2 - Critical values for Cramér-von Mises test
|-
| ||colspan="5" style="text-align:center;"|<math>\alpha </math>
|-
|<math>M</math>|| 0.20|| 0.15|| 0.10|| 0.05|| 0.01
|-
|2|| 0.138|| 0.149|| 0.162|| 0.175|| 0.186
|-
|3|| 0.121|| 0.135|| 0.154|| 0.184||0.23
|-
|4|| 0.121|| 0.134|| 0.155|| 0.191||0.28
|-
|5|| 0.121|| 0.137|| 0.160|| 0.199||0.30
|-
|6|| 0.123|| 0.139|| 0.162|| 0.204||0.31
|-
|7|| 0.124|| 0.140|| 0.165|| 0.208||0.32
|-
|8|| 0.124|| 0.141|| 0.165|| 0.210||0.32
|-
|9|| 0.125|| 0.142|| 0.167|| 0.212||0.32
|-
|10|| 0.125|| 0.142|| 0.167|| 0.212||0.32
|-
|11|| 0.126|| 0.143|| 0.169|| 0.214||0.32
|-
|12|| 0.126|| 0.144|| 0.169|| 0.214||0.32
|-
|13|| 0.126|| 0.144|| 0.169|| 0.214||0.33
|-
|14|| 0.126|| 0.144|| 0.169|| 0.214||0.33
|-
|15|| 0.126|| 0.144|| 0.169|| 0.215||0.33
|-
|16|| 0.127|| 0.145|| 0.171|| 0.216|| 0.33
|-
|17|| 0.127|| 0.145|| 0.171|| 0.217|| 0.33
|-
|18|| 0.127|| 0.146|| 0.171|| 0.217|| 0.33
|-
|19|| 0.127|| 0.146|| 0.171|| 0.217|| 0.33
|-
|20|| 0.128|| 0.146|| 0.172|| 0.217|| 0.33
|-
|30|| 0.128|| 0.146|| 0.172|| 0.218|| 0.33
|-
|60|| 0.128|| 0.147|| 0.173|| 0.220|| 0.33
|-
|100|| 0.129|| 0.147|| 0.173|| 0.220|| 0.34
|}
 
For application of the Cramér-von Mises critical values, refer to Sections 5.5.1 and 10.1.6.1.

Latest revision as of 23:06, 23 August 2012

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