Template:Laplace trend test rsa

Laplace Trend Test
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, $$U$$, using the following equation:


 * $$U=\frac{\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{X}_{i}}}{N}-\tfrac{T}{2}}{T\sqrt{\tfrac{1}{12N}}}$$

where:
 * •	 $$T$$ = total operating time (termination time)
 * •	 $${{X}_{i}}$$ = age of the system at the  $${{i}^{th}}$$  successive failure
 * •	 $$N$$ = total number of failures

The test statistic $$U$$  is approximately a standard normal random variable. The critical value is read from the Standard Normal tables with a given significance level, $$\alpha $$. Example 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 $$U$$  for System 1 using Eqn. (Utatistic).


 * $$U=-2.6121$$

From the Standard Normal tables with a significance level of 0.10, the critical value is equal to 1.645. If $$-1.6451.645$$  then a deteriorating trend would exist.