Life of Incandescent Light Bulbs: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
No edit summary
Line 38: Line 38:
|}
|}


'''Do the following:'''
'''Do the following:'''  


#Plot the data on a Weibull probability plot and obtain the Weibull model parameters.  
#Plot the data on a Weibull probability plot and obtain the Weibull model parameters.  
#Compute the B10 life of the bulbs.
#Compute the B10 life of the bulbs.


<br>


The median ranks&nbsp; for&nbsp;the &nbsp;the <math>{{j}^{th}}</math>&nbsp;failure out of N&nbsp;units is obtained by solving the cumulative binomial equation for <math>Z</math> . This however requires numerical solution.&nbsp; Tables of median ranks can be used in lieu of the solution.


Median ranks are used to obtain an estimate of the unreliability, <math>Q({T_j})</math> for each failure. It is the value that the true probability of failure, <math>Q({{T}_{j}}),</math> should have at the <math>{{j}^{th}}</math> failure out of a sample of <math>N</math> units at a <math>50%</math> confidence level. This essentially means that this is our best estimate for the unreliability. Half of the time the true value will be greater than the 50% confidence estimate, the other half of the time the true value will be less than the estimate. This estimate is based on a solution of the binomial equation. The rank can be found for any percentage point, <math>P</math>, greater than zero and less than one, by solving the cumulative binomial equation for <math>Z</math> . This represents the rank, or unreliability estimate, for the <math>{{j}^{th}}</math> failure[15; 16] in the following equation for the cumulative binomial:  
 
 
 
 
represents the rank, or unreliability estimate, for the&nbsp; failure[15; 16] in the following equation for the cumulative binomial:  


<math>P=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
<math>P=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
Line 58: Line 64:
   k  \\
   k  \\
\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>  
\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>  


[[Category:Weibull_Examples]]
[[Category:Weibull_Examples]]

Revision as of 03:31, 25 June 2011

This example uses time-to-failure data from a life test done on incandescent light bulbs. The observed times-to-failure are given in the next table.   

Observed times-to-failure for ten bulbs in hours.
Order Number Hours-to-failure
1 361
2 680
3  721
4 905
5 1010
6  1090
7 1157
8 1330
9 1400
10 1695

Do the following:

  1. Plot the data on a Weibull probability plot and obtain the Weibull model parameters.
  2. Compute the B10 life of the bulbs.


The median ranks  for the  the [math]\displaystyle{ {{j}^{th}} }[/math] failure out of N units is obtained by solving the cumulative binomial equation for [math]\displaystyle{ Z }[/math] . This however requires numerical solution.  Tables of median ranks can be used in lieu of the solution.



represents the rank, or unreliability estimate, for the  failure[15; 16] in the following equation for the cumulative binomial:

[math]\displaystyle{ P=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix} N \\ k \\ \end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}} }[/math]


where [math]\displaystyle{ N }[/math] is the sample size and [math]\displaystyle{ j }[/math] the order number. The median rank is obtained by solving this equation for [math]\displaystyle{ Z }[/math] at [math]\displaystyle{ P=0.50, }[/math]

[math]\displaystyle{ 0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix} N \\ k \\ \end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}} }[/math]

Retrieved from "https://www.reliawiki.com/index.php?title=Life_of_Incandescent_Light_Bulbs&oldid=3397"