Life of Incandescent Light Bulbs

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.

Median Rank Tables

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]