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.
| 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:
- Plot the data on a Weibull probability plot and obtain the Weibull model parameters.
- Compute the B10 life of the bulbs.
Median ranks are used to obtain an estimate of the unreliability, [math]\displaystyle{ Q({T_j}) }[/math] for each failure. It is the value that the true probability of failure, [math]\displaystyle{ Q({{T}_{j}}), }[/math] should have at the [math]\displaystyle{ {{j}^{th}} }[/math] failure out of a sample of [math]\displaystyle{ N }[/math] units at a [math]\displaystyle{ 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]\displaystyle{ P }[/math], greater than zero and less than one, by solving the cumulative binomial equation for [math]\displaystyle{ Z }[/math] . This represents the rank, or unreliability estimate, for the [math]\displaystyle{ {{j}^{th}} }[/math] 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]