General Examples Using the Weibull Distribution

Weibull Distribution Example 7

In this example, we will determine the median rank value used for plotting the sixth failure from a sample size of ten. This will be used to illustrate two of the built-in functions in Weibull++'s Quick Statistical Reference.

Solution to Weibull Distribution Example 7

First, open the Quick Statistical Reference by clicking its icon.

or by selecting Quick Statistical Reference from the Tools menu.

In this example N = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

Calculate the value of F0.50:10:12 by using the Inverse F-Distribution Values option from the Quick Statistical Reference, or F0.50;10;12 = 0.9886 as shown next:

Consequently:

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Weibull Distribution Example 8

What is the unreliability of the units in Example 1 for a mission duration of 30 hours, starting the mission at age zero? To replicate the results in this reference with Weibull++, choose RRX (Rank Regression on X) as the calculation method.

Solution to Weibull Distribution Example 8

First, use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off . Then, a good estimate of the sought unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP). To activate the Quick Calculation Pad either click its icon on the Data Analysis toolbar, or Data Folio Control Panel

or select it from the Tools menu. Once activated, the QCP will appear on your desktop. Switch to the Basic Calculations page of the QCP, select Std. Prob. Calculations from the Options For Calculations box, and enter 30 hours in the Mission End Time box, making sure that the Results as Probability of Failure option is selected in the Results Option box.

Click Calculate to get the result. The result is shown in the figure above, or

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

Weibull Distribution Example 9

What is the reliability for a new mission of t = 10 hours duration, starting the new mission at the age of T = 30 hours, for the same data as Example 7?

Solution to Weibull Distribution Example 9

The conditional reliability is given by:

or:

Again, the Quick Calculation Pad can provide this result directly and more accurately than the plot.

Weibull Distribution Example 10

What is the longest mission that this equipment should undertake for a reliability of 90%?

Solution to Weibull Distribution Example 10

Using the QCP again, choose Warranty (Time) Information and enter the Required Reliability, 0.90, and click Calculate. The result is 15.9933 hours.

Weibull Distribution Example 11

Assume that ten identical units (N = 10) are being reliability tested at the same application and operation stress levels. Six of these units fail during this test after operating the following numbers of hours, Tj: 150, 105, 83, 123, 64 and 46. The test is stopped at the sixth failure. Find the parameters of the Weibull pdf that represents these data.

Solution to Weibull Distribution Example 11

Open a new Data Folio choosing Times-to-failure data, My data set contains suspensions (right censored data) and I want to enter data in groups.

Enter the data in the appropriate columns. Note that there are four suspensions, as only six of the ten units were tested to failure (the next figure shows the data as entered). Use the three-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Weibull Distribution Example 12

Suppose that we have run an experiment with eight units being tested and the following is a table of their last inspection times and times-to-failure:

Table 6.5 - The test data for Example 16

Analyze the data using several different parameter estimation techniques and compare the results.

Solution to Weibull Distribution Example 12

This data set can be entered into Weibull++ by opening a new Data Folio and choosing Times-to-failure and My data set contains interval and/or left censored data.

The data is entered as follows,

The computed parameters using maximum likelihood are:

using RRX or rank regression on X:

and using RRY or rank regression on Y:

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Weibull Distribution Example 13

ACME company manufactures widgets, and is currently engaged in reliability testing a new widget design. Nineteen units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. Table 6.6 contains the collected data.

Table 6.6 - Widget test data

Solution to Weibull Distribution Example 13

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Further, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

This data set can be entered into Weibull++ using Times-to-failure and My data set contains suspensions (right censored data).

We will use the two-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

using RRX:

and using RRY:

Weibull Distribution Example 14

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. Table 6.8 contains the data.

Table 6.8 - The test data for Example 13

Solution to Weibull Distribution Example 14

This data set can be entered into Weibull++ by selecting the Times-to-failure and My data set contains suspensions (right censored data), My data set contains interval and/or left censored data and I want to enter data in groups options.

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data.

The computed parameters using MLE are:

using RRX:

and using RRY:

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

See Also:
The Weibull Distribution

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