View source for Template:Example: Normal Distribution Likelihood Ratio Bound (Reliability)

'''Normal Distribution Likelihood Ratio Bound Example (Reliability)'''

For the data given in Example 5, determine the two-sided 80% confidence bounds on the reliability estimate for  <math>t=30</math> .  The ML estimate for the reliability at  <math>t=30</math>  is 45.739%.

'''Solution'''
In this example, we are trying to determine the two-sided 80% confidence bounds on the reliability estimate of 45.739%. This is accomplished by substituting  <math>t=30</math>  and  <math>\alpha =0.8</math>  into the likelihood ratio equation for normal distribution, and varying  <math>\sigma </math>  until the maximum and minimum values of  <math>R</math>  are found. The following table gives the values of  <math>R</math>  based on given values of  <math>\sigma </math> .

[[Image:tablerbasedonsigma.png|center|350px| ]]

This data set is represented graphically in the following contour plot:

[[Image:WB.9 reliability v sigma.png|center|350px| ]] 

As can be determined from the table, the lowest calculated value for  <math>R</math>  is 24.776%, while the highest is 68.000%. These represent the 80% two-sided confidence limits on the reliability at  <math>t=30</math>