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| '''Normal Distribution Likelihood Ratio Bound Example (Time)'''
| | #REDIRECT [[The Normal Distribution]] |
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| For the data given in Example 5, determine the two-sided 80% confidence bounds on the time estimate for a reliability of 40%. The ML estimate for the time at <math>R(t)=40%</math> is 31.637.
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| '''Solution'''
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| In this example, we are trying to determine the two-sided 80% confidence bounds on the time estimate of 31.637. This is accomplished by substituting <math>R=0.40</math> and <math>\alpha =0.8</math> into the likelihood ratio equation for the normal distribution, and varying <math>\sigma </math> until the maximum and minimum values of <math>t</math> are found. The following table gives the values of <math>t</math> based on given values of <math>\sigma </math> .
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| [[Image:tabletbasedonsigma.gif|center|400px| ]]
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| <math></math>
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| This data set is represented graphically in the following contour plot:
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| [[Image:WB.9 time v sigma contour.png|center|400px| ]]
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| As can be determined from the table, the lowest calculated value for <math>t</math> is 25.046, while the highest is 39.250. These represent the 80% confidence limits on the time at which reliability is equal to 40%.
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