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| '''Lognormal Distribution Likelihood Ratio Bound Example (Reliability)'''
| | #REDIRECT [[The Lognormal Distribution]] |
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| For the data given in [[Lognormal Example 5 Data|Example 5]], determine the two-sided 75% confidence bounds on the reliability estimate for <math>t=65</math> . The ML estimate for the reliability at <math>t=65</math> is 64.261%.
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| '''Solution'''
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| In this example, we are trying to determine the two-sided 75% confidence bounds on the reliability estimate of 64.261%. This is accomplished by substituting <math>t=65</math> and <math>\alpha =0.75</math> into the likelihood function, 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> .
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| <center><math>\begin{matrix}
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| {{\sigma'}} & {{R}_{1}} & {{R}_{2}} & {{\sigma'}} & {{R}_{1}} & {{R}_{2}} \\
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| 0.24 & 61.107% & 75.910% & 0.37 & 43.573% & 78.845% \\
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| 0.25 & 55.906% & 78.742% & 0.38 & 43.807% & 78.180% \\
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| 0.26 & 55.528% & 80.131% & 0.39 & 44.147% & 77.448% \\
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| 0.27 & 50.067% & 80.903% & 0.40 & 44.593% & 76.646% \\
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| 0.28 & 48.206% & 81.319% & 0.41 & 45.146% & 75.767% \\
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| 0.29 & 46.779% & 81.499% & 0.42 & 45.813% & 74.802% \\
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| 0.30 & 45.685% & 81.508% & 0.43 & 46.604% & 73.737% \\
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| 0.31 & 44.857% & 81.387% & 0.44 & 47.538% & 72.551% \\
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| 0.32 & 44.250% & 81.159% & 0.45 & 48.645% & 71.212% \\
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| 0.33 & 43.827% & 80.842% & 0.46 & 49.980% & 69.661% \\
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| 0.34 & 43.565% & 80.446% & 0.47 & 51.652% & 67.789% \\
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| 0.35 & 43.444% & 79.979% & 0.48 & 53.956% & 65.299% \\
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| 0.36 & 43.450% & 79.444% & {} & {} & {} \\
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| \end{matrix}</math></center>
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| This data set is represented graphically in the following contour plot:
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| [[Image:WB.10 reliability v sigma.png|center|250px| ]]
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| As can be determined from the table, the lowest calculated value for <math>R</math> is 43.444%, while the highest is 81.508%. These represent the two-sided 75% confidence limits on the reliability at <math>t=65</math> .
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