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| '''Lognormal Distribution Likelihood Ratio Bound Example (Time)'''
| | #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 time estimate for a reliability of 80%. The ML estimate for the time at <math>R(t)=80%</math> is 55.718.
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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 time estimate of 55.718. This is accomplished by substituting <math>R=0.80</math> and <math>\alpha =0.75</math> into the likelihood function, 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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| <center><math>\begin{matrix}
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| {{\sigma' }} & {{t}_{1}} & {{t}_{2}} & {{\sigma' }} & {{t}_{1}} & {{t}_{2}} \\
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| 0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031 \\
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| 0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454 \\
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| 0.26 & 53.093 & 65.079 & 0.39 & 44.200 & 62.809 \\
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| 0.27 & 51.811 & 65.576 & 0.40 & 43.963 & 62.093 \\
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| 0.28 & 50.711 & 65.881 & 0.41 & 43.786 & 61.304 \\
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| 0.29 & 49.743 & 66.041 & 0.42 & 43.674 & 60.436 \\
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| 0.30 & 48.881 & 66.085 & 0.43 & 43.634 & 59.481 \\
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| 0.31 & 48.106 & 66.028 & 0.44 & 43.681 & 58.426 \\
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| 0.32 & 47.408 & 65.883 & 0.45 & 43.832 & 57.252 \\
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| 0.33 & 46.777 & 65.657 & 0.46 & 44.124 & 55.924 \\
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| 0.34 & 46.208 & 65.355 & 0.47 & 44.625 & 54.373 \\
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| 0.35 & 45.697 & 64.983 & 0.48 & 45.517 & 52.418 \\
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| 0.36 & 45.242 & 64.541 & {} & {} & {} \\
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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 time vs sigma.png|center|350px| ]]
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| As can be determined from the table, the lowest calculated value for <math>t</math> is 43.634, while the highest is 66.085. These represent the two-sided 75% confidence limits on the time at which reliability is equal to 80%.
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