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| | | #REDIRECT [[Lognormal Distribution Examples]] |
| ==General Examples==
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| '''Example 9:'''
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| {{Example: Lognormal General Example Interval Data}}
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| '''Example 10:'''
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| {{Example: Lognormal General Example Complete Data}}
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| ===Example 11===
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| From Kececioglu [19, p. 406]. Nine identical units are tested continuously to failure and their times-to-failure were recorded at 30.4, 36.7, 53.3, 58.5, 74.0, 99.3, 114.3, 140.1, and 257.9 hours.
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| ====Solution to Example 11====
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| The results published were obtained by using the unbiased model.
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| Published Results (using MLE):
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=4.3553 \\
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| {{\widehat{\sigma }}_{{{T}'}}}=0.67677 \\
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| \end{matrix}</math>
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| This same data set can be entered into Weibull++ by creating a data sheet capable of handling non-grouped time-to-failure data. Since the results shown above are unbiased, the Use Unbiased Std on Normal Data option in the User Setup must be selected in order to duplicate these results.
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| Weibull++ computed parameters for maximum likelihood are:
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=4.3553 \\
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| {{\widehat{\sigma }}_{{{T}'}}}=0.6768 \\
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| \end{matrix}</math>
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| ===Example 12===
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| From Kececioglu [20, p. 347]. Fifteen identical units were tested to failure and following is a table of their times-to-failure:
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| <center><math>\text{Table 9}\text{.5 - Data of Example 11}</math></center>
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| <center><math>\begin{matrix}
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| \text{Data Point Index} & \text{Time-to-Failure, hr} \\
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| \text{1} & \text{62}\text{.5} \\
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| \text{2} & \text{91}\text{.9} \\
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| \text{3} & \text{100}\text{.3} \\
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| \text{4} & \text{117}\text{.4} \\
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| \text{5} & \text{141}\text{.1} \\
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| \text{6} & \text{146}\text{.8} \\
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| \text{7} & \text{172}\text{.7} \\
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| \text{8} & \text{192}\text{.5} \\
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| \text{9} & \text{201}\text{.6} \\
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| \text{10} & \text{235}\text{.8} \\
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| \text{11} & \text{249}\text{.2} \\
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| \text{12} & \text{297}\text{.5} \\
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| \text{13} & \text{318}\text{.3} \\
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| \text{14} & \text{410}\text{.6} \\
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| \text{15} & \text{550}\text{.5} \\
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| \end{matrix}</math></center>
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| ====Solution to Example 12====
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| Published results (using probability plotting):
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=5.22575 \\
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| {{\widehat{\sigma }}_{{{T}'}}}=0.62048. \\
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| \end{matrix}</math>
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| Weibull++ computed parameters for rank regression on X are:
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=5.2303 \\
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| {{\widehat{\sigma }}_{{{T}'}}}=0.6283. \\
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| \end{matrix}</math>
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| The small differences are due to the precision errors when fitting a line manually, whereas in Weibull++ the line was fitted mathematically.
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| ===Example 13===
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| From Nelson [30, p. 324]. Ninety-six locomotive controls were tested, 37 failed and 59 were suspended after running for 135,000 miles. Table 9.6 (at the end of this chapter) shows their times-to-failure.
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| ====Solution to Example 13====
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| The distribution used in the publication was the base-10 lognormal.
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| Published results (using MLE):
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=2.2223 \\
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| {{\widehat{\sigma }}_{{{T}'}}}=0.3064 \\
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| \end{matrix}</math>
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| Published 95% confidence limits on the parameters:
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\
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| {{\widehat{\sigma }}_{{{T}'}}}=\left\{ 0.2365,0.3970 \right\} \\
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| \end{matrix}</math>
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| Published variance/covariance matrix:
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| ::<math>\left[ \begin{matrix}
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| \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0020 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma }}}_{{{T}'}}})=0.001 \\
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| {} & {} & {} \\
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| \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma }}}_{{{T}'}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma }}}_{{{T}'}}} \right)=0.0016 \\
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| \end{matrix} \right]</math>
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| To replicate the published results (since Weibull++ uses a lognormal to the base <math>e</math> ), take the base-10 logarithm of the data and estimate the parameters using the Normal distribution and MLE.
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| • Weibull++ computed parameters for maximum likelihood are:
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=2.2223 \\
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| {{\widehat{\sigma }}_{{{T}'}}}=0.3064 \\
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| \end{matrix}</math>
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| • Weibull++ computed 95% confidence limits on the parameters:
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\
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| {{\widehat{\sigma }}_{{{T}'}}}=\left\{ 0.2395,0.3920 \right\} \\
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| \end{matrix}</math>
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|
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| • Weibull++ computed/variance covariance matrix:
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| ::<math>\left[ \begin{matrix}
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| \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0019 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma }}}_{{{T}'}}})=0.0009 \\
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| {} & {} & {} \\
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| \widehat{Cov}({\mu }',{{{\hat{\sigma }}}_{{{T}'}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma }}}_{{{T}'}}} \right)=0.0015 \\
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| \end{matrix} \right]</math>
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| {|align="center" border="1" cellspacing="1"
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| |-
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| |colspan="4" style="text-align:center"|Table 9.6 - Nelson's Locomotive Data
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| |-
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| !
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| !Number in State
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| !F or S
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| !Time
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| |-
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| |1||1||F||22.5
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| |-
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| |2||1||F||37.5
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| |-
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| |3||1||F||46
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| |-
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| |4||1||F||48.5
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| |-
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| |5||1||F||51.5
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| |-
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| |6||1||F||53
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| |-
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| |7||1||F||54.5
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| |-
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| |8||1||F||57.5
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| |-
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| |9||1||F||66.5
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| |-
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| |10||1||F||68
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| |-
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| |11||1||F||69.5
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| |-
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| |12||1||F||76.5
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| |-
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| |13||1||F||77
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| |-
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| |14||1||F||78.5
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| |-
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| |15||1||F||80
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| |-
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| |16||1||F||81.5
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| |-
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| |17||1||F||82
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| |-
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| |18||1||F||83
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| |-
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| |19||1||F||84
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| |-
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| |20||1||F||91.5
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| |-
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| |21||1||F||93.5
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| |-
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| |22||1||F||102.5
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| |-
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| |23||1||F||107
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| |-
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| |24||1||F||108.5
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| |-
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| |25||1||F||112.5
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| |-
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| |26||1||F||113.5
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| |-
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| |27||1||F||116
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| |-
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| |28||1||F||117
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| |-
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| |29||1||F||118.5
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| |-
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| |30||1||F||119
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| |-
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| |31||1||F||120
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| |-
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| |32||1||F||122.5
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| |-
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| |33||1||F||123
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| |-
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| |34||1||F||127.5
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| |-
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| |35||1||F||131
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| |-
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| |36||1||F||132.5
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| |-
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| |37||1||F||134
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| |-
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| |38||59||S||135
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| |}
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