Template:Example: Lognormal Distribution Likelihood Ratio Bound (Reliability) - Revision history

Lisa Hacker: Redirected page to The Lognormal Distribution

2012-08-13T03:43:04Z

Redirected page to The Lognormal Distribution

← Older revision		Revision as of 03:43, 13 August 2012
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	~~'''Lognormal Distribution Likelihood Ratio Bound Example (Reliability)'''~~		#REDIRECT [[The Lognormal Distribution]]

	~~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%.

	~~'''Solution'''~~

	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> .

	~~<center><math>\begin{matrix}~~
	~~{{\sigma'}} & {{R}_{1}} & {{R}_{2}} & {{\sigma'}} & {{R}_{1}} & {{R}_{2}} \\~~
	~~0.24 & 61.107% & 75.910% & 0.37 & 43.573% & 78.845% \\~~
	~~0.25 & 55.906% & 78.742% & 0.38 & 43.807% & 78.180% \\~~
	~~0.26 & 55.528% & 80.131% & 0.39 & 44.147% & 77.448% \\~~
	~~0.27 & 50.067% & 80.903% & 0.40 & 44.593% & 76.646% \\~~
	~~0.28 & 48.206% & 81.319% & 0.41 & 45.146% & 75.767% \\~~
	~~0.29 & 46.779% & 81.499% & 0.42 & 45.813% & 74.802% \\~~
	~~0.30 & 45.685% & 81.508% & 0.43 & 46.604% & 73.737% \\~~
	~~0.31 & 44.857% & 81.387% & 0.44 & 47.538% & 72.551% \\~~
	~~0.32 & 44.250% & 81.159% & 0.45 & 48.645% & 71.212% \\~~
	~~0.33 & 43.827% & 80.842% & 0.46 & 49.980% & 69.661% \\~~
	~~0.34 & 43.565% & 80.446% & 0.47 & 51.652% & 67.789% \\~~
	~~0.35 & 43.444% & 79.979% & 0.48 & 53.956% & 65.299% \\~~
	~~0.36 & 43.450% & 79.444% & {} & {} & {} \\~~
	~~\end{matrix}</math></center>~~

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

	~~[[Image:WB.10 reliability v sigma.png\|center\|350px\|~~ ]]

	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> .

Richard House at 05:02, 8 August 2012

2012-08-08T05:02:54Z

← Older revision		Revision as of 05:02, 8 August 2012
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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%.		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%.


	'''Solution'''		'''Solution'''

	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> .		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> .


	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	0.36 & 43.450% & 79.444% & {} & {} & {} \\		0.36 & 43.450% & 79.444% & {} & {} & {} \\
	\end{matrix}</math></center>		\end{matrix}</math></center>


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

	[[Image:WB.10 reliability v sigma.png\|center\|~~250px~~\| ]]		[[Image:WB.10 reliability v sigma.png\|center\|350px\| ]]

	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> .		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> .

Nicolette Young at 21:21, 25 April 2012

2012-04-25T21:21:35Z

← Older revision		Revision as of 21:21, 25 April 2012
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	This data set is represented graphically in the following contour plot:		This data set is represented graphically in the following contour plot:

	[[Image:WB.10 reliability v sigma.png\|center\|~~400px~~\| ]]		[[Image:WB.10 reliability v sigma.png\|center\|250px\| ]]

	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> .		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> .

Nicolette Young at 19:15, 14 March 2012

2012-03-14T19:15:40Z

← Older revision		Revision as of 19:15, 14 March 2012
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	This data set is represented graphically in the following contour plot:		This data set is represented graphically in the following contour plot:

	[[Image:~~ldachp9ex7~~.~~gif\|thumb~~\|center\|400px\| ]]		[[Image:WB.10 reliability v sigma.png\|center\|400px\| ]]

	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> .		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> .

Harry Guo at 23:27, 13 February 2012

2012-02-13T23:27:56Z

← Older revision		Revision as of 23:27, 13 February 2012
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	'''Lognormal Distribution Likelihood Ratio Bound Example (Reliability)'''		'''Lognormal Distribution Likelihood Ratio Bound Example (Reliability)'''

	For the data given in 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%.		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%.


	'''Solution'''		'''Solution'''

	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 ~~Eqn. (lognormliketr)~~, and varying <math>{{\sigma ~~}_{{{T}~~'}}}</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 ~~}_{{{T}~~'}}}</math> .		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> .


	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	{{\sigma ~~}_{{{T}~~'}}} & {{R}_{1}} & {{R}_{2}} & {{\sigma ~~}_{{{T}~~'}}} & {{R}_{1}} & {{R}_{2}} \\		{{\sigma'}} & {{R}_{1}} & {{R}_{2}} & {{\sigma'}} & {{R}_{1}} & {{R}_{2}} \\
	0.24 & 61.107% & 75.910% & 0.37 & 43.573% & 78.845% \\		0.24 & 61.107% & 75.910% & 0.37 & 43.573% & 78.845% \\
	0.25 & 55.906% & 78.742% & 0.38 & 43.807% & 78.180% \\		0.25 & 55.906% & 78.742% & 0.38 & 43.807% & 78.180% \\

Harry Guo: Created page with ''''Lognormal Distribution Likelihood Ratio Bound Example (Reliability)''' For the data given in Example 5, determine the two-sided 75% confidence bounds on the reliability estim…'

2012-02-13T18:53:24Z

Created page with ''''Lognormal Distribution Likelihood Ratio Bound Example (Reliability)''' For the data given in Example 5, determine the two-sided 75% confidence bounds on the reliability estim…'

New page

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

For the data given in 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%.

'''Solution'''

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 Eqn. (lognormliketr), and varying <math>{{\sigma }_{{{T}'}}}</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 }_{{{T}'}}}</math> .

<center><math>\begin{matrix}
{{\sigma }_{{{T}'}}} & {{R}_{1}} & {{R}_{2}} & {{\sigma }_{{{T}'}}} & {{R}_{1}} & {{R}_{2}} \\
0.24 & 61.107% & 75.910% & 0.37 & 43.573% & 78.845% \\
0.25 & 55.906% & 78.742% & 0.38 & 43.807% & 78.180% \\
0.26 & 55.528% & 80.131% & 0.39 & 44.147% & 77.448% \\
0.27 & 50.067% & 80.903% & 0.40 & 44.593% & 76.646% \\
0.28 & 48.206% & 81.319% & 0.41 & 45.146% & 75.767% \\
0.29 & 46.779% & 81.499% & 0.42 & 45.813% & 74.802% \\
0.30 & 45.685% & 81.508% & 0.43 & 46.604% & 73.737% \\
0.31 & 44.857% & 81.387% & 0.44 & 47.538% & 72.551% \\
0.32 & 44.250% & 81.159% & 0.45 & 48.645% & 71.212% \\
0.33 & 43.827% & 80.842% & 0.46 & 49.980% & 69.661% \\
0.34 & 43.565% & 80.446% & 0.47 & 51.652% & 67.789% \\
0.35 & 43.444% & 79.979% & 0.48 & 53.956% & 65.299% \\
0.36 & 43.450% & 79.444% & {} & {} & {} \\
\end{matrix}</math></center>

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

[[Image:ldachp9ex7.gif|thumb|center|400px| ]]

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> .