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

Lisa Hacker: Redirected page to The Lognormal Distribution

2012-08-13T03:38:05Z

Redirected page to The Lognormal Distribution

← Older revision		Revision as of 03:38, 13 August 2012
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	~~'''Lognormal Distribution Likelihood Ratio Bound Example (Time)'''~~		#REDIRECT [[The Lognormal Distribution]]

	~~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.~~

	~~'''Solution'''~~

	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> .

	~~<center><math>\begin{matrix}~~
	~~{{\sigma' }} & {{t}_{1}} & {{t}_{2}} & {{\sigma' }} & {{t}_{1}} & {{t}_{2}} \\~~
	~~0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031 \\~~
	~~0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454 \\~~
	~~0.26 & 53.093 & 65.079 & 0.39 & 44.200 & 62.809 \\~~
	~~0.27 & 51.811 & 65.576 & 0.40 & 43.963 & 62.093 \\~~
	~~0.28 & 50.711 & 65.881 & 0.41 & 43.786 & 61.304 \\~~
	~~0.29 & 49.743 & 66.041 & 0.42 & 43.674 & 60.436 \\~~
	~~0.30 & 48.881 & 66.085 & 0.43 & 43.634 & 59.481 \\~~
	~~0.31 & 48.106 & 66.028 & 0.44 & 43.681 & 58.426 \\~~
	~~0.32 & 47.408 & 65.883 & 0.45 & 43.832 & 57.252 \\~~
	~~0.33 & 46.777 & 65.657 & 0.46 & 44.124 & 55.924 \\~~
	~~0.34 & 46.208 & 65.355 & 0.47 & 44.625 & 54.373 \\~~
	~~0.35 & 45.697 & 64.983 & 0.48 & 45.517 & 52.418 \\~~
	~~0.36 & 45.242 & 64.541 & {} & {} & {} \\~~
	~~\end{matrix}</math></center>~~

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

	~~[[Image:WB.10 time vs sigma.png\|center\|350px\|~~ ]]

	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%.

Richard House at 05:02, 8 August 2012

2012-08-08T05:02:14Z

← 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 time estimate for a reliability of 80%. The ML estimate for the time at <math>R(t)=80%</math> is 55.718.		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.


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

	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> .		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> .


	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	0.36 & 45.242 & 64.541 & {} & {} & {} \\		0.36 & 45.242 & 64.541 & {} & {} & {} \\
	\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 time vs sigma.png\|center\|~~250px~~\| ]]		[[Image:WB.10 time vs sigma.png\|center\|350px\| ]]

	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%.		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%.

Nicolette Young at 21:21, 25 April 2012

2012-04-25T21:21:55Z

← 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 time vs sigma.png\|center\|~~400px~~\| ]]		[[Image:WB.10 time vs sigma.png\|center\|250px\| ]]

	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%.		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%.

Nicolette Young at 19:14, 14 March 2012

2012-03-14T19:14:05Z

← Older revision		Revision as of 19:14, 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:~~ldachp9ex6~~.~~gif\|thumb~~\|center\|400px\| ]]		[[Image:WB.10 time vs sigma.png\|center\|400px\| ]]

	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%.		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%.

Harry Guo at 23:25, 13 February 2012

2012-02-13T23:25:45Z

← Older revision		Revision as of 23:25, 13 February 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 time estimate for a reliability of 80%. The ML estimate for the time at <math>R(t)=80%</math> is 55.718.		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.


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

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


	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	{{\sigma ~~}_{{{T}~~'}}} & {{t}_{1}} & {{t}_{2}} & {{\sigma ~~}_{{{T}~~'}}} & {{t}_{1}} & {{t}_{2}} \\		{{\sigma' }} & {{t}_{1}} & {{t}_{2}} & {{\sigma' }} & {{t}_{1}} & {{t}_{2}} \\
	0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031 \\		0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031 \\
	0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454 \\		0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454 \\

Harry Guo at 23:24, 13 February 2012

2012-02-13T23:24:05Z

← Older revision		Revision as of 23:24, 13 February 2012
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	'''Lognormal Distribution Likelihood Ratio Bound Example (Time)'''		'''Lognormal Distribution Likelihood Ratio Bound Example (Time)'''

	For the data given in 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.		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.

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

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2012-02-13T18:51:30Z

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

For the data given in 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.

'''Solution'''

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

<center><math>\begin{matrix}
{{\sigma }_{{{T}'}}} & {{t}_{1}} & {{t}_{2}} & {{\sigma }_{{{T}'}}} & {{t}_{1}} & {{t}_{2}} \\
0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031 \\
0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454 \\
0.26 & 53.093 & 65.079 & 0.39 & 44.200 & 62.809 \\
0.27 & 51.811 & 65.576 & 0.40 & 43.963 & 62.093 \\
0.28 & 50.711 & 65.881 & 0.41 & 43.786 & 61.304 \\
0.29 & 49.743 & 66.041 & 0.42 & 43.674 & 60.436 \\
0.30 & 48.881 & 66.085 & 0.43 & 43.634 & 59.481 \\
0.31 & 48.106 & 66.028 & 0.44 & 43.681 & 58.426 \\
0.32 & 47.408 & 65.883 & 0.45 & 43.832 & 57.252 \\
0.33 & 46.777 & 65.657 & 0.46 & 44.124 & 55.924 \\
0.34 & 46.208 & 65.355 & 0.47 & 44.625 & 54.373 \\
0.35 & 45.697 & 64.983 & 0.48 & 45.517 & 52.418 \\
0.36 & 45.242 & 64.541 & {} & {} & {} \\
\end{matrix}</math></center>

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

[[Image:ldachp9ex6.gif|thumb|center|400px| ]]

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%.