Logistic Confidence Bounds Example - Revision history

Lisa Hacker at 21:17, 18 September 2023

2023-09-18T21:17:07Z

← Older revision		Revision as of 21:17, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Logistic\|~~Reliability ~~Growth and Repairable System Analysis Reference]~~]''.		''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

David J. Groebel at 16:57, 20 June 2014

2014-06-20T16:57:55Z

← Older revision		Revision as of 16:57, 20 June 2014
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	The next figure shows a graph of the reliability plotted with 2-sided 90% confidence bounds.		The next figure shows a graph of the reliability plotted with 2-sided 90% confidence bounds.

	[[Image:rga8.6.png\|center\|~~400px\|Logistic Reliability vs. Time plot with 2-sided 90% confidence bounds.~~]]		[[Image:rga8.6.png\|center\|450px]]
	</li>		</li>
	</ol>		</ol>

David J. Groebel at 18:21, 29 May 2014

2014-05-29T18:21:34Z

← Older revision		Revision as of 18:21, 29 May 2014
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Logistic\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Logistic\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

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	#The parameters <math>b\,\!</math> and <math>k\,\!</math>.		#The parameters <math>b\,\!</math> and <math>k\,\!</math>.
	#Reliability at month 5.		#Reliability at month 5.


	'''Solution'''		'''Solution'''
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	<li>The values of <math>\hat{b}\,\!</math> and <math>\hat{k}\,\!</math> that were estimated from the least squares analysis in the reliability data example are:		<li>The values of <math>\hat{b}\,\!</math> and <math>\hat{k}\,\!</math> that were estimated from the least squares analysis in the reliability data example are:

	::<math>\begin{align}		:<math>\begin{align}
	\widehat{b}= & 3.3991 \\		\widehat{b}= & 3.3991 \\
	\widehat{\alpha }= & 0.7398		\widehat{\alpha }= & 0.7398
	\end{align}\,\!</math>		\end{align}\,\!</math>

			Thus, the 2-sided 90% confidence bounds on parameter <math>b\,\!</math> are:

	~~:Thus, the 2-sided 90% confidence bounds on parameter <math>b\,\!</math> are:~~		:<math>\begin{align}

	::<math>\begin{align}
	{{b}_{lower}}= & 2.5547 \\		{{b}_{lower}}= & 2.5547 \\
	{{b}_{upper}}= & 4.5225		{{b}_{upper}}= & 4.5225
	\end{align}\,\!</math>		\end{align}\,\!</math>

			The 2-sided 90% confidence bounds on parameter <math>k\,\!</math> are:

	~~:The 2-sided 90% confidence bounds on parameter <math>k\,\!</math> are:~~		:<math>\begin{align}

	::<math>\begin{align}
	{{k}_{lower}}= & 0.6798 \\		{{k}_{lower}}= & 0.6798 \\
	{{k}_{upper}}= & 0.7997		{{k}_{upper}}= & 0.7997
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	<li>First, calculate the reliability estimation at month 5:		<li>First, calculate the reliability estimation at month 5:

	::<math>\begin{align}		:<math>\begin{align}
	{{R}_{5}}= & \frac{1}{1+b{{e}^{-5k}}} \\		{{R}_{5}}= & \frac{1}{1+b{{e}^{-5k}}} \\
	= & 0.9224		= & 0.9224
	\end{align}\,\!</math>		\end{align}\,\!</math>

			Thus, the 2-sided 90% confidence bounds on reliability at month 5 are:
	:Thus, the 2-sided 90% confidence bounds on reliability at month 5 are:

	::<math>\begin{align}		::<math>\begin{align}
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	{{[{{R}_{5}}]}_{upper}}= & 0.9955		{{[{{R}_{5}}]}_{upper}}= & 0.9955
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The next figure shows a graph of the reliability plotted with 2-sided 90% confidence bounds.		The next figure shows a graph of the reliability plotted with 2-sided 90% confidence bounds.

Chris Kahn at 20:36, 15 May 2014

2014-05-15T20:36:30Z

← Older revision		Revision as of 20:36, 15 May 2014
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	For the data given <noinclude>in the [[Reliability Data - Logistic Model]] example</noinclude><includeonly>above for the reliability data example</includeonly>, calculate the 2-sided 90% confidence bounds under the Logistic model for the following:		For the data given <noinclude>in the [[Reliability Data - Logistic Model]] example</noinclude><includeonly>above for the reliability data example</includeonly>, calculate the 2-sided 90% confidence bounds under the Logistic model for the following:

	~~:1)~~ The parameters <math>b\,\!</math> and <math>k\,\!</math>.		#The parameters <math>b\,\!</math> and <math>k\,\!</math>.
			#Reliability at month 5.
	~~:2)~~ Reliability at month 5.


	'''Solution'''		'''Solution'''
			<ol>
	~~:1)~~ The values of <math>\hat{b}\,\!</math> and <math>\hat{k}\,\!</math> that were estimated from the least squares analysis in the reliability data example are:		<li>The values of <math>\hat{b}\,\!</math> and <math>\hat{k}\,\!</math> that were estimated from the least squares analysis in the reliability data example are:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}\,\!</math>		\end{align}\,\!</math>

			</li>
	~~:2)~~ First, calculate the reliability estimation at month 5:		<li>First, calculate the reliability estimation at month 5:

	::<math>\begin{align}		::<math>\begin{align}
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	:Thus, the 2-sided 90% confidence bounds on reliability at month 5 ~~iare~~:		:Thus, the 2-sided 90% confidence bounds on reliability at month 5 are:

	::<math>\begin{align}		::<math>\begin{align}
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	[[Image:rga8.6.png\|center\|400px\|Logistic Reliability vs. Time plot with 2-sided 90% confidence bounds.]]		[[Image:rga8.6.png\|center\|400px\|Logistic Reliability vs. Time plot with 2-sided 90% confidence bounds.]]
			</li>
			</ol>

Kate Racaza: Created page with '{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. For the data given…'

2014-01-21T21:18:36Z

Created page with '<noinclude>{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. </noinclude> For the data given…'

New page

<noinclude>{{Banner RGA Examples}}
''This example appears in the [[Logistic|Reliability Growth and Repairable System Analysis Reference book]]''.
</noinclude>

For the data given <noinclude>in the [[Reliability Data - Logistic Model]] example</noinclude><includeonly>above for the reliability data example</includeonly>, calculate the 2-sided 90% confidence bounds under the Logistic model for the following:

:1) The parameters <math>b\,\!</math> and <math>k\,\!</math>.

:2) Reliability at month 5.

'''Solution'''

:1) The values of <math>\hat{b}\,\!</math> and <math>\hat{k}\,\!</math> that were estimated from the least squares analysis in the reliability data example are:

::<math>\begin{align}
\widehat{b}= & 3.3991 \\
\widehat{\alpha }= & 0.7398
\end{align}\,\!</math>

:Thus, the 2-sided 90% confidence bounds on parameter <math>b\,\!</math> are:

::<math>\begin{align}
{{b}_{lower}}= & 2.5547 \\
{{b}_{upper}}= & 4.5225
\end{align}\,\!</math>

:The 2-sided 90% confidence bounds on parameter <math>k\,\!</math> are:

::<math>\begin{align}
{{k}_{lower}}= & 0.6798 \\
{{k}_{upper}}= & 0.7997
\end{align}\,\!</math>

:2) First, calculate the reliability estimation at month 5:

::<math>\begin{align}
{{R}_{5}}= & \frac{1}{1+b{{e}^{-5k}}} \\
= & 0.9224
\end{align}\,\!</math>

:Thus, the 2-sided 90% confidence bounds on reliability at month 5 iare:

::<math>\begin{align}
{{[{{R}_{5}}]}_{lower}}= & 0.8493 \\
{{[{{R}_{5}}]}_{upper}}= & 0.9955
\end{align}\,\!</math>

The next figure shows a graph of the reliability plotted with 2-sided 90% confidence bounds.

[[Image:rga8.6.png|center|400px|Logistic Reliability vs. Time plot with 2-sided 90% confidence bounds.]]