Template:Example: Normal Distribution RRY - Revision history

Lisa Hacker: Redirected page to The Normal Distribution

2012-08-13T01:55:47Z

Redirected page to The Normal Distribution

← Older revision		Revision as of 01:55, 13 August 2012
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	~~'''Normal Distribution RRY Example'''~~		#REDIRECT [[The Normal Distribution]]

	~~Fourteen units were reliability tested and the following life test data were obtained:~~


	~~{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"~~
	\|-
	~~\|colspan="2" style="text-align:center"\|The test data~~
	\|-
	~~!Data point index~~
	~~!Time-to-failure~~
	\|-
	~~\|1 \|\|5~~
	\|-
	~~\|2 \|\|10~~
	\|-
	~~\|3 \|\|15~~
	\|-
	~~\|4 \|\|20~~
	\|-
	~~\|5 \|\|25~~
	\|-
	~~\|6 \|\|30~~
	\|-
	~~\|7\|\|35~~
	\|-
	~~\|8\|\|40~~
	\|-
	~~\|9\|\|50~~
	\|-
	~~\|10\|\|60~~
	\|-
	~~\|11\|\|70~~
	\|-
	~~\|12\|\|80~~
	\|-
	~~\|13\|\|90~~
	\|-
	~~\|14\|\|100~~
	\|}


	~~Assuming the data follow a normal distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math> , using rank regression on Y.~~

	~~'''Solution'''~~

	~~Construct a table like the one shown next.~~

	~~<center><math>\overset{{}}{\mathop{\text{Table}\text{ - Least Squares Analysis}}}\,</math></center>~~

	~~<center><math>\begin{matrix}~~
	~~\text{N} & \text{T}_{i} & \text{F(T}_{i}\text{)} & \text{y}_{i} & \text{T}_{i}^{2} & \text{y}_{i}^{2} & \text{T}_{i}\text{ y}_{i} \\~~
	~~\text{1} & \text{5} & \text{0}\text{.0483} & \text{-1}\text{.6619} & \text{25} & \text{2}\text{.7619} & \text{-8}\text{.3095} \\~~
	~~\text{2} & \text{10} & \text{0}\text{.1170} & \text{-1}\text{.1901} & \text{100} & \text{1}\text{.4163} & \text{-11}\text{.9010} \\~~
	~~\text{3} & \text{15} & \text{0}\text{.1865} & \text{-0}\text{.8908} & \text{225} & \text{0}\text{.7935} & \text{-13}\text{.3620} \\~~
	~~\text{4} & \text{20} & \text{0}\text{.2561} & \text{-0}\text{.6552} & \text{400} & \text{0}\text{.4292} & \text{-13}\text{.1030} \\~~
	~~\text{5} & \text{25} & \text{0}\text{.3258} & \text{-0}\text{.4512} & \text{625} & \text{0}\text{.2036} & \text{-11}\text{.2800} \\~~
	~~\text{6} & \text{30} & \text{0}\text{.3954} & \text{-0}\text{.2647} & \text{900} & \text{0}\text{.0701} & \text{-7}\text{.9422} \\~~
	~~\text{7} & \text{35} & \text{0}\text{.4651} & \text{-0}\text{.0873} & \text{1225} & \text{0}\text{.0076} & \text{-3}\text{.0542} \\~~
	~~\text{8} & \text{40} & \text{0}\text{.5349} & \text{0}\text{.0873} & \text{1600} & \text{0}\text{.0076} & \text{3}\text{.4905} \\~~
	~~\text{9} & \text{50} & \text{0}\text{.6046} & \text{0}\text{.2647} & \text{2500} & \text{0}\text{.0701} & \text{13}\text{.2370} \\~~
	~~\text{10} & \text{60} & \text{0}\text{.6742} & \text{0}\text{.4512} & \text{3600} & \text{0}\text{.2036} & \text{27}\text{.0720} \\~~
	~~\text{11} & \text{70} & \text{0}\text{.7439} & \text{0}\text{.6552} & \text{4900} & \text{0}\text{.4292} & \text{45}\text{.8605} \\~~
	~~\text{12} & \text{80} & \text{0}\text{.8135} & \text{0}\text{.8908} & \text{6400} & \text{0}\text{.7935} & \text{71}\text{.2640} \\~~
	~~\text{13} & \text{90} & \text{0}\text{.8830} & \text{1}\text{.1901} & \text{8100} & \text{1}\text{.4163} & \text{107}\text{.1090} \\~~
	~~\text{14} & \text{100} & \text{0}\text{.9517} & \text{1}\text{.6619} & \text{10000} & \text{2}\text{.7619} & \text{166}\text{.1900} \\~~
	~~\mathop{}_{}^{} & \text{630} & {} & \text{0} & \text{40600} & \text{11}\text{.3646} & \text{365}\text{.2711} \\~~
	~~\end{matrix}</math></center>~~


	~~:*The median rank values ( <math>F({{t}_{i}})</math> ) can be found in rank tables, available in many statistical texts, or they can be estimated by using the Quick Statistical Reference in Weibull++.~~
	:*The <math>{{y}_{i}}</math> values were obtained from standardized normal distribution's area tables by entering for <math>F(z)</math> and getting the corresponding <math>z</math> value ( <math>{{y}_{i}}</math> ). As with the median rank values, these standard normal values can be obtained with the Quick Statistical Reference.

	~~Given the values in the table above, calculate <math>\widehat{a}</math> and <math>\widehat{b}</math> using:~~

	~~::<math>\begin{align}~~
	& \widehat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{T}_{i}}{{y}_{i}}-(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{T}_{i}})(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}})/14}{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{2}-{{(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{T}_{i}})}^{2}}/14} \\
	~~& & \\~~
	~~& \widehat{b}= & \frac{365.2711-(630)(0)/14}{40,600-{{(630)}^{2}}/14}=0.02982~~
	~~\end{align}</math>~~

	~~and:~~

	::<math>\widehat{a}=\overline{y}-\widehat{b}\overline{T}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\widehat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{t}_{i}}}{N}</math>

	~~or:~~

	~~::<math>\widehat{a}=\frac{0}{14}-(0.02982)\frac{630}{14}=-1.3419</math>~~

	~~Therefore:~~

	~~::<math>\widehat{\sigma}=\frac{1}{\hat{b}}=\frac{1}{0.02982}=33.5367</math>~~

	~~and:~~

	~~::<math>\widehat{\mu }=-\widehat{a}\cdot \widehat{\sigma }=-(-1.3419)\cdot 33.5367\simeq 45</math>~~

	~~or <math>\widehat{\mu }=45</math> hours <math>.</math>~~

	~~The correlation coefficient can be estimated using:~~

	~~::<math>\widehat{\rho }=0.979</math>~~

	~~The preceding example can be repeated using Weibull++ .~~

	~~:*Create a new folio for Times-to-Failure data, and enter the data given in this example.~~
	~~:*Choose Normal from the Distributions list.~~
	~~:*Go to the Analysis page and select Rank Regression on Y (RRY).~~
	~~:*Click the Calculate icon located on the Main page.~~

	[[~~Image:Normal RRY Setting.png\|center\|550px\| ]]~~

	The ~~probability plot is shown next.~~

	~~[[Image:~~Normal ~~RRY Plot.png\|center\|550px\|~~ ]]

Richard House at 02:35, 8 August 2012

2012-08-08T02:35:59Z

← Older revision		Revision as of 02:35, 8 August 2012
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	:*The <math>{{y}_{i}}</math> values were obtained from standardized normal distribution's area tables by entering for <math>F(z)</math> and getting the corresponding <math>z</math> value ( <math>{{y}_{i}}</math> ). As with the median rank values, these standard normal values can be obtained with the Quick Statistical Reference.		:*The <math>{{y}_{i}}</math> values were obtained from standardized normal distribution's area tables by entering for <math>F(z)</math> and getting the corresponding <math>z</math> value ( <math>{{y}_{i}}</math> ). As with the median rank values, these standard normal values can be obtained with the Quick Statistical Reference.

	Given the values in ~~Table 9.2~~, calculate <math>\widehat{a}</math> and <math>\widehat{b}</math> using:		Given the values in the table above, calculate <math>\widehat{a}</math> and <math>\widehat{b}</math> using:

	::<math>\begin{align}		::<math>\begin{align}

Richard House at 02:29, 8 August 2012

2012-08-08T02:29:24Z

← Older revision		Revision as of 02:29, 8 August 2012
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	{\|align="center" border=1 cellspacing=1		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|-		\|-
	\|colspan="2" style="text-align:center"\|~~Table -~~The test data		\|colspan="2" style="text-align:center"\|The test data
	\|-		\|-
	!Data point index		!Data point index
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	:*Click the Calculate icon located on the Main page.		:*Click the Calculate icon located on the Main page.

	[[Image:Normal RRY Setting.png~~\|thumb~~\|center\|~~250px~~\| ]]		[[Image:Normal RRY Setting.png\|center\|550px\| ]]

	The probability plot is shown next.		The probability plot is shown next.

	[[Image:Normal RRY Plot.png~~\|thumb~~\|center\|~~250px~~\| ]]		[[Image:Normal RRY Plot.png\|center\|550px\| ]]

Nicolette Young at 18:58, 25 April 2012

2012-04-25T18:58:56Z

← Older revision		Revision as of 18:58, 25 April 2012
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	:*Click the Calculate icon located on the Main page.		:*Click the Calculate icon located on the Main page.

	[[Image:Normal RRY Setting.png\|thumb\|center\|~~400px~~\| ]]		[[Image:Normal RRY Setting.png\|thumb\|center\|250px\| ]]

	The probability plot is shown next.		The probability plot is shown next.

	[[Image:Normal RRY Plot.png\|thumb\|center\|~~400px~~\| ]]		[[Image:Normal RRY Plot.png\|thumb\|center\|250px\| ]]

Harry Guo at 23:33, 10 February 2012

2012-02-10T23:33:51Z

← Older revision		Revision as of 23:33, 10 February 2012
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	The preceding example can be repeated using Weibull++ .		The preceding example can be repeated using Weibull++ .

	:*Create a new folio for Times-to-Failure data, and enter the data given in ~~Table 9.1~~.		:*Create a new folio for Times-to-Failure data, and enter the data given in this example.
	:*Choose Normal from the Distributions list.		:*Choose Normal from the Distributions list.
	:*Go to the Analysis page and select Rank Regression on Y (RRY).		:*Go to the Analysis page and select Rank Regression on Y (RRY).

Harry Guo at 23:32, 10 February 2012

2012-02-10T23:32:42Z

← Older revision		Revision as of 23:32, 10 February 2012
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	{\|align="center" border=1 cellspacing=1		{\|align="center" border=1 cellspacing=1
	\|-		\|-
	\|colspan="2" style="text-align:center"\|Table ~~9.1~~ -The test data		\|colspan="2" style="text-align:center"\|Table -The test data
	\|-		\|-
	!Data point index		!Data point index
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	Construct a table like the one shown next.		Construct a table like the one shown next.

	<center><math>\overset{{}}{\mathop{\text{Table 9}\text{.2 - Least Squares Analysis}}}\,</math></center>		<center><math>\overset{{}}{\mathop{\text{Table}\text{ - Least Squares Analysis}}}\,</math></center>

	<center><math>\begin{matrix}		<center><math>\begin{matrix}

Harry Guo at 23:27, 10 February 2012

2012-02-10T23:27:56Z

← Older revision		Revision as of 23:27, 10 February 2012
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	{\|align="center" border=1 cellspacing=1		{\|align="center" border=1 cellspacing=1
	\|-		\|-
	\|colspan="2" style="text-align:center"\|Table 8.1 -The test data		\|colspan="2" style="text-align:center"\|Table 9.1 -The test data
	\|-		\|-
	!Data point index		!Data point index
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	Construct a table like the one shown next.		Construct a table like the one shown next.

	<center><math>\overset{{}}{\mathop{\text{Table 8}\text{.2 - Least Squares Analysis}}}\,</math></center>		<center><math>\overset{{}}{\mathop{\text{Table 9}\text{.2 - Least Squares Analysis}}}\,</math></center>

	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	:*The <math>{{y}_{i}}</math> values were obtained from standardized normal distribution's area tables by entering for <math>F(z)</math> and getting the corresponding <math>z</math> value ( <math>{{y}_{i}}</math> ). As with the median rank values, these standard normal values can be obtained with the Quick Statistical Reference.		:*The <math>{{y}_{i}}</math> values were obtained from standardized normal distribution's area tables by entering for <math>F(z)</math> and getting the corresponding <math>z</math> value ( <math>{{y}_{i}}</math> ). As with the median rank values, these standard normal values can be obtained with the Quick Statistical Reference.

	Given the values in Table 8.2, calculate <math>\widehat{a}</math> and <math>\widehat{b}</math> using:		Given the values in Table 9.2, calculate <math>\widehat{a}</math> and <math>\widehat{b}</math> using:

	::<math>\begin{align}		::<math>\begin{align}
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	The preceding example can be repeated using Weibull++ .		The preceding example can be repeated using Weibull++ .

	:*Create a new folio for Times-to-Failure data, and enter the data given in Table 8.1.		:*Create a new folio for Times-to-Failure data, and enter the data given in Table 9.1.
	:*Choose Normal from the Distributions list.		:*Choose Normal from the Distributions list.
	:*Go to the Analysis page and select Rank Regression on Y (RRY).		:*Go to the Analysis page and select Rank Regression on Y (RRY).

Harry Guo at 21:01, 10 February 2012

2012-02-10T21:01:23Z

← Older revision		Revision as of 21:01, 10 February 2012
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	:*Click the Calculate icon located on the Main page.		:*Click the Calculate icon located on the Main page.

	[[Image:~~weibullfolio1~~.png\|thumb\|center\|400px\| ]]		[[Image:Normal RRY Setting.png\|thumb\|center\|400px\| ]]

	The probability plot is shown next.		The probability plot is shown next.

	[[Image:~~weibullfolioplot1st~~.png\|thumb\|center\|400px\| ]]		[[Image:Normal RRY Plot.png\|thumb\|center\|400px\| ]]

Harry Guo at 18:45, 10 February 2012

2012-02-10T18:45:04Z

← Older revision		Revision as of 18:45, 10 February 2012
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	{\|align="center" border=1 cellspacing=1		{\|align="center" border=1 cellspacing=1
	\|-		\|-
	\|colspan="2" style="text-align:center"\|Table 8.1 -The test data ~~for Example 2~~		\|colspan="2" style="text-align:center"\|Table 8.1 -The test data
	\|-		\|-
	!Data point index		!Data point index
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	Assuming the data follow a normal distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math> , using rank regression on Y.		Assuming the data follow a normal distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math> , using rank regression on Y.

	~~====~~Solution ~~to Example 2====~~		'''Solution'''

	Construct a table like the one shown next.		Construct a table like the one shown next.

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	:• The median rank values ( <math>F({{T}_{i}})</math> ) can be found in rank tables, available in many statistical texts, or they can be estimated by using the Quick Statistical Reference in Weibull++.		:*The median rank values ( <math>F({{t}_{i}})</math> ) can be found in rank tables, available in many statistical texts, or they can be estimated by using the Quick Statistical Reference in Weibull++.
	:• The <math>{{y}_{i}}</math> values were obtained from standardized normal distribution's area tables by entering for <math>F(z)</math> and getting the corresponding <math>z</math> value ( <math>{{y}_{i}}</math> ). As with the median rank values, these standard normal values can be obtained with the Quick Statistical Reference.		:*The <math>{{y}_{i}}</math> values were obtained from standardized normal distribution's area tables by entering for <math>F(z)</math> and getting the corresponding <math>z</math> value ( <math>{{y}_{i}}</math> ). As with the median rank values, these standard normal values can be obtained with the Quick Statistical Reference.
	Given the values in Table 8.2, calculate <math>\widehat{a}</math> and <math>\widehat{b}</math> using ~~Eqns. (aan) and (bbn)~~:
			Given the values in Table 8.2, calculate <math>\widehat{a}</math> and <math>\widehat{b}</math> using:

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

	:and:		and:

	::<math>\widehat{a}=\overline{y}-\widehat{b}\overline{T}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\widehat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{T}_{i}}}{N}</math>		::<math>\widehat{a}=\overline{y}-\widehat{b}\overline{T}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\widehat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{t}_{i}}}{N}</math>

	:or:		or:

	::<math>\widehat{a}=\frac{0}{14}-(0.02982)\frac{630}{14}=-1.3419</math>		::<math>\widehat{a}=\frac{0}{14}-(0.02982)\frac{630}{14}=-1.3419</math>

	Therefore~~, from Eqn. (bn)~~:		Therefore:

	::<math>\widehat{\sigma}=\frac{1}{\hat{b}}=\frac{1}{0.02982}=33.5367</math>		::<math>\widehat{\sigma}=\frac{1}{\hat{b}}=\frac{1}{0.02982}=33.5367</math>

	:and ~~from Eqn. (an)~~:		and:

	::<math>\widehat{\mu }=-\widehat{a}\cdot \widehat{\sigma }=-(-1.3419)\cdot 33.5367\simeq 45</math>		::<math>\widehat{\mu }=-\widehat{a}\cdot \widehat{\sigma }=-(-1.3419)\cdot 33.5367\simeq 45</math>
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	or <math>\widehat{\mu }=45</math> hours <math>.</math>		or <math>\widehat{\mu }=45</math> hours <math>.</math>

	The correlation coefficient can be estimated using ~~Eqn. (RHOn)~~:		The correlation coefficient can be estimated using:

	::<math>\widehat{\rho }=0.979</math>		::<math>\widehat{\rho }=0.979</math>
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	The preceding example can be repeated using Weibull++ .		The preceding example can be repeated using Weibull++ .

	:• Create a new folio for Times-to-Failure data, and enter the data given in Table 8.1.		:*Create a new folio for Times-to-Failure data, and enter the data given in Table 8.1.
	:• Choose Normal from the Distributions list.		:*Choose Normal from the Distributions list.
	:• Go to the Analysis page and select Rank Regression on Y (RRY).		:*Go to the Analysis page and select Rank Regression on Y (RRY).
	:• Click the Calculate icon located on the Main page.		:*Click the Calculate icon located on the Main page.

	[[Image:weibullfolio1.png\|thumb\|center\|400px\| ]]		[[Image:weibullfolio1.png\|thumb\|center\|400px\| ]]

Harry Guo: Created page with ''''Normal Distribution RRY Example''' Fourteen units were reliability tested and the following life test data were obtained: {|align="center" border=1 cellspacing=1 |- |colsp…'

2012-02-10T18:41:16Z

Created page with ''''Normal Distribution RRY Example''' Fourteen units were reliability tested and the following life test data were obtained: {|align="center" border=1 cellspacing=1 |- |colsp…'

New page

'''Normal Distribution RRY Example'''

Fourteen units were reliability tested and the following life test data were obtained:

{|align="center" border=1 cellspacing=1
|-
|colspan="2" style="text-align:center"|Table 8.1 -The test data for Example 2
|-
!Data point index
!Time-to-failure
|-
|1 ||5
|-
|2 ||10
|-
|3 ||15
|-
|4 ||20
|-
|5 ||25
|-
|6 ||30
|-
|7||35
|-
|8||40
|-
|9||50
|-
|10||60
|-
|11||70
|-
|12||80
|-
|13||90
|-
|14||100
|}

Assuming the data follow a normal distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math> , using rank regression on Y.

====Solution to Example 2====
Construct a table like the one shown next.

<center><math>\overset{{}}{\mathop{\text{Table 8}\text{.2 - Least Squares Analysis}}}\,</math></center>

<center><math>\begin{matrix}
\text{N} & \text{T}_{i} & \text{F(T}_{i}\text{)} & \text{y}_{i} & \text{T}_{i}^{2} & \text{y}_{i}^{2} & \text{T}_{i}\text{ y}_{i} \\
\text{1} & \text{5} & \text{0}\text{.0483} & \text{-1}\text{.6619} & \text{25} & \text{2}\text{.7619} & \text{-8}\text{.3095} \\
\text{2} & \text{10} & \text{0}\text{.1170} & \text{-1}\text{.1901} & \text{100} & \text{1}\text{.4163} & \text{-11}\text{.9010} \\
\text{3} & \text{15} & \text{0}\text{.1865} & \text{-0}\text{.8908} & \text{225} & \text{0}\text{.7935} & \text{-13}\text{.3620} \\
\text{4} & \text{20} & \text{0}\text{.2561} & \text{-0}\text{.6552} & \text{400} & \text{0}\text{.4292} & \text{-13}\text{.1030} \\
\text{5} & \text{25} & \text{0}\text{.3258} & \text{-0}\text{.4512} & \text{625} & \text{0}\text{.2036} & \text{-11}\text{.2800} \\
\text{6} & \text{30} & \text{0}\text{.3954} & \text{-0}\text{.2647} & \text{900} & \text{0}\text{.0701} & \text{-7}\text{.9422} \\
\text{7} & \text{35} & \text{0}\text{.4651} & \text{-0}\text{.0873} & \text{1225} & \text{0}\text{.0076} & \text{-3}\text{.0542} \\
\text{8} & \text{40} & \text{0}\text{.5349} & \text{0}\text{.0873} & \text{1600} & \text{0}\text{.0076} & \text{3}\text{.4905} \\
\text{9} & \text{50} & \text{0}\text{.6046} & \text{0}\text{.2647} & \text{2500} & \text{0}\text{.0701} & \text{13}\text{.2370} \\
\text{10} & \text{60} & \text{0}\text{.6742} & \text{0}\text{.4512} & \text{3600} & \text{0}\text{.2036} & \text{27}\text{.0720} \\
\text{11} & \text{70} & \text{0}\text{.7439} & \text{0}\text{.6552} & \text{4900} & \text{0}\text{.4292} & \text{45}\text{.8605} \\
\text{12} & \text{80} & \text{0}\text{.8135} & \text{0}\text{.8908} & \text{6400} & \text{0}\text{.7935} & \text{71}\text{.2640} \\
\text{13} & \text{90} & \text{0}\text{.8830} & \text{1}\text{.1901} & \text{8100} & \text{1}\text{.4163} & \text{107}\text{.1090} \\
\text{14} & \text{100} & \text{0}\text{.9517} & \text{1}\text{.6619} & \text{10000} & \text{2}\text{.7619} & \text{166}\text{.1900} \\
\mathop{}_{}^{} & \text{630} & {} & \text{0} & \text{40600} & \text{11}\text{.3646} & \text{365}\text{.2711} \\
\end{matrix}</math></center>

:• The median rank values ( <math>F({{T}_{i}})</math> ) can be found in rank tables, available in many statistical texts, or they can be estimated by using the Quick Statistical Reference in Weibull++.
:• The <math>{{y}_{i}}</math> values were obtained from standardized normal distribution's area tables by entering for <math>F(z)</math> and getting the corresponding <math>z</math> value ( <math>{{y}_{i}}</math> ). As with the median rank values, these standard normal values can be obtained with the Quick Statistical Reference.
Given the values in Table 8.2, calculate <math>\widehat{a}</math> and <math>\widehat{b}</math> using Eqns. (aan) and (bbn):

::<math>\begin{align}
& \widehat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{T}_{i}}{{y}_{i}}-(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{T}_{i}})(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}})/14}{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{2}-{{(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{T}_{i}})}^{2}}/14} \\
& & \\
& \widehat{b}= & \frac{365.2711-(630)(0)/14}{40,600-{{(630)}^{2}}/14}=0.02982
\end{align}</math>

:and:

::<math>\widehat{a}=\overline{y}-\widehat{b}\overline{T}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\widehat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{T}_{i}}}{N}</math>

:or:

::<math>\widehat{a}=\frac{0}{14}-(0.02982)\frac{630}{14}=-1.3419</math>

Therefore, from Eqn. (bn):

::<math>\widehat{\sigma}=\frac{1}{\hat{b}}=\frac{1}{0.02982}=33.5367</math>

:and from Eqn. (an):

::<math>\widehat{\mu }=-\widehat{a}\cdot \widehat{\sigma }=-(-1.3419)\cdot 33.5367\simeq 45</math>

or <math>\widehat{\mu }=45</math> hours <math>.</math>

The correlation coefficient can be estimated using Eqn. (RHOn):

::<math>\widehat{\rho }=0.979</math>

The preceding example can be repeated using Weibull++ .

:• Create a new folio for Times-to-Failure data, and enter the data given in Table 8.1.
:• Choose Normal from the Distributions list.
:• Go to the Analysis page and select Rank Regression on Y (RRY).
:• Click the Calculate icon located on the Main page.

[[Image:weibullfolio1.png|thumb|center|400px| ]]

The probability plot is shown next.

[[Image:weibullfolioplot1st.png|thumb|center|400px| ]]