Normal Distribution Examples

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These examples appear in the Life Data Analysis Reference book book.

The following examples illustrate the different types of life data that can be analyzed in Weibull++ using the Normal distribution. For more information on the different types of life data, see Life Data Classification.

Times-to-Failure (Complete) Data

Six units are tested to failure. The following hours-to-failure data are obtained: 12125, 11260, 12080, 12825, 13550 and 14670 hours. Assuming that the data are normally distributed, do the following:

Objectives

1. Find the parameters for the data set, using the Rank Regression on X (RRX) parameter estimation method

2. Obtain the probability plot for the data with 90%, two-sided Type 1 confidence bounds.

3. Obtain the [math]\displaystyle{ pdf }[/math] plot for the data.

4. Using the Quick Calculation Pad, determine the reliability for a mission of 11,000 hours, as well as the upper and lower two-sided 90% confidence limit on this reliability.

5. Using the Quick Calculation Pad, determine the MTTF, as well as the upper and lower two-sided 90% confidence limit on this MTTF.

6. Obtain tabulated values for the failure rate for 10 different mission end times. The mission end times are 1,000 to 10,000 hours, using increments of 1,000 hours.

Solution

The following figure shows the data as entered in Weibull++, as well as the calculated parameters.

The following figures show the probability plot with the 90% two-sided confidence bounds and the pdf plot.

Both the reliability and MTTF can be easily obtained from the QCP. The QCP, with results, for both cases is shown in the next two figures.

To obtain tabulated values for the failure rate, use the Analysis Workbook or General Spreadsheet features that are included in Weibull++. (For more information on these features, please refer to the Weibull++ User's Guide. For a step-by-step example on creating Weibull++ reports, please see the Weibull++/ALTA QuickStart Guide.) The following worksheet shows the mission times and the corresponding failure rates.

Interval Censored Data

Eight units are being reliability tested, and the following is a table of their times-to-failure:

Non-Grouped Interval Data
Data point index	Last Inspected	State End Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

This is a sequence of interval times-to-failure data. Using the normal distribution and the maximum likelihood (MLE) parameter estimation method, the computed parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align} }[/math]

For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align} }[/math]

If we analyze the data set with the rank regression on y (RRY) parameter estimation method, the computed parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align} }[/math]

The following plot shows the results if the data were analyzed using the rank regression on X (RRX) method.

Times-to- Failure with Right Censored (Suspension) Data

Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Non-Grouped Data Times-to-Failure Data with Suspensions
Data point index	Last Inspected	State End Time
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Using the normal distribution and the maximum likelihood (MLE) parameter estimation method, the computed parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align} }[/math]

If we analyze the data set with the rank regression on x (RRX) method, the computed parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align} }[/math]

For the rank regression on y (RRY) method, the parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align} }[/math]

Times-to- Failure Data Without Suspensions

Eight units are being reliability tested and the following is a table of their times-to-failure:

Non-Grouped Data Times-to-Failure Data
Data point index	State F or S	State End Time
1	F	2
2	F	5
3	F	11
4	F	23
5	F	29
6	F	37
7	F	43
8	F	59

Using the normal distribution and the maximum likelihood (MLE) parameter estimation method, the computed parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align} }[/math]

If we analyze the data set with the rank regression on x (RRX) method, the computed parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align} }[/math]

For the rank regression on y (RRY) method, the parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align} }[/math]

Times-to-Failure with All Types of Censored Data

Suppose our data set includes left and right censored, interval censored and complete data, as shown in the following table.

Grouped Data Times-to-Failure with Suspensions and Intervals (Interval, Left and Right Censored)
Data point index	Number in State	Last Inspection	State (S or F)	State End Time
1	1	10	F	10
2	1	20	S	20
3	2	0	F	30
4	2	40	F	40
5	1	50	F	50
6	1	60	S	60
7	1	70	F	70
8	2	20	F	80
9	1	10	F	85
10	1	100	F	100

Using the normal distribution and the maximum likelihood (MLE) parameter estimation method, the computed parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42 \end{align} }[/math]

If we analyze the data set with the rank regression on x (RRX) method, the computed parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17 \end{align} }[/math]

For the rank regression on y (RRY) method, the parameters are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07 \end{align} }[/math]