Probability and Statistics for Reliability Benbow and Broome (Ch 4 and Ch 5)

Probability and Statistics Probability and Statistics for Reliabilityfor Reliability

Benbow and Broome (Ch 4 and Ch 5)Benbow and Broome (Ch 4 and Ch 5)

Presented by Dr. Joan BurtnerPresented by Dr. Joan Burtner

Certified Quality Engineer Certified Quality Engineer

Associate Professor ofAssociate Professor of

Industrial Engineering and Industrial Industrial Engineering and Industrial ManagementManagement

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 22

OverviewOverview

Chapter 4 Basic ConceptsChapter 4 Basic Concepts Measures of central tendencyMeasures of central tendency Measures of dispersionMeasures of dispersion Discrete and continuous probability distributionsDiscrete and continuous probability distributions Statistical process controlStatistical process control

Chapter 5 Statistical InferenceChapter 5 Statistical Inference Point estimate for failure ratePoint estimate for failure rate Confidence intervalsConfidence intervals Parametric hypothesis testingParametric hypothesis testing Nonparametric hypothesis testingNonparametric hypothesis testing Type I and Type II errorsType I and Type II errors Bayes’s theorem for reliabilityBayes’s theorem for reliability

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Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 33

Statistical Analysis Statistical Analysis

Measures of Central TendencyMeasures of Central Tendency MeanMean MedianMedian ModeMode

Measures of Dispersion (aka Measures of Dispersion (aka Variation or Spread) Variation or Spread) RangeRange Standard DeviationStandard Deviation VarianceVariance

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 44

Probability Probability Distributions Distributions Widely-used discrete distributionsWidely-used discrete distributions

PoissonPoisson BinomialBinomial Negative BinomialNegative Binomial HypergeometricHypergeometric

Widely-used continuous distributions Widely-used continuous distributions NormalNormal ExponentialExponential WeibullWeibull LognormalLognormal

Skewness and KurtosisSkewness and Kurtosis

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 55

Statistical Process Statistical Process Control (SPC) Control (SPC) Central tool is the control chartCentral tool is the control chart

Provides an early signal when a process changesProvides an early signal when a process changes Basic chart consists of an upper control limit, Basic chart consists of an upper control limit,

lower control limit, and process meanlower control limit, and process mean Trial control charts are based on historic dataTrial control charts are based on historic data The process is monitored and control limits are The process is monitored and control limits are

modified as neededmodified as needed Evaluation of control charts is based on probability Evaluation of control charts is based on probability

distribution of the characteristic being monitoreddistribution of the characteristic being monitored Normal (variables)Normal (variables) Binomial or Poisson (attributes)Binomial or Poisson (attributes)

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 66

SPC - Theory of SPC - Theory of Variation Variation Common CauseCommon Cause

Stable and predictable causes of variationStable and predictable causes of variation Inherent in all processesInherent in all processes Managers, not workers, are responsible Managers, not workers, are responsible

for common cause variationfor common cause variation Special Cause Special Cause

Unexpected or abnormal causes of Unexpected or abnormal causes of variationvariation

May result in sudden or extreme May result in sudden or extreme departures from normaldepartures from normal

May also result in gradual shifts (trends)May also result in gradual shifts (trends)

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 77

SPC - Control Chart SPC - Control Chart Types Types Control ChartsControl Charts

Variables – based on continuous dataVariables – based on continuous data• X bar and R (mean and range)X bar and R (mean and range)• X bar and S (mean and standard X bar and S (mean and standard

deviation)deviation) Attributes - based on discrete dataAttributes - based on discrete data

• P (proportion)P (proportion)• C (count)C (count)• U (count per unit)U (count per unit)

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 88

Control Chart Control Chart Calculations for Xbar Calculations for Xbar and R Chartsand R Charts Xbar and R Xbar and R

Control Chart Control Chart ConstantsConstants

RAXLCL

XCL

RAXUCL

RDLCL

RCL

RDUCL

X

X

X

R

R

R

2

2

3

4

Control Chart Control Chart CalculationsCalculations

n d 2 A 2 D 3 D 4

2 1.128 1.88 0 3.267

3 1.693 1.023 0 2.575

4 2.059 0.729 0 2.282

5 2.326 0.577 0 2.115

6 2.534 0.483 0 2.004

7 2.704 0.419 0.076 1.924

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 99

Control Chart Control Chart Interpretation Interpretation We will use Minitab to We will use Minitab to

build / interpret control chartsbuild / interpret control charts

Building Control Charts Building Control Charts Collect at least 25 samplesCollect at least 25 samples Enter data in Minitab using Enter data in Minitab using

appropriate formattingappropriate formatting Use pull-down menu to select the Use pull-down menu to select the

desired type of chartdesired type of chart Interpretation of Control ChartsInterpretation of Control Charts

Use Minitab to identify the testsUse Minitab to identify the testsSample

Sa

mp

le R

an

ge

454137332925211713951

0.4

0.3

0.2

0.1

0.0

_R=0.1972

UCL=0.4170

LCL=0

Line 2 Range Chart

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 1010

Parametric Hypothesis Parametric Hypothesis Testing Testing (used for ‘known’ distributions)(used for ‘known’ distributions)

Basic Hypothesis Testing for MeansBasic Hypothesis Testing for Means One Sample t or Z TestsOne Sample t or Z Tests Two Sample t or Z TestsTwo Sample t or Z Tests

Hypothesis Tests for Population Standard Hypothesis Tests for Population Standard DeviationDeviation

Hypothesis Tests for Population ProportionHypothesis Tests for Population Proportion Advanced Designs for Hypothesis Testing Advanced Designs for Hypothesis Testing

(Covered in Chapter 6 of Benbow and (Covered in Chapter 6 of Benbow and Broome)Broome) One Factor ANOVAOne Factor ANOVA Two Factor ANOVATwo Factor ANOVA Full Factorial Experiments Full Factorial Experiments

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 1111

Nonparametric Nonparametric Hypothesis Testing Hypothesis Testing Kruskal-WallisKruskal-Wallis

Nonparametric equivalent to one factor ANOVANonparametric equivalent to one factor ANOVA Does not require the assumption that the population is Does not require the assumption that the population is

normalnormal Hypothesizes about medians unless population known to Hypothesizes about medians unless population known to

be “mound-shaped and symmetric”be “mound-shaped and symmetric” Minitab hypotheses- mediansMinitab hypotheses- medians Benbow and Broome hypotheses - meansBenbow and Broome hypotheses - means

Wilcoxon Signed Rank TestWilcoxon Signed Rank Test Nonparametric equivalent to single sample test for meanNonparametric equivalent to single sample test for mean Used when we can’t assume that the population is Used when we can’t assume that the population is

normalnormal Used when we can’t assume the Central Limit Theorem Used when we can’t assume the Central Limit Theorem

applicableapplicable Examples in MinitabExamples in Minitab

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 1212

ReferencesReferences

Course Text:Course Text: Benbow, D.W. and Broome, H.W., Ed. (2009). Benbow, D.W. and Broome, H.W., Ed. (2009).

The Certified Reliability Engineer HandbookThe Certified Reliability Engineer Handbook . . Milwaukee,WI: ASQ Quality Press.Milwaukee,WI: ASQ Quality Press.

Additional Sources Additional Sources Christensen, E.H., Coombes-Betz, K.M., and Christensen, E.H., Coombes-Betz, K.M., and

Stein, M.S. (2006). Stein, M.S. (2006). The Certified Quality Process The Certified Quality Process Analyst HandbookAnalyst Handbook. Milwaukee: ASQ Quality . Milwaukee: ASQ Quality Press.Press.

Westcott, R.T., Ed. (2006). Westcott, R.T., Ed. (2006). Certified Manager of Certified Manager of Quality/Organizational Excellence HandbookQuality/Organizational Excellence Handbook (3 (3rdrd ed.). Milwaukee: ASQ Quality Press.ed.). Milwaukee: ASQ Quality Press.

Fall 2010 ETM 591 ISE Fall 2010 ETM 591 ISE 427 427

Dr. Joan Burtner, Associate Professor of Industrial EngineeringDr. Joan Burtner, Associate Professor of Industrial Engineering Slide Slide 1313

Contact InformationContact Information

Email: [email protected]: [email protected] US Mail:US Mail:

Mercer University School of Mercer University School of Engineering Engineering

1400 Coleman Avenue 1400 Coleman Avenue

Macon, GAMacon, GA Phone: (478) 301- 4127Phone: (478) 301- 4127