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

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Probability and Statistics for ReliabilityBenbow
and Broome (Ch 4 and Ch 5)

• Presented by Dr. Joan Burtner
• Certified Quality Engineer
• Associate Professor of
• Industrial Engineering and Industrial Management

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Overview

• Chapter 4 Basic Concepts
• Measures of central tendency
• Measures of dispersion
• Discrete and continuous probability distributions
• Statistical process control
• Chapter 5 Statistical Inference
• Point estimate for failure rate
• Confidence intervals
• Parametric hypothesis testing
• Nonparametric hypothesis testing
• Type I and Type II errors
• Bayess theorem for reliability

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Statistical Analysis

• Measures of Central Tendency
• Mean
• Median
• Mode
• Measures of Dispersion (aka Variation or Spread)
• Range
• Standard Deviation
• Variance

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Probability Distributions

• Widely-used discrete distributions
• Poisson
• Binomial
• Negative Binomial
• Hypergeometric
• Widely-used continuous distributions
• Normal
• Exponential
• Weibull
• Lognormal
• Skewness and Kurtosis

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Statistical Process Control (SPC)

• Central tool is the control chart
• Provides an early signal when a process changes
• Basic chart consists of an upper control limit,
  lower control limit, and process mean
• Trial control charts are based on historic data
• The process is monitored and control limits are
  modified as needed
• Evaluation of control charts is based on
  probability distribution of the characteristic
  being monitored
• Normal (variables)
• Binomial or Poisson (attributes)

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SPC - Theory of Variation

• Common Cause
• Stable and predictable causes of variation
• Inherent in all processes
• Managers, not workers, are responsible for common
  cause variation
• Special Cause
• Unexpected or abnormal causes of variation
• May result in sudden or extreme departures from
  normal
• May also result in gradual shifts (trends)

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SPC - Control Chart Types

• Control Charts
• Variables based on continuous data
• X bar and R (mean and range)
• X bar and S (mean and standard deviation)
• Attributes - based on discrete data
• P (proportion)
• C (count)
• U (count per unit)

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Control Chart Calculations for Xbar and R Charts

• Xbar and R Control Chart Constants

• Control Chart Calculations

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
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Control Chart Interpretation We will use Minitab
to build / interpret control charts

• Building Control Charts
• Collect at least 25 samples
• Enter data in Minitab using appropriate
  formatting
• Use pull-down menu to select the desired type of
  chart
• Interpretation of Control Charts
• Use Minitab to identify the tests

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Parametric Hypothesis Testing (used for known
distributions)

• Basic Hypothesis Testing for Means
• One Sample t or Z Tests
• Two Sample t or Z Tests
• Hypothesis Tests for Population Standard
  Deviation
• Hypothesis Tests for Population Proportion
• Advanced Designs for Hypothesis Testing (Covered
  in Chapter 6 of Benbow and Broome)
• One Factor ANOVA
• Two Factor ANOVA
• Full Factorial Experiments

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Nonparametric Hypothesis Testing

• Kruskal-Wallis
• Nonparametric equivalent to one factor ANOVA
• Does not require the assumption that the
  population is normal
• Hypothesizes about medians unless population
  known to be mound-shaped and symmetric
• Minitab hypotheses- medians
• Benbow and Broome hypotheses - means
• Wilcoxon Signed Rank Test
• Nonparametric equivalent to single sample test
  for mean
• Used when we cant assume that the population is
  normal
• Used when we cant assume the Central Limit
  Theorem applicable
• Examples in Minitab

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References

• Course Text
• Benbow, D.W. and Broome, H.W., Ed. (2009). The
  Certified Reliability Engineer Handbook .
  Milwaukee,WI ASQ Quality Press.
• Additional Sources
• Christensen, E.H., Coombes-Betz, K.M., and Stein,
  M.S. (2006). The Certified Quality Process
  Analyst Handbook. Milwaukee ASQ Quality Press.
• Westcott, R.T., Ed. (2006). Certified Manager of
  Quality/Organizational Excellence Handbook (3rd
  ed.). Milwaukee ASQ Quality Press.

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Contact Information

• Email Burtner_J_at_Mercer.edu
• US Mail
• Mercer University School of Engineering
• 1400 Coleman Avenue
• Macon, GA
• Phone [phone]