Engineering Statistics Part 2

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Engineering Statistics Part 2

• Hypothesis testing
• Quality control
• Acceptance sampling
• Goodness of fit

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Hypothesis Testing

• Basic idea
• Perform statistical analysis of data to determine
  if a particular hypothesis should be accepted
• Use results of the hypothesis test for decision
  making
• General procedure
• Formulate the hypothesis
• Formulate an alternative to the hypothesis
• Choose a significance level a that represents the
  probability of rejecting a true hypothesis
• Perform statistical analysis on samples to test
  the validity of the hypothesis
• Either accept or reject the hypothesis based on
  the test

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One-Sided and Two-Sided Alternatives

• Let q be an unknown parameter in a distribution
  for which we hypothesize q q0
• Alternatives
• Testing errors
• a probability of rejecting a true hypothesis
  significance level
• b probability of accepting a false hypothesis

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Right-Handed Test Procedure

• Hypothesis q q0
• Alternative q q1 gt q0
• Consider a critical c gt q0
• Given samples x1,,xn , compute parameter
  estimate
• Accept hypothesis if
• Testing errors
• Usually specify a, then compute c b

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Mean Test for Normal Distributions

• Students t-distribution
• Let X1,,Xn be independent normal random
  variables, each with same mean m variance s2
• Then the random variable T follows a
  t-distribution with m n-1
• Data n random samples x1, x2,, xn
• Left-handed mean test
• Hypothesis mean is m0 instead of m1 lt m0
• Select significance level a
• Compute observed value of T as
• Determine c from Table A9 with m n-1 as
• Accept hypothesis if t gt c otherwise reject

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Hypothesis Test Example

• Measurements of polymer molecular weight
• Hypothesis m0 1.3 instead of m1 1.2
• Significance level a 0.10
• Degrees of freedom m 9
• Critical value
• Sample t
• Reject hypothesis

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Variance Test for Normal Distributions

• Chi-squared distribution
• Let X1,,Xn be independent normal random
  variables, each with same mean m variance s2
• Then the random variable Y follows a chi-squared
  distribution with m n-1
• Right-handed variance test
• Hypothesis variance is m02 instead of m12 gt m02
• Compute s2 as
• Select significance level a
• Determine c from Table A10 with m n-1 as
• Compute critical value of s2 as
• Accept hypothesis if s2 lt c otherwise reject

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Quality Control

• Control chart
• Statistical method to determine if production
  process is functioning normally
• Most common tool for quality control
• Data n random samples x1, x2,, xn
• Mean control chart
• Assume knowledge of expected mean m0
• Lower upper control limits for a 1
• Variance s2 assumed known or computed from
  samples
• Process is out of control if sample mean falls
  outside control limits
• Variance control chart
• Assume knowledge of expected variance s2
• Upper control limit for a 1
• Utilize chi-squared distribution (Table A10 in
  Appendix 5)
• Process is out of control if sample variance is
  above upper control limit

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Polymerization Reactor Control

• Lab measurements
• Viscosity monomer content of polymer every four
  hours
• Three monomer content measurements 23, 18,
  22
• Mean content control
• Expected mean known variance m0 20, s2 1
• Control limits
• Sample mean 21 ? process is in control

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Acceptance Sampling

• Problem statement
• Produce a lot of N items
• Determine number of defects x from n samples
• Reject lot if x gt c (the acceptance number)
• Probability
• Let the event A be accept the lot
• Let M number of defects in lot of N items
• Define fraction defective q M/N
• Governed by hypergeometric distribution (see
  text)
• If q ltlt 1 n ltlt N, Poisson distribution provides
  good approximation
• Can determine sampling plan (n c) to hypothesis
  test for errors a b (see text)

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Goodness of Fit

• Problem statement
• Hypothesize that n random samples x1, x2,, xn
  have been drawn from a particular distribution
  with a distribution function F(x)
• If the following sample distribution fits F(x)
  sufficiently well, then accept hypothesis
• Chi-squared test
• Subdivide x-axis into K intervals I1, I2,, IK
  such that each interval contains at least 5
  sample values. Determine the number bj of sample
  values in each interval.
• Using the assumed F(x), compute the probability
  pj that the random variable X under consideration
  assumes any value in the interval IJ

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Goodness of Fit cont.

• Chi-square test cont.
• Compute the number of samples values ej expected
  in IJ if the hypothesis is true ej npj
• Compute the deviation
• Choose a significance level a
• Determine the value c from the chi-square
  distribution with m K-1 if the distribution is
  completely known and with m K-r-1 if the
  maximum likelihood estimates of the r parameters
  are used.

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Goodness of Fit Example

• Maximum likelihood estimates

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Goodness of Fit Example cont.