PROBABILITY AND STATISTICS REVIEW

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PROBABILITY AND STATISTICS REVIEW

• Describing Summarizing Data
•   
• The Binomial Distribution
• The Poisson Distribution
• The Normal Distribution
• Sampling Distributions

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RANDOM VARIABLES AND REALITY

• All processes are subject to variability
• Unassignable variation
• Variation inherent in the process which cannot be
  reduced without process redesign
• Randomness
• Assignable variation
• Process variation which represents deviation from
  expectations and may be traced (assigned) to an
  external factor

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RANDOM VARIABLES AND REALITY

• A key objective in QM is separating assignable
  from unassignable variation
• Assignable variation can be quickly eliminated
• First need to model the unassignable variation
• A reference probability distribution is selected
  whose behavior matches that of the quality
  characteristic generated by the process (coin
  flipping)
• If the process produces a different distribution,
  we know that assignable variation is present
• Selecting the correct distribution for a quality
  characteristic is a critical step

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SPOTTING PROBLEMS USING A REFERENCE DISTRIBUTION
(HYPOTHESIS TESTING)

• After a reference distribution describing the
  population is selected, samples of the quality
  characteristic are checked periodically for
  assignable variation
• Quality data is collected
• Sample statistics are calculated
• The sample statistics are compared with those
  from the reference distribution
•  Does this data come from this distribution?
• If yes, conclude there is no assignable variation
  (in control)
• If no, conclude there is assignable variation
  (out of control)

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DESCRIBING SUMMARIZING DATA

• How can data (Xi) from a population of size N or
  from a sample of size n be summarily described?
• Graphically
• Histogram
• Numerically
• Mean
• Variance (Standard Deviation)2

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DESCRIBING SUMMARIZING DATA

• Mean

• Variance (Standard Deviation)2

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THE BINOMIAL DISTRIBUTION

• Models "go/no-go" attribute quality
  characteristics
• n -- Sample size
• p -- Probability of a nonconformity
• X -- Number of nonconformities in sample X 0,
  1, . , n
• X is a binomial random variable with following
  distribution

• The mean and variance are

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THE BINOMIAL DISTRIBUTION

• Example improperly sealed orange juice cans
• n 100
• p 0.02

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THE BINOMIAL DISTRIBUTION

• Example improperly sealed orange juice cans
• n 100
• p 0.02

• The mean and variance are

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THE POISSON DISTRIBUTION

• Models integer-valued quality characteristics
  that range from 0 to infinity
• c -- Constant rate of nonconformities per item
• X -- number of nonconformities per itemx 0, 1,
  2, . . .
• X is a Poisson random variable with following
  distribution

• The mean and variance are

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THE POISSON DISTRIBUTION

• Example scratches per table top
• c 2

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THE POISSON DISTRIBUTION

• Example scratches per table top
• c 2

• The mean and variance are

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THE POISSON DISTRIBUTION

• For future reference
• When n is large and p is small and np lt 5,
• The Poisson can be used to approximate the
  binomial distribution if we set 
• c np
• 2 (100)(0.02)
• Why do this? Binomial is unwieldy for large n

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THE NORMAL DISTRIBUTION

• Models continuous quality characteristics that
  range from negative to positive infinity
• µ -- Mean
• s2 -- Variance
• X -- Value of quality characteristic - ? lt X lt
  ?
•  
• X is a normal random variable with the following
  distribution

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THE NORMAL DISTRIBUTION

• All normal distributions have same shape

• If X is a standard normal random variable
• And Y is any normal random variable with mean µ
  and standard deviation s
• PROB(X ? z) PROB(Y ? µ zs)

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THE NORMAL DISTRIBUTION
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THE NORMAL DISTRIBUTION

• Since PROB(X ? z) PROB(Y ? µ zs)
•  
• If Y N?, ? AND X N0, 1, then
• Let A be any constant 
• Let z be chosen so that A ? z? (I.E. SET z
  (A - ?)/?)
• Then PROB(Y ? A) PROB(Y ? ? z?)
• PROB(X ? z)

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THE NORMAL DISTRIBUTION

• Example weighing filled bags of sugar
• m 10
• s 0.1
• What is the likelihood of a bag weighing more
  than 10.2 pounds?

• PROB(Y ? A) PROB(Y ? ? z?) PROB(X ? z)
• PROB(Y ? 10.2) PROB(Y ? 10 2(0.1)) PROB(X ?
  2)
• Or we could calculate

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SAMPLING DISTRIBUTIONS

• If we know the distribution of a random variable
  --
• We can construct a distribution of any function
  of the random variables
• Averages
• Ranges
• Proportions
• Standard deviations

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SAMPLING DISTRIBUTIONS THE BINOMIAL DISTRIBUTION

• If X is binomially distributed with
• n -- Sample size
• p -- Probability of a nonconformity
• We know the mean and variance are then

• But the sample proportion pi Xi/n has the
  following mean and variance

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SAMPLING DISTRIBUTIONS THE BINOMIAL DISTRIBUTION

• Example improperly sealed orange juice cans
• n 100
• p 0.02

• The mean and variance of the number nonconforming
  are

• The mean and variance of the fraction
  nonconforming are

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SAMPLING DISTRIBUTIONS THE POISSON DISTRIBUTION

• If X is Poisson distributed with
• c -- Constant rate of nonconformities per item
• n Size of item
• u Constant rate of nonconformities per unit of
  size
• We know the mean and variance are then

• But the number of nonconformities per unit of
  size ui Xi/n has the following mean and
  variance

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SAMPLING DISTRIBUTIONS THE POISSON DISTRIBUTION

• Example scratches per square foot of table top
• c 2
• n 10

• The mean and variance of the number nonconforming
  per item are

• The mean and variance of the number nonconforming
  per unit of size are

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SAMPLING DISTRIBUTIONS THE NORMAL DISTRIBUTION

• If X is normally distributed with
• µ -- Mean
• s2 -- Variance
• Let samples of size n be collected

• Each of these statistics has a distribution with
  a mean and standard deviation

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SAMPLING DISTRIBUTIONS THE NORMAL DISTRIBUTION

• For the sample means

• For the sample ranges

• For the sample standard deviations

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SAMPLING DISTRIBUTIONS THE NORMAL DISTRIBUTION
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SAMPLING DISTRIBUTIONS THE NORMAL DISTRIBUTION

• Example weighing filled bags of sugar
• m 10
• s 0.1
• n 5
• The distribution of sample means is as follows

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SAMPLING DISTRIBUTIONS THE NORMAL DISTRIBUTION

• Example weighing filled bags of sugar
• m 10
• s 0.1
• n 5
• The distribution of sample ranges is as follows

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SAMPLING DISTRIBUTIONS THE NORMAL DISTRIBUTION

• Example weighing filled bags of sugar
• m 10
• s 0.1
• n 5
• The distribution of sample standard deviations is
  as follows

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