Modeling Process Quality

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Chapter 2

• Modeling Process Quality

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2-1. Describing Variation

• Graphical displays of data are important tools
  for investigating samples and populations.
• Displays can include stem and leaf plots,
  histograms, box plots, and dot diagrams.
• Graphical displays give an indication of the
  overall distribution of the data.

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2-1.1 The Stem-and-Leaf Plot

• 17 558
• 18 357
• 19 00445589
• 20 1399
• 21 00238
• 22 005
• 23 5678
• 24 1555899
• 25 158

• The numbers on the left are the stems
• The values on the right are the leaves
• The smallest number in this set of data is 175
• The median is 211

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More about it

• The previous is actually an ordered stem-and-leaf
  display
• Ordered from lowest to highest
• Terms used in describing data can be introduced
  using the example
• Percentiles
• kth percentile

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More about it

• 50th percentile is at 211
• 20 observations are below and 20 observations are
  above
• So, its also the sample median
• 10th percentile is at 184
• 4 observations are below and 36 observations are
  above

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More about it

• Quartiles
• 1st quartile
• 25 of the observations are below the value

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More about it

• 1st quartile is at 194.5
• 10 observations are below and 30 observations are
  above
• (194 195)/2 194.5
• 3rd quartile is at 239.5
• 30 observations are below and 10 observations are
  above
• (238 241)/2 239.5

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More about it

• Interquartile range, or IQR, is 45
• Q3 Q1 239.5 194.5

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2-1.2 The Frequency Distribution and
Histogram

• Frequency Distribution
• Arrangement of data by magnitude
• More compact than a stem-and-leaf display
• Graphs of observed frequencies are called
  histograms.

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2-1.2 The Frequency Distribution and
Histogram

• Histogram

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Suggestions for histograms

• Use between 4 and 20 bins
• Guideline is SQRT (n)
• Make the bins of uniform width
• Start the lower limit for the first bin just
  slightly below the smallest data value
• lt5 overflow

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Graphical Displays

• What is the overall shape of the data?
• Are there any unusual observations?
• Where is the center or average of the data
  located?
• What is the spread of the data? Is the data
  spread out or close to the center?

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2-1.3 Numerical Summary of Data

• Important summary statistics for a distribution
• of data can include
• Sample mean,
• We often write Xbar
• Sample variance, S2
• Sample standard deviation, S
• Sample median, M

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2-1.3 Numerical Summary of Data

• For the data shown in the previous histogram and
  stem and leaf plot, the summary statistics are
• n Mean Median Var StDev
• 40 215.50 211.00 634.5 25.19

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Use Excel
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Understanding variance

• Sample 1
• 1, 3, 5
• Mean 3
• S2 12 32 52 3(32)/2 4
• S 2

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Understanding variance

• Sample 2
• 1, 5, 9
• Mean 5
• S2 16
• S 4

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Understanding variance

• Sample 3
• 101, 103, 105
• Mean 103
• S2 4
• S 2

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2-1.4 The Box Plot

• The Box Plot is a graphical display that
• provides important quantitative
• information about a data set. Some of
• this information is
• Location or central tendency
• Spread or variability
• Departure from symmetry
• Identification of outliers

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Example 2-2 Data on hole diameters

• 120.5, 120.9, 120.3, 121.3, 120.4, 120.2, 120.1,
  120.5, 120.7, 121.1, 120.9, 120.8
• Minimum
• 120.1
• Maximum
• 121.3

• 1st quartile
• 120.35
• 3rd quartile
• 120.9
• Median
• 120.6

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2-1.4 The Box Plot
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2-1.5 Sample Computer Output
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2-1.6 Probability Distributions

• Definitions
• Sample A collection of measurements selected
  from some larger source or population.
• Probability Distribution A mathematical model
  that relates the value of the variable with the
  probability of occurrence of that value in the
  population.
• Random Variable Variable that can take on
  different values in the population according to
  some random mechanism.

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2-1.6 Probability Distributions

• Two Types of Probability Distributions
• Continuous When a variable being measured is
  expressed on a continuous scale, its probability
  distribution is called a continuous distribution.
  The probability distribution of piston-ring
  diameters is continuous.
• Discrete When the parameter being measured can
  only take on certain values, such as the integers
  0, 1, 2, , the probability distribution is
  called a discrete distribution. The distribution
  of the number of nonconformities would be a
  discrete distribution.

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2-2 Important Discrete Distributions

• 2-2.1 The Hypergeometric Distribution
• 2-2.2 The Binomial Distribution
• 2-2.3 The Poisson Distribution

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Hypergeometric distribution

• Finite population of N items
• D (where D lt N) have a characteristic of interest
• Sample of size n is taken
• Probability that x of n have the characteristic
  of interest
• Concepts are used in acceptance sampling

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Hypergeometric distribution

• Example
• Lot contains 100 items
• 5 of the lot are nonconforming
• Sample 10 from the lot
• Probability that not more than one is
  nonconforming
• P(x lt 1) P(x 0) P(x 1)

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Hypergeometric distribution

• Example, cont.
• P(x 0)
• 5!/(0!5!) 95!/(10!85!)/100!/(10!90!)
• P(x 1)
• 5!/(1!4!) 95!/(9!86!)/100!/(10!90!)
• P(x lt 1) .923

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2-2.2 The Binomial Distribution

• A quality characteristic follows a binomial
• distribution if
• 1. All trials are independent.
• 2. Each outcome is either a success or a
  failure.
• The probability of success on any trial is given
  as p. The probability of a failure is 1- p.
• 4. The probability of a success is constant.

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2-2.2 The Binomial Distribution

• The binomial distribution with parameters
• n 0 and 0 lt p lt 1, is
• The mean and variance of the binomial
  distribution are

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Example

• The probability that the Braves win a game at
  home against the Mets is 0.52
• What is the probability that the Braves win
  exactly 2 of 3 in the next home stand with the
  Mets
• 3!/(2!1!)(.52)2 (.48) .389

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2-2.3 The Poisson Distribution

• The Poisson distribution is
• Where the parameter ? gt 0. The mean and variance
  of the Poisson distribution are

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2-2.3 The Poisson Distribution

• The Poisson distribution is useful in quality
  engineering
• Typical model for the number of defects or
  nonconformities that occur in a unit of product.
• Any random phenomenon that occurs on a per unit
  basis is often well approximated by the Poisson
  distribution.

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Example

• The expected number of surface blemishes on the
  door of a new Lexus 400 is distributed as a
  Poisson random variable with a mean of 1.2
• What is the probability that there are 2 or more
  blemishes on a door

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Example, cont.

• P(x gt2) 1 P(x0) P(x1)
• P(x0) (e-1.21.20)/0! .301
• P(x1) (e-1.21.21)/1! .361
• P(xgt2) 1 - .301 - .361 .339

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2-3 Important Continuous Distributions

• 2-3.1 The Normal Distribution
• 2-3.2 The Exponential Distribution

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2-3.1 The Normal Distribution

• The normal distribution
• is an important
• continuous distribution.
• Symmetric, bell-shaped
• Mean, ?
• Standard deviation, ?
• N(m, s2)

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2-3.1 The Normal Distribution

• For a population that is
• normally distributed
• approx. 68 of the data will lie within 1
  standard deviation of the mean
• approx. 95 of the data will lie within 2
  standard deviations of the mean, and
• approx. 99.7 of the data will lie within 3
  standard deviations of the mean.

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2-3.1 The Normal Distribution

• Standard normal distribution
• Many situations will involve data that is
  normally distributed. We will often want to find
  probabilities of events occurring or percentages
  of nonconformities, etc. A standardized normal
  random variable is

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2-3.1 The Normal Distribution

• Standard normal distribution
• Z is normally distributed with mean 0 and
  standard deviation, 1.
• Use the standard normal distribution to find
  probabilities when the original population or
  sample of interest is normally distributed.
• Tabulated.

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2-3.2 The Normal Distribution

• Example 2-5
• The tensile strength of paper is modeled by a
  normal
• distribution with a mean of 35 lbs/in2 and a
  standard
• deviation of 2 lbs/in2.
• What is the probability that the tensile strength
  of a selected item is less than 40 lbs/in2?
• If the specifications require the tensile
  strength to exceed 30 lbs/in2, what is the
  probability that a selected item is scrapped?

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Example, cont.

• Determine P(xlt40)
• Z (40 35)/2 2.5
• So, F(2.5) .99379
• And, P(xlt40) .99379
• Determine P(xgt30)
• Z (30 35)/2 -2.5
• Note, F(-2.5) 1 - F(2.5)
• So, P(xgt30) 1 - .99379 .00621

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Another example

• Given XN(10,9)
• Find a such that P(x gt a) .05
• From Appendix II, Z 1.645
• Z (a m)/s
• a Zs m
• So, a 3(1.645) 10 14.935

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Linear combinations

• If normally distributed random variables are
  combined, the result is a normally distributed
  random variable whose mean is the sum of the
  individual means and whose variance is the sum of
  the individual variances

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Linear combinations

• If Y a1x1 a2x2 anxn
• Then, my a1m1 a2m2 anmn
• And sy2 a12s12 a22s22 an2sn2

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Example

• Four rods with the following distributions N(m,
  s2) in cm are laid end to end
• N( 40, 4), N(36, 3), N(42, 7), N(100, 11)
• What is the distribution of the combination?
• N(218, 25), my 218 cm, sy2 25, sy 5

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Example, cont.

• What is the probability that the assembly will be
  longer than 220 cm?
• Z (220 218)/5 .4
• So, P(Ygt220) 1 - .65542 .34458
• What is the probability that the assembly will be
  between 216 and 220 cm?
• 1 - .34458 - .34458 .31084

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Central limit theorem

• If x1, x2, , xn are independent random variables
  mi, and variance si2, and if y x1 x2
  xn, then
• y Smi/SQRT(Ssi2) is distributed N(0, 1) as n
  approaches infinity
• The further the deviation from a symmetric,
  unimodal distribution, the larger the value of n
  required to achieve normality

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Example

• If six U(0, 1) random number are added, the mean
  is subtracted, and the result is divided by
  SQRT(.5), a N(0, 1) random variable is generated
• .2345, .1987, .7762, .8150, .5337, .3462
• (2.9043 3)/.7071 -.1353
• Note, si2 1/12

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2-3.3 The Exponential Distribution

• The exponential distribution is widely used in
  the field of reliability engineering.
• The exponential distribution is
• The mean and variance are

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2-4 Some Useful Approximations

• In certain quality control problems, it is
  sometimes useful to approximate one probability
  distribution with another. This is particularly
  useful if the original distribution is difficult
  to manipulate analytically.
• Some approximations
• Binomial approximation to the hypergeometric
• Poisson approximation to the binomial
• Normal approximation to the binomial

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Binomial approximation to the hypergeometric

• If n/N (the sampling fraction is small (say n/N lt
  .1), then the binomial distribution with p D/N
  and n is a good approximation to the
  hypergeometric

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Example

• A lot of 200 units contains 5 nonconforming units
• What is the probability that a sample of 10 will
  contain no nonconforming units?
• N 200, n 10, n/N .05 lt .1
• Using the hypergeometric
• P(x0) (C5,0)(C195,10)/ C200,10
• P(x0) .7717

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Example, cont.

• Using the binomial approximation with p
  D/N 5/200 .025, and n 10
• P(x0) C10,0(.025)0(.975)10 .7763

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Poisson approximation to the binomial

• When p approaches 0 and n approaches infinity
  with l np constant
• Good when p lt .1

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Example

• Using the data from the last example with p
  .025 and n 10, l np .25
• Then, p(0) (e-.252.50)/0! .7788
• (Compare to .7763 from the last example)

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Normal approximation to the binomial

• If the number of trials, n, is large
  (say n gt 10), and p is close to ½, the normal can
  be used to approximate the binomial, with a
  continuity correction

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Example

• p .5
• n 20
• np 10
• np(1 p) 5, SQRT np(1 p) 2.236
• P(8) ?
• Using the binomial
• p(8) C20,8(.5)8(.5)12 .12

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Example, cont.

• Using the normal approximation
• P(7.5 lt a lt 8.5) F(8.5 10)/2.236 - F(7.5
  10)/2.236
• F(-.671) F(-1.118) .8682 - .7489
• .1193
• (Compare to .12)

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Suggestion

• Work enough odd numbered exercises so that you
  understand this chapter
• If it says Derive you can skip it
• If it says Continuation of work the previous
  exercise as well

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End