Introduction to Statistical Quality Control, 5th edition

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• We may formally test the hypothesis
• H0 p1 p2
• H1 p1 p2 where
• p1
• P2 maybe estimated by
• 
  

Z07.10 comparing this to the upper 0.05 point of
Z, Z07.10 Z0.051.645. Consequently we reject
H0 and conclude that there has been a significant
decrease in the process fallout
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From inspection of Fig 6.4 we see that all the
points would fall insided the revised UCL.
Therefore, we conclude that the process is in
control
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The continued operation of this control chart for
the next five shifts is shown in Fig 6.5 and data
is shown in Table 6.3
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Design of Fraction Nonconforming Chart

• Three parameters must be specified
• The sample size
• The frequency of sampling
• The width of the control limits
• Common to base chart on 100 inspection of all
  process output over time
  
• Rational subgroups may also play role in
  determining sampling frequency

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Design of Fraction Nonconforming Chart contd

• Method 1 If p is very small, we can choose the
  sample size n so that the probability of finding
  at least one nonconforming unit per sample is at
  least ?.
• For example, suppose that p0.01 and we want the
  probability of at least one nonconforming unit in
  the sample to be at least 0.95.
  
• PD1 0.95
• using the poisson approximation to the binomial,
  we find from the cumulative poisson table that
  ?np must exceed 3. np3 n ?/p3/0.01300
  
• From table ?3 is corresponding to the Px0
  which is, Then P(x 1)1-P(x0) 1- 0.050.95

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Design of Fraction Nonconforming Chart contd

• Method 2 Duncan has suggested that the sample
  size should be large enough that we have
  approximately a 50 chance of detecting a process
  shift of some specified amount.
• For example, suppose that p0.01, and we want
  the probability of detecting a shift to p0.05 to
  be 0.5. Assuming that the normal approximation to
  the binomial applies, we should choose n so that
  the upper control limit exactly coincides with
  the fraction nonconforming in the out of control
  state. If d is the magnitude of the process
  shift, then n must satisfy

Therefore, n(L/d)2 p(1-p) In our example, p0.01
, d0.05-0.010.04
n(3/0.04)2 (0.01)(0.99)56
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Design of Fraction Nonconforming Chart contd

• Method 3If the in control value of the fraction
  nonconforming is small, another useful criterion
  is to choose n large enough so that the control
  chart will have a positive control limit. This
  ensures that we will have a mechan to force us
  to investigate one or more samples that contain
  an unusual small number of nonconforming items.
  Since we wish to have

This implies
For example, if p0.05
Thus, if n 172 units, the control chart will
have a positive lower control limit
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Variable Sample Size contd
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The probability of type II error for the
nonconforming control chart may be computed from
It displays the probability of incorrectly
accepting the hypothesis of statistical control
(ß error) against the process fraction
nonconforming
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Where nUCL represents the largest integer to
the nUCL, and nLCL represents the smallest
integer to the nLCL
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X Total nonconformities n inspection unit
observed number of nonconformities per unit
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With complex products such as automobiles
computers, or major appliances, we usually find
that many different types of nonconformities can
occur. Not all of these types of defects are
equally important. One possible demerit scheme is
defined as follows.
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Recall that
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For the u-chart, we may generate the OC curve
from ß XC Where nLCL denotes the smallest int
eger greater than or equal to nLCL, andnUCL
devotes the larghest integer less than or equal
to nUCL
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• When defect levels or in general, count rates, in
  a process become very lowsay, under 1000
  occurrences per millionthere will be very long
  periods of time between the occurrence of a
  nonconforming unit. Conventional c and u charts
  become ineffective as count rates are driven into
  the low parts per million (ppm) range.
• The time-between-events control chart has been
  very effective as a process-control procedure for
  processes with low defect levels. If the
  events of interest occur according to a Poisson
  distribution, the probability distribution of the
  time between events is the exponential
  distribution.
• Constructing a time-between-events control chart
  is essentially equivalent to control charting an
  exponentially distributed variable.

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