Chapter 20 Statistical Methods for Quality Control

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Chapter 20Statistical Methods for Quality Control

• Statistical Process Control

• Acceptance Sampling

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Quality

• Quality is the totality of features and
  characteristics of a product or service that
  bears on its ability to satisfy given needs.

• Organizations recognize that they
  must strive for high levels of quality.

• They have increased the emphasis on
  methods for monitoring and maintaining
  quality.

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Quality Terminology
QA
refers to the entire system of
policies, procedures, and
guide- lines established by an organization to
achieve and maintain quality.
Quality Assurance
Quality Engineering
Its
objective is to include quality
in the design of products
and processes and to identify potential
quality problems prior to production.
QC
consists of making a series
of inspections and measure-
ments to determine whether quality standards
are being met.
Quality Control
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Statistical Process Control (SPC)

• Output of the production process is sampled and
  inspected.

• Using SPC methods, it can be determined whether
• variations in output are due to common causes or
  assignable causes.

The goal is decide whether the process can be
continued or should be adjusted to achieve a
desired quality level.
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Causes of Process Output Variation
Common Causes

• randomly occurring variations in materials,
• humidity, temperature, . . .
• variations the manufacturer cannot control
• process is in statistical control
• process does not need to be adjusted

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Causes of Process Output Variation
Assignable Causes

• non-random variations in output due to tools
• wearing out, operator error, incorrect
  machine
• settings, poor quality raw material, . . .
• variations the manufacturer can control
• process is out of control
• corrective action should be taken

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SPC Hypotheses

• SPC procedures are based on hypothesis-testing
  methodology.

Null Hypothesis
H0 is formulated in terms of the production
process being in control.
Alternative Hypothesis
Ha is formulated in terms of the production
process being out of control.
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Decisions and State of the Process

• Type I and Type II Errors

State of Production Process
H0 True In-Control
H0 False Out-of-Control
Decision
Correct Decision
Type II Error Allow out-of-control process to
continue
Accept H0 Continue Process
Correct Decision
Type I Error Adjust in-control process
Reject H0 Adjust Process
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Control Charts

• SPC uses graphical displays known as control
  charts to monitor a production process.

• Control charts provide a basis for deciding
  whether the variation in the output is due to
  common causes (in control) or assignable causes
  (out of control).

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Control Charts

• Two important lines on a control chart are the
  upper control limit (UCL) and lower control limit
  (LCL).

• These lines are chosen so that when the process
  is in control, there will be a high probability
  that the sample finding will be between the two
  lines.

• Values outside of the control limits provide
  strong evidence that the process is out of
  control.

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Variables Control Charts
R Chart
This chart is used to monitor the range of
the measurements in the sample.
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Attributes Control Charts
p Chart
This chart is used to monitor the proportion
defective in the sample..
np Chart
This chart is used to monitor the number of
defective items in the sample.
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x Chart Structure
Upper Control Limit
UCL
Center Line
Process Mean When in Control
LCL
Time
Lower Control Limit
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Control Limits for an x Chart

• Process Mean m and Standard Deviation s Known

where n sample size
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• Example Granite Rock Co.

When Granite Rocks packaging process is
in control, the weight of bags of cement filled
by the process is normally distributed with a
mean of 50 pounds and a standard deviation
of 1.5 pounds. What should be the control
limits for samples of 9 bags?
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????? 50, ? 1.5, n 9
UCL 50 3(.5) 51.5
LCL 50 - 3(.5) 48.5
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Control Limits for an x Chart

• Process Mean and Standard Deviation Unknown

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Factors for x Control Chart

• Factors Table (Partial)

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Control Limits for an R Chart

• Process Mean and Standard Deviation Unknown

__
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Factors for R Control Chart

• Factors Table (Partial)

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R Chart
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Control Limits for an R Chart Process Mean and
Standard Deviation Unknown

• Example Granite Rock Co.

Suppose Granite does not know the true
mean and standard deviation for its bag filling
process. It wants to develop x and R charts
based on twenty samples of 5 bags each.
The twenty samples, collected when the
process was in control, resulted in an
overall sample mean of 50.01 pounds and an
average range of .322 pounds.
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Control Limits for an R Chart Process Mean and
Standard Deviation Unknown
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Control Limits for an R Chart Process Mean and
Standard Deviation Unknown
UCL
LCL
Sample Number
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(No Transcript)
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UCL
Sample Mean
LCL
Sample Number
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Control Limits for a p Chart
assuming np gt 5 n(1-p) gt 5
Note If computed LCL is negative, set LCL 0
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Control Limits For a p Chart

• Example Granite Rock Co.

• Every check cashed or deposited at
• Norwest Bank must be encoded with
• the amount of the check before it can
• begin the Federal Reserve clearing
• process. The accuracy of the check
• encoding process is of utmost
• importance. If there is any discrepancy
• between the amount a check is made
• out for and the encoded amount, the check is
• defective.

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Control Limits For a p Chart

• Example Granite Rock Co.

• Twenty samples, each consisting of 250
• checks, were selected and examined
• when the encoding process was known
• to be operating correctly. The number
• of defective checks found in the 20
• samples are listed below.

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Control Limits For a p Chart

• Suppose Norwest does not know the proportion
  of defective checks, p, for the encoding
  process when it is in control.

• We will treat the data (20 samples) collected as
  one large sample and compute the average number
  of defective checks for all the data. That value
  can then be used to estimate p.

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Control Limits For a p Chart

• Estimated p 80/((20)(250)) 80/5000 .016

Note that the computed LCL is negative.
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p Chart for Norwest Bank
UCL
LCL
Sample Number
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Control Limits for an np Chart
assuming np gt 5 n(1-p) gt 5
Note If computed LCL is negative, set LCL 0
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Interpretation of Control Charts
The location and pattern of points in a control
chart enable us to determine, with a small
probability of error, whether a process is in
statistical control.
A primary indication that a process may be out
of control is a data point outside the control
limits.
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Interpretation of Control Charts
Certain patterns of points within the control
limits can be warning signals of quality
problems
a large number of points on one side of center
line
six or seven points in a row that indicate
either an increasing or decreasing trend
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Acceptance Sampling
Acceptance sampling is a statistical method
that enables us to base the accept-reject
decision on the inspection of a sample of items
from the lot.
The items of interest can be incoming shipments
of raw materials or purchased parts as well as
finished goods from final assembly.
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Acceptance Sampling
Acceptance sampling has advantages over 100
inspection including
usually less expensive
less product damage due to less handling
fewer inspectors required
provides only approach possible if destructive
testing must be used
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Acceptance Sampling Procedure
Lot received
Sample selected
Sampled items inspected for quality
Results compared with specified quality
characteristics
Quality is not satisfactory
Quality is satisfactory
Accept the lot
Reject the lot
Send to production or customer
Decide on disposition of the lot
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Acceptance Sampling
Acceptance sampling is based on
hypothesis-testing methodology.
The hypotheses are
H0 Good-quality lot
Ha Poor-quality lot
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The Outcomes of Acceptance Sampling

• Type I and Type II Errors

State of the Lot
H0 True Good-Quality Lot
H0 False Poor-Quality Lot
Decision
Correct Decision
Type II Error Accepting a Poor-quality lot
Accept H0 Accept the Lot
Correct Decision
Type I Error Rejecting a Good-quality lot
Reject H0 Reject the Lot
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Producer Risk Vs Consumer Risk

• The probability of a Type I error creates a risk
  for the producer of the lot and is known as the
  Producers Risk. For example a producers risk
  of .10 indicates a 10 chance that a good quality
  lot will be erroneously rejected.
• Type II error creates a risk for the consumer of
  the lot know as Consumers risk. For example a
  consumer risk of .05 means a 5 chance that a
  poor quality lot will be erroneously accepted and
  therefore used in production or shipped to the
  costumer.

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Probability of Accepting a Lot

• Binomial Probability Function for Acceptance
  Sampling

where n sample size p proportion of
defective items in lot x number of defective
items in sample f(x) probability of x
defective items in sample
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Example Acceptance Sampling
An inspector takes a sample of 20 items from a
lot. His policy is to accept a lot if no more
than 2 defective items are found in the
sample. Assuming that 5 percent of a lot is
defective, what is the probability that he
will accept a lot? Reject a lot?
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Example Acceptance Sampling
n 20, c 2, and p .05
P(Accept Lot) f(0) f(1) f(2)
P(Accept Lot) .3585 .3774 .1887 .9246
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Example Acceptance Sampling
n 20, c 2, and p .05
P(Reject Lot) 1 P(Accept Lot)
1 - .9246 .0754
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Selecting an Acceptance Sampling Plan
In formulating a plan, managers must specify
two values for the fraction defective in the
lot.
a The producers risk the probability that a
lot with p0 defectives will be rejected
b The consumers risk the probability that a
lot with p1 defectives will be accepted
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Selecting an Acceptance Sampling Plan
Then, the values of n and c are selected that
result in an acceptance sampling plan that comes
closest to meeting both the a and b
requirements specified.
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Operating Characteristic Curve
a
n 15, c 0 p0 .03, p1 .15 a .3667,
b .0874
Probability of Accepting the Lot
(1 - a)
p1
p0
b
0 5 10
15 20 25
Percent Defective in the Lot
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Multiple Sampling Plans

• A multiple sampling plan uses two or more stages
  of sampling.

• At each stage the decision possibilities are

• stop sampling and accept the lot,

• stop sampling and reject the lot, or

• continue sampling.

• Multiple sampling plans often result in a smaller
  total sample size than single-sample plans with
  the same Type I error and Type II error
  probabilities.

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A Two-Stage Acceptance Sampling Plan
First Stage
Inspect n1 items
Find x1 defective items in this sample
Yes
Accept the lot
x1 lt c1 ?
No
Yes
Reject the lot
x1 gt c2 ?
Second Stage
No
Inspect n2 additional items
Find x2 defective items in this sample
Yes
No
x1 x2 lt c3 ?
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End of Chapter 20