1 1 Slide © 2003 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edwards University.

1 1 Slide

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© 2003 Thomson/South-Western© 2003 Thomson/South-Western

Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University

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UCLUCL

CLCL

LCLLCL

Statistical Methods for Quality ControlStatistical Methods for Quality Control

Statistical Process ControlStatistical Process Control Acceptance SamplingAcceptance Sampling

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Quality TerminologyQuality Terminology

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

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Quality TerminologyQuality Terminology

Quality assuranceQuality assurance refers to the entire system refers to the entire system of policies, procedures, and guidelines of policies, procedures, and guidelines established by an organization to achieve and established by an organization to achieve and maintain quality.maintain quality.

The objective of The objective of quality engineeringquality engineering is to is to include quality in the design of products and include quality in the design of products and processes and to identify potential quality processes and to identify potential quality problems prior to production.problems prior to production.

Quality controlQuality control consists of making a series of consists of making a series of inspections and measurements to determine inspections and measurements to determine whether quality standards are being met.whether quality standards are being met.

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Statistical Process Control (SPC)Statistical Process Control (SPC)

The goal of SPC is to determine whether the The goal of SPC is to determine whether the process can be continued or whether it should process can be continued or whether it should be adjusted to achieve a desired quality level.be adjusted to achieve a desired quality level.

If the variation in the quality of the production If the variation in the quality of the production output is due to output is due to assignable causesassignable causes (operator (operator error, worn-out tooling, bad raw material, . . . ) error, worn-out tooling, bad raw material, . . . ) the process should be adjusted or corrected as the process should be adjusted or corrected as soon as possible.soon as possible.

If the variation in output is due to If the variation in output is due to common common causescauses (variation in materials, humidity, (variation in materials, humidity, temperature, . . . ) which the manager cannot temperature, . . . ) which the manager cannot control, the process does not need to be control, the process does not need to be adjusted.adjusted.

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SPC HypothesesSPC Hypotheses

SPC procedures are based on hypothesis-SPC procedures are based on hypothesis-testing methodology.testing methodology.

The The null hypothesisnull hypothesis HH00 is formulated in terms is formulated in terms of the production process being in control.of the production process being in control.

The The alternative hypothesisalternative hypothesis HHaa is formulated in is formulated in terms of the process being out of control.terms of the process being out of control.

As with other hypothesis-testing procedures, As with other hypothesis-testing procedures, both a both a Type I errorType I error (adjusting an in-control (adjusting an in-control process) and a process) and a Type II errorType II error (allowing an out- (allowing an out-of-control process to continue) are possible.of-control process to continue) are possible.

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Decisions and State of the ProcessDecisions and State of the Process

Type I and Type II ErrorsType I and Type II Errors

State of Production State of Production ProcessProcess

DecisionDecision

CorrectCorrectDecisionDecision

Type II ErrorType II ErrorAllow out-of-controlAllow out-of-controlprocess to continueprocess to continue

CorrectCorrect

DecisionDecision

Type I ErrorType I ErrorAdjust in-controlAdjust in-control

processprocess

AdjustAdjustProcessProcess

ContinueContinueProcessProcess

HH0 0 TrueTrueIn ControlIn Control

HHa a TrueTrue

Out of ControlOut of Control

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Control ChartsControl Charts

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

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

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Control ChartsControl Charts

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

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

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

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Types of Control ChartsTypes of Control Charts

An An xx chart chart is used if the quality of the output is is used if the quality of the output is measured in terms of a variable such as measured in terms of a variable such as length, weight, temperature, and so on.length, weight, temperature, and so on.

xx represents the mean value found in a sample represents the mean value found in a sample of the output.of the output.

An An RR chart chart is used to monitor the range of the is used to monitor the range of the measurements in the sample.measurements in the sample.

A A pp chart chart is used to monitor the proportion is used to monitor the proportion defective in the sample.defective in the sample.

An An npnp chart chart is used to monitor the number of is used to monitor the number of defective items in the sample.defective items in the sample.

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xx Chart Structure Chart Structure

UCLUCL

LCLLCL

Process MeanProcess MeanWhen in ControlWhen in Control

Center LineCenter Line

TimeTime

x

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Control Limits for an Control Limits for an xx Chart Chart

Process Mean and Standard Deviation KnownProcess Mean and Standard Deviation Known

UCL = 3 xUCL = 3 x

LCL = 3 xLCL = 3 x

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Example: Granite Rock Co.Example: Granite Rock Co.

Control Limits for an Control Limits for an xx Chart: Process Mean Chart: Process Mean

and Standard Deviation Knownand Standard Deviation Known

The weight of bags of cement filled by The weight of bags of cement filled by Granite’s packaging process is normally Granite’s packaging process is normally distributed with a mean of 50 pounds and a distributed with a mean of 50 pounds and a standard deviation of 1.5 pounds.standard deviation of 1.5 pounds.

What should be the control limits for What should be the control limits for samples of 9 bags?samples of 9 bags?

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Control Limits for an Control Limits for an xx Chart: Process Mean Chart: Process Mean

and Standard Deviation Knownand Standard Deviation Known

= 50, = 50, = 1.5, = 1.5, nn = 9 = 9

UCL = 50 + 3(.5) = 51.5UCL = 50 + 3(.5) = 51.5

LCL = 50 - 3(.5) = 48.5LCL = 50 - 3(.5) = 48.5

x n 15

9 05. . x n 15

9 05. .

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Control Limits for an Control Limits for an xx Chart Chart

Process Mean and Standard Deviation UnknownProcess Mean and Standard Deviation Unknown

where:where:

xx = overall sample mean = overall sample mean

RR = average range = average range

AA22 = a constant that depends on = a constant that depends on nn; taken ; taken fromfrom

“ “Factors for Control Charts” tableFactors for Control Charts” table

UCL = x A R 2UCL = x A R 2

==__

LCL = x A R 2LCL = x A R 2

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Factors for Factors for xx and and RR Control Charts Control Charts

Factors Table (Partial)Factors Table (Partial)

n d2 A 2 d3 D3 D4

5 2.326 0.577 0.864 0 2.1146 2.534 0.483 0.848 0 2.0047 2.704 0.419 0.833 0.076 1.9248 2.847 0.373 0.820 0.136 1.8649 2.970 0.337 0.808 0.184 1.81610 3.078 0.308 0.797 0.223 1.777. . . . . .. . . . . .

n d2 A 2 d3 D3 D4

5 2.326 0.577 0.864 0 2.1146 2.534 0.483 0.848 0 2.0047 2.704 0.419 0.833 0.076 1.9248 2.847 0.373 0.820 0.136 1.8649 2.970 0.337 0.808 0.184 1.81610 3.078 0.308 0.797 0.223 1.777. . . . . .. . . . . .

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UCL = UCL = RDRD44

LCL = LCL = RDRD33

where:where:

RR = average range = average range

DD33, , DD44 = constants that depend on = constants that depend on nn; found ; found in “Factors for in “Factors for Control Charts” Control Charts” table table

Control Limits for an Control Limits for an RR Chart Chart

__

__

__

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Factors for Factors for xx and and RR Control Charts Control Charts

Factors Table (Partial)Factors Table (Partial)

n d2 A 2 d3 D3 D4

5 2.326 0.577 0.864 0 2.1146 2.534 0.483 0.848 0 2.0047 2.704 0.419 0.833 0.076 1.9248 2.847 0.373 0.820 0.136 1.8649 2.970 0.337 0.808 0.184 1.81610 3.078 0.308 0.797 0.223 1.777. . . . . .. . . . . .

n d2 A 2 d3 D3 D4

5 2.326 0.577 0.864 0 2.1146 2.534 0.483 0.848 0 2.0047 2.704 0.419 0.833 0.076 1.9248 2.847 0.373 0.820 0.136 1.8649 2.970 0.337 0.808 0.184 1.81610 3.078 0.308 0.797 0.223 1.777. . . . . .. . . . . .

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Control Limits for Control Limits for xx and and RR Charts: Process Charts: Process MeanMean

and Standard Deviation Unknownand Standard Deviation Unknown

Suppose Granite does not know the true Suppose Granite does not know the true mean and standard deviation for its bag filling mean and standard deviation for its bag filling process. It wants to develop process. It wants to develop xx and and RR charts charts based on twenty samples of 5 bags each. based on twenty samples of 5 bags each.

The twenty samples resulted in an The twenty samples resulted in an overall sample mean of 50.01 pounds and an overall sample mean of 50.01 pounds and an average range of .322 pounds.average range of .322 pounds.

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Control Limits for Control Limits for RR Chart: Process Mean Chart: Process Mean


xx = 50.01, = 50.01, RR = .322, = .322, nn = 5 = 5

UCL = UCL = RDRD44 = .322(2.114) = .322(2.114) = .681= .681

LCL = LCL = RDRD33 = .322(0) = .322(0) = = 00


__==

__

__

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RR Chart Chart

A B C D E FR Chart for Granite Rock Co.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0 5 10 15 20Sample Number

Sam

ple

Ran

ge

R

LCL

UCL

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Control Limits for Control Limits for xx Chart: Process Mean Chart: Process Mean


xx = 50.01, = 50.01, RR = .322, = .322, nn = 5 = 5

UCL = UCL = xx + + AA22RR = 50.01 + .577(.322) = 50.01 + .577(.322) = 50.196= 50.196

LCL = LCL = x x - - AA22R R = 50.01 - .577(.322) = = 50.01 - .577(.322) = 49.82449.824


==

==

==

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xx Chart Chart

x Chart for Granite Rock Co.

49.7

49.8

49.9

50.0

50.1

50.2

50.3


Sa

mp

leM

ea

n

UCL

LCL

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Control Limits for a Control Limits for a pp Chart Chart

where:where:

assuming:assuming:

npnp >> 5 5

nn(1-(1-pp) ) >> 5 5

Note: If computed LCL is negative, set LCL = Note: If computed LCL is negative, set LCL = 00

UCL = p p 3UCL = p p 3

LCL = p p 3LCL = p p 3

pp p

n

( )1 pp p

n

( )1

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Example: Norwest BankExample: Norwest Bank

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

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TwTwenty samples, each consisting of 250 enty samples, each consisting of 250 checks, were selected and examined when the checks, were selected and examined when the encoding process was known to be operating encoding process was known to be operating correctly. The number of defective checks correctly. The number of defective checks found in the samples follow.found in the samples follow.

4 1 5 3 2 7 4 5 2 3

2 8 5 3 6 4 2 5 3 6

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

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

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Estimated Estimated pp = 80/((20)(250)) = 80/5000 = 80/((20)(250)) = 80/5000 = .016= .016

(1 ) .016(1 .016) .015744.007936

250 250p

p pn

(1 ) .016(1 .016) .015744

.007936250 250p

p pn

UCL = 3 .016 3(.007936) .039808pp UCL = 3 .016 3(.007936) .039808pp

LCL = 3 .016 3(.007936) -.007808 0pp LCL = 3 .016 3(.007936) -.007808 0pp

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pp Chart Chart

p Chart for Norwest Bank

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045


Sam

ple

Pro

po

rtio

n p

UCL

LCL

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Control Limits for an Control Limits for an npnp Chart Chart

assuming:assuming:

npnp >> 5 5

nn(1-(1-pp) ) >> 5 5

Note: If computed LCL is negative, set Note: If computed LCL is negative, set LCL = 0LCL = 0

UCL = np np p 3 1( )UCL = np np p 3 1( )

LCL = np np p 3 1( )LCL = np np p 3 1( )

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Interpretation of Control ChartsInterpretation of Control Charts

The location and pattern of points in a control The location and pattern of points in a control chart enable us to determine, with a small chart enable us to determine, with a small probability of error, whether a process is in probability of error, whether a process is in statistical control.statistical control.

A primary indication that a process may be out A primary indication that a process may be out of control is a of control is a data point outside the control data point outside the control limitslimits..

Certain patternsCertain patterns of points within the control of points within the control limits can be warning signals of quality limits can be warning signals of quality problems:problems:

• Large number of points on one side of Large number of points on one side of center line.center line.

• Six or seven points in a row that indicate Six or seven points in a row that indicate either an increasing or decreasing trend.either an increasing or decreasing trend.

• . . . and other patterns.. . . and other patterns.

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Acceptance SamplingAcceptance Sampling

Acceptance samplingAcceptance sampling is a statistical method is a statistical method that enables us to base the accept-reject that enables us to base the accept-reject decision on the inspection of a sample of items decision on the inspection of a sample of items from the lot.from the lot.

Acceptance sampling has advantages over Acceptance sampling has advantages over 100% inspection including: less expensive, 100% inspection including: less expensive, less product damage, fewer people less product damage, fewer people involved, . . . and more.involved, . . . and more.

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Acceptance Sampling ProcedureAcceptance Sampling Procedure

Lot receivedLot received

Sample selectedSample selected

Sampled itemsSampled itemsinspected for qualityinspected for quality

Results compared withResults compared withspecified quality characteristicsspecified quality characteristics

Accept the lotAccept the lot Reject the lotReject the lot

Send to productionSend to productionor customeror customer

Decide on dispositionDecide on dispositionof the lotof the lot

Quality is Quality is notnot satisfactorysatisfactory

Quality isQuality issatisfactorysatisfactory

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Acceptance SamplingAcceptance Sampling

Acceptance sampling is based on hypothesis-Acceptance sampling is based on hypothesis-testing methodology.testing methodology.

The hypothesis are:The hypothesis are:

HH00: Good-quality lot: Good-quality lot

HHaa: Poor-quality lot: Poor-quality lot

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The Outcomes of Acceptance SamplingThe Outcomes of Acceptance Sampling

Type I and Type II ErrorsType I and Type II Errors

State of the LotState of the Lot

DecisionDecision

CorrectCorrectDecisionDecision

Type II ErrorType II ErrorConsumer’s RiskConsumer’s Risk

CorrectCorrect

DecisionDecisionType I ErrorType I Error

Producer’s RiskProducer’s RiskRejectReject HH00

Reject the Lot Reject the Lot

AcceptAccept HH00

Accept the LotAccept the Lot

HH0 0 TrueTrueGood-Quality LotGood-Quality Lot

HHa a TrueTrue

Poor-Quality LotPoor-Quality Lot

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Binomial Probability Function for Acceptance Binomial Probability Function for Acceptance SamplingSampling

where:where:

nn = sample size = sample size

pp = proportion of defective items in lot = proportion of defective items in lot

xx = number of defective items in sample = number of defective items in sample

ff((xx) = probability of ) = probability of xx defective items in defective items in samplesample

Probability of Accepting a LotProbability of Accepting a Lot

f xn

x n xp px n x( )

!!( )!

( )( )

1f xn

x n xp px n x( )

!!( )!

( )( )

1

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Example: Acceptance Sampling Example: Acceptance Sampling

An inspector takes a sample of 20 items from a lot.An inspector takes a sample of 20 items from a lot.

Her policy is to accept a lot if no more than 2 defectiveHer policy is to accept a lot if no more than 2 defective

items are found in the sample.items are found in the sample.

Assuming that 5 percent of a lot is defective, what isAssuming that 5 percent of a lot is defective, what is

the probability that she will accept a lot? Reject a lot?the probability that she will accept a lot? Reject a lot?

n n = 20, = 20, cc = 2, and = 2, and pp = .05 = .05

PP(Accept Lot) = (Accept Lot) = ff(0) + (0) + ff(1) + (1) + ff(2)(2)

= .3585 + .3774 + .1887= .3585 + .3774 + .1887

= .9246= .9246

PP(Reject Lot) = 1 - .9246 (Reject Lot) = 1 - .9246

= .0754 = .0754

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Example: Acceptance SamplingExample: Acceptance Sampling

Using the Tables of Binomial ProbabilitiesUsing the Tables of Binomial Probabilities

pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5020 0 .3585 .1216 .0388 .0115 .0032 .0008 .0002 .0000 .0000 .0000

1 .3774 .2702 .1368 .0576 .0211 .0068 .0020 .0005 .0001 .00002 .1887 .2852 .2293 .1369 .0669 .0278 .0100 .0031 .0008 .00023 .0596 .1901 .2428 .2054 .1339 .0716 .0323 .0123 .0040 .00114 .0133 .0898 .1821 .2182 .1897 .1304 .0738 .0350 .0139 .00465 .0022 .0319 .1028 .1746 .2023 .1789 .1272 .0746 .0365 .01486 .0003 .0089 .0454 .1091 .1686 .1916 .1712 .1244 .0746 .03707 .0000 .0020 .0160 .0545 .1124 .1643 .1844 .1659 .1221 .07398 .0000 .0004 .0046 .0222 .0609 .1144 .1614 .1797 .1623 .12019 .0000 .0001 .0011 .0074 .0271 .0654 .1158 .1597 .1771 .1602

pn x .05 .10 .15 .20 .25 .30 .35 .40 .45 .5020 0 .3585 .1216 .0388 .0115 .0032 .0008 .0002 .0000 .0000 .0000

1 .3774 .2702 .1368 .0576 .0211 .0068 .0020 .0005 .0001 .00002 .1887 .2852 .2293 .1369 .0669 .0278 .0100 .0031 .0008 .00023 .0596 .1901 .2428 .2054 .1339 .0716 .0323 .0123 .0040 .00114 .0133 .0898 .1821 .2182 .1897 .1304 .0738 .0350 .0139 .00465 .0022 .0319 .1028 .1746 .2023 .1789 .1272 .0746 .0365 .01486 .0003 .0089 .0454 .1091 .1686 .1916 .1712 .1244 .0746 .03707 .0000 .0020 .0160 .0545 .1124 .1643 .1844 .1659 .1221 .07398 .0000 .0004 .0046 .0222 .0609 .1144 .1614 .1797 .1623 .12019 .0000 .0001 .0011 .0074 .0271 .0654 .1158 .1597 .1771 .1602

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Selecting an Acceptance Sampling PlanSelecting an Acceptance Sampling Plan

In formulating a plan, managers must specify In formulating a plan, managers must specify two values for the fraction defective in the lot.two values for the fraction defective in the lot.

• aa = the probability that a lot with = the probability that a lot with pp00

defectives will defectives will be rejected. be rejected.

• b b = the probability that a lot with = the probability that a lot with pp11

defectives will defectives will be accepted. be accepted. Then, the values of Then, the values of nn and and cc are selected that are selected that

result in an acceptance sampling plan that result in an acceptance sampling plan that comes closest to meeting both the comes closest to meeting both the aa and and bb requirements specified. requirements specified.

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Operating Characteristic CurveOperating Characteristic Curve

.10.10

.20.20

.30.30

.40.40

.50.50

.60.60

.70.70

.80.80

.90.90P

rob

ab

ilit

y o

f A

ccep

tin

g t

he L

ot

Pro

bab

ilit

y o

f A

ccep

tin

g t

he L

ot

0 5 10 15 20 25 0 5 10 15 20 25

1.001.00

Percent Defective in the Lot

pp00 pp11

(1 - (1 - ))

nn = 15, = 15, cc = 0 = 0

pp00 = .03, = .03, pp11 = .15 = .15

= .3667, = .3667, = .0874 = .0874

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Multiple Sampling PlansMultiple Sampling Plans

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

At each stage the At each stage the decision possibilitiesdecision possibilities are: are:

• stop sampling and accept the lot,stop sampling and accept the lot,

• stop sampling and reject the lot, orstop sampling and reject the lot, or

• continue sampling.continue sampling. Multiple sampling plans often result in a Multiple sampling plans often result in a

smaller total sample sizesmaller total sample size than single-sample than single-sample plans with the same Type I error and Type II plans with the same Type I error and Type II error probabilities.error probabilities.

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A Two-Stage Acceptance Sampling PlanA Two-Stage Acceptance Sampling Plan

Inspect Inspect nn11 items items

Find Find xx11 defective items in this sample defective items in this sample

xx11 << cc11 ? ?

xx11 >> cc22 ? ?

Inspect Inspect nn22 additional items additional items

AcceptAcceptthe lotthe lot

RejectRejectthe lotthe lot

xx1 1 + + xx22 << cc33 ? ?

Find Find xx22 defective items in this sample defective items in this sample

YesYes

YesYesNoNo

NoNo

NoNoYesYes

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End of Chapter End of Chapter

1 1 Slide © 2003 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edwards University.

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