1 1 slide 2003 south-western/thomson learning slides prepared by john s. loucks st. edwards...

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© 2003 South-Western/Thomson Learning™© 2003 South-Western/Thomson Learning™

Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University

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UCLUCL

CLCL

LCLLCL

Chapter 19Chapter 19Statistical Methods for Quality ControlStatistical Methods for Quality Control

Statistical Process ControlStatistical Process Control Acceptance SamplingAcceptance Sampling

3 Slide


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.”

4 Slide


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.

5 Slide


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.

6 Slide


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.

7 Slide


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

DecisionDecisionCorrectCorrectDecisionDecision

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

CorrectCorrectDecisionDecision

Type I ErrorType I ErrorAdjust in-controlAdjust in-control

processprocessAdjustAdjust

ProcessProcess

ContinueContinueProcessProcess

HH0 0 TrueTrueIn ControlIn Control

HHa a TrueTrueOut of ControlOut of Control

8 Slide


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).

9 Slide


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.

10 Slide


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.

11 Slide


xx Chart Structure Chart Structure

UCLUCL

LCLLCL

Process MeanProcess MeanWhen in ControlWhen in Control

Center LineCenter Line

TimeTime

x

12 Slide


Control Limits for an Control Limits for an xx Chart Chart

Process Mean and Standard Deviation KnownProcess Mean and Standard Deviation Known

UCL = 3 xLCL = 3 x

13 Slide


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

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

14 Slide



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

15 Slide


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 from; taken from “ “Factors for Control Charts” tableFactors for Control Charts” table

Control Limits for an Control Limits for an xx Chart Chart

2UCL = x A R

==__

2LCL = x A R

16 Slide


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

Factors Table (Partial)Factors Table (Partial)

n d2 A 2 d3 D3 D45 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. . . . . .. . . . . .

17 Slide


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

__

____

18 Slide


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

Factors Table (Partial)Factors Table (Partial)

n d2 A 2 d3 D3 D45 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. . . . . .. . . . . .

19 Slide



Control Limits for Control Limits for xx and and RR Charts: Process Charts: Process MeanMeanand Standard Deviation Unknownand Standard Deviation UnknownSuppose Granite does not know the true mean Suppose Granite does not know the true mean and standard deviation for its bag filling and standard deviation for its bag filling process. In this case, the process mean and process. In this case, the process mean and standard deviation must be estimated by using standard deviation must be estimated by using samples that are selected from the process samples that are selected from the process when it is assumed to be operating in control. when it is assumed to be operating in control. Granite will develop Granite will develop xx and and RR charts based on charts based on twenty samples of 5 bags each. twenty samples of 5 bags each.

20 Slide


““Data” WorksheetData” Worksheet

Using Excel to ConstructUsing Excel to Constructan an RR Chart and an Chart and an xx Chart Chart

Note: Rows 12-21 are not shown.Note: Rows 12-21 are not shown.

A B C D E F1 Sample Observ. 1 Observ. 2 Observ. 3 Observ. 4 Observ. 52 1 50.22 49.91 50.05 50.12 49.793 2 50.13 50.09 50.17 50.23 50.044 3 50.30 50.11 49.91 49.89 50.095 4 49.91 50.20 50.16 50.05 50.246 5 50.11 49.89 49.94 50.17 49.887 6 49.78 50.07 49.78 49.79 50.048 7 49.84 49.92 50.11 50.04 49.819 8 50.08 49.88 49.90 49.97 49.78

10 9 49.89 50.10 50.14 49.99 49.9011 10 50.11 50.16 50.20 50.06 50.06

21 Slide


Formula Worksheet (for Formula Worksheet (for RR control limits) control limits)


Note: Rows 7-19 are hidden.Note: Rows 7-19 are hidden.

A B C D E1 Samp. R LCL Mean UCL2 1 =MAX(Data!B2:F2)-MIN(Data!B2:F2) =$D$24*$B$22 =$B$22 =$D$25*$B$223 2 =MAX(Data!B3:F3)-MIN(Data!B3:F3) =$D$24*$B$22 =$B$22 =$D$25*$B$224 3 =MAX(Data!B4:F4)-MIN(Data!B4:F4) =$D$24*$B$22 =$B$22 =$D$25*$B$225 4 =MAX(Data!B5:F5)-MIN(Data!B5:F5) =$D$24*$B$22 =$B$22 =$D$25*$B$226 5 =MAX(Data!B6:F6)-MIN(Data!B6:F6) =$D$24*$B$22 =$B$22 =$D$25*$B$2220 19 =MAX(Data!B20:F20)-MIN(Data!B20:F20) =$D$24*$B$22 =$B$22 =$D$25*$B$2221 20 =MAX(Data!B21:F21)-MIN(Data!B21:F21) =$D$24*$B$22 =$B$22 =$D$25*$B$2222 Rbar =AVERAGE($B$2:$B$21)2324 D3 025 D4 2.114

22 Slide


Value Worksheet (for Value Worksheet (for RR control limits) control limits)



A B C D E1 Sample R LCL Mean UCL2 1 0.43 0 0.32 0.68073 2 0.19 0 0.32 0.68074 3 0.41 0 0.32 0.68075 4 0.33 0 0.32 0.68076 5 0.29 0 0.32 0.680720 19 0.49 0 0.32 0.680721 20 0.28 0 0.32 0.680722 Rbar 0.3222324 D3 025 D4 2.114

23 Slide


Constructing an Constructing an RR Chart ChartStep 1Step 1 Select cells A29:E48 Select cells A29:E48Step 2Step 2 Select the Select the Chart WizardChart Wizard button on main button on main toolbartoolbarStep 3Step 3 When the Chart Wizard-Step 1 0f 4- When the Chart Wizard-Step 1 0f 4-Chart TypeChart Type dialog box appears:dialog box appears: Choose Choose XY (Scatter)XY (Scatter) in the in the Chart TypeChart Type list list Choose Choose Scatter with data points connected Scatter with data points connected by lines by lines from the Chart subtype display from the Chart subtype display Select Select Next >Next >Step 4Step 4 When the Chart Wizard-Step 2 of 4- When the Chart Wizard-Step 2 of 4- ChartChart Source DataSource Data dialog box appears: dialog box appears: Select Select Next >Next >


24 Slide


Constructing an Constructing an RR Chart ChartStep 5Step 5 When the When the Chart Wizard-Step 3 of 4-Chart Wizard-Step 3 of 4-Chart Chart Options Options dialog box dialog box appears:appears:

Type Type R Chart for Granite Rock R Chart for Granite Rock Co.Co. in the in the Chart titleChart title box box

Type Type Sample NumberSample Number in the in the Value Value (X)(X) axis boxaxis box

Type Type Sample Range RSample Range R in the in the Value (Y)Value (Y) axis boxaxis box

… … continuedcontinued


25 Slide


Constructing an Constructing an RR Chart Chart Select the Select the Legend Legend tab and thentab and thenRemove the check in the Remove the check in the Show LegendShow Legend box box Select the Select the Gridlines Gridlines tab and thentab and thenRemove the check in the Remove the check in the Major GridlinesMajor Gridlines box box Select Select Next >Next >Step 6Step 6 When the When the Chart Wizard- Step 4 of 4 – Chart Wizard- Step 4 of 4 – ChartChart LocationLocation dialog box appears: dialog box appears:Specify a location for the chartSpecify a location for the chartSelect Select FinishFinish


26 Slide


Control Limits for Control Limits for RR Chart: Process Mean Chart: Process Meanand Standard Deviation Unknownand Standard Deviation Unknown

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


__==

__

__

27 Slide


RR Chart ChartA B C D E FR Chart for Granite Rock Co.

0.000.100.200.300.400.500.600.700.80

0 5 10 15 20Sample Number

Sam

ple

Rang

e R

LCL

UCL


28 Slide


Formula Worksheet (for Formula Worksheet (for xx control limits) control limits)



A B C D E28 Sample x bar LCL Mean UCL29 1 =AVERAGE(Data!B2:F2) =$B$49-$D$51*$B$22 =$D$51 =$B$49+$D$51*$B$2230 2 =AVERAGE(Data!B3:F3) =$B$49-$D$51*$B$22 =$D$51 =$B$49+$D$51*$B$2231 3 =AVERAGE(Data!B4:F4) =$B$49-$D$51*$B$22 =$D$51 =$B$49+$D$51*$B$2232 4 =AVERAGE(Data!B5:F5) =$B$49-$D$51*$B$22 =$D$51 =$B$49+$D$51*$B$2233 5 =AVERAGE(Data!B6:F6) =$B$49-$D$51*$B$22 =$D$51 =$B$49+$D$51*$B$2247 19 =AVERAGE(Data!B20:F20) =$B$49-$D$51*$B$22 =$D$51 =$B$49+$D$51*$B$2248 20 =AVERAGE(Data!B21:F21) =$B$49-$D$51*$B$22 =$D$51 =$B$49+$D$51*$B$2249 Mean =AVERAGE(B29:B48)5051 A2 0.57752

29 Slide


Value Worksheet (for Value Worksheet (for xx control limits) control limits)


A B C D E28 Sample x bar LCL Mean UCL29 1 50.018 49.824 50.01 50.19530 2 50.132 49.824 50.01 50.19531 3 50.060 49.824 50.01 50.19532 4 50.112 49.824 50.01 50.19533 5 49.998 49.824 50.01 50.19547 19 49.972 49.824 50.01 50.19548 20 49.966 49.824 50.01 50.19549 Mean 50.0105051 A2 0.577


30 Slide


Control Limits for Control Limits for xx Chart: Process Mean Chart: Process Meanand Standard Deviation Unknownand Standard Deviation Unknown

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


==

==

==

31 Slide


xx Chart Chart

x Chart for Granite Rock Co.

49.7

49.8

49.9

50.0

50.1

50.2

50.3


Sam

ple

Mea

n

UCL

LCL


32 Slide


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 3LCL = p p 3

pp p

n

( )1

33 Slide


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.

34 Slide



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 32 8 5 3 6 4 2 5 3 6

35 Slide



Control Limits for a Control Limits for a pp Chart ChartSuppose 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..

36 Slide



Control Limits for a Control Limits for a pp Chart Chart

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

(1 ) .016(1 .016) .015744 .007936250 250pp p

n

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

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

37 Slide


Using Excel to Construct a Using Excel to Construct a pp Chart Chart

Formula WorksheetFormula WorksheetA B C D E F

1 SampleNumber Defect.

Proportion Defective LCL CL UCL

2 1 4 =B2/250 =MAX(0,$C$22-3*$C$23) =$C$22 =$C$22+3*$C$233 2 1 =B3/250 =MAX(0,$C$22-3*$C$23) =$C$22 =$C$22+3*$C$234 3 5 =B4/250 =MAX(0,$C$22-3*$C$23) =$C$22 =$C$22+3*$C$235 4 3 =B5/250 =MAX(0,$C$22-3*$C$23) =$C$22 =$C$22+3*$C$2320 19 3 =B20/250 =MAX(0,$C$22-3*$C$23) =$C$22 =$C$22+3*$C$2321 20 6 =B21/250 =MAX(0,$C$22-3*$C$23) =$C$22 =$C$22+3*$C$2322 =AVERAGE(C2:C21)23 =SQRT(C22*(1-C22)/250)

Estimated p Estim. Std. Error


38 Slide


Using Excel to Construct a Using Excel to Construct a pp Chart Chart

Value WorksheetValue Worksheet


A B C D E F

1 SampleNumber Defect.

Proportion Defective LCL CL UCL

2 1 4 0.016 0 0.01600 0.0317433 2 1 0.004 0 0.01600 0.0317434 3 5 0.020 0 0.01600 0.0317435 4 3 0.012 0 0.01600 0.03174320 19 3 0.012 0 0.01600 0.03174321 20 6 0.024 0 0.01600 0.03174322 0.0160023 0.00794

Estimated p Estim. Std. Error

39 Slide



pp Chart Chartp 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

Prop

ortio

n p

UCL

LCL

40 Slide


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( )LCL = np np p 3 1( )

41 Slide


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.

42 Slide


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.

43 Slide


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

44 Slide


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

45 Slide


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


Type II ErrorType II ErrorConsumer’s RiskConsumer’s Risk


Type I ErrorType I ErrorProducer’s RiskProducer’s Risk

RejectReject HH00

Reject the Lot Reject the Lot

AcceptAccept HH00

Accept the LotAccept the Lot

HH0 0 TrueTrueGood-Quality LotGood-Quality Lot

HHa a TrueTruePoor-Quality LotPoor-Quality Lot

46 Slide


Binomial Probability Function for Acceptance Binomial Probability Function for Acceptance SamplingSampling

where:where:nn = sample size = sample sizepp = proportion of defective items in lot = proportion of defective items in lotxx = 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 x nx n x

p px n x( ) !!( )! ( )( )

1

47 Slide


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 defectiveitems 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 isthe 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

48 Slide


Example: Acceptance SamplingExample: Acceptance Sampling

Using the Tables of Binomial ProbabilitiesUsing the Tables of Binomial Probabilitiesp

n 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

49 Slide


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.

50 Slide


Operating Characteristic CurveOperating Characteristic Curve

.10

.20

.30

.40

.50

.60

.70

.80

.90Pr

obab

ility

of A

ccep

ting

the

Lot

0 5 10 15 20 25

1.00

Percent Defective in the Lot

pp00 pp11

(1 - (1 - ))

nn = 15, = 15, cc = 0 = 0pp00 = .03, = .03, pp11 = .15 = .15 = .3667, = .3667, = .0874 = .0874

51 Slide


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.

52 Slide


A Two-Stage Acceptance Sampling PlanA Two-Stage Acceptance Sampling PlanInspect 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

53 Slide


End of Chapter 19End of Chapter 19

1 1 slide 2003 south-western/thomson learning slides prepared by john s. loucks st. edwards...

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