Statistics for Business and Economics: bab 20

7/24/2019 Statistics for Business and Economics: bab 20

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Slides Prepared by

JOHN S. LOUCKSSt. Edwards University

2002 South-Western College Publishing/Thomson Learning


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UCL

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LCL

Chapter 20Statistical Methods or !ualit" Control

Statistical Process Control

#cceptance Sampling


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$$

!ualit" Terminolog"

!ualit" is %the totalit" o eatures andcharacteristics o a product or ser&ice thatbears on its abilit" to satis" gi&en needs'(


4/43

)

!ualit" Terminolog"

!ualit" assurance reers to the entire s"stem

o policies* procedures* and guidelinesestablished b" an organi+ation to achie&e andmaintain ,ualit"'

The obecti&e o ,ualit" engineering is to

include ,ualit" in the design o products andprocesses and to identi" potential ,ualit"problems prior to production'

!ualit" control consists o ma.ing a series o

inspections and measurements to determinehether ,ualit" standards are being met'


5/43

Statistical Process Control SPC

The goal o SPC is to determine hether the

process can be continued or hether it shouldbe adusted to achie&e a desired ,ualit" le&el'

3 the &ariation in the ,ualit" o the productionoutput is due to assignable causes operator

error* orn-out tooling* bad ra material* ' ' ' the process should be adusted or corrected assoon as possible'

3 the &ariation in output is due to common

causes &ariation in materials* humidit"*temperature* ' ' ' hich the manager cannotcontrol* the process does not need to beadusted'


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44

SPC 5"potheses

SPC procedures are based on h"pothesis-

testing methodolog"'

The null h"pothesis H0is ormulated in terms

o the production process being in control'

The alternati&e h"pothesis Hais ormulated in

terms o the process being out o control'

#s ith other h"pothesis-testing procedures*both a T"pe 3 error adusting an in-controlprocess and a T"pe 33 error alloing an out-o-

control process to continue are possible'


7/43 6

7ecisions and State o the Process

T"pe 3 and T"pe 33 8rrors

State o Production Process

7ecision

Correct7ecision

T"pe 33 8rror#llo out-o-controlprocess to continue

Correct

7ecision

T"pe 3 8rror#dust in-control

process

AdjustProcess

ContinueProcess

H0 True

In Control

Ha True

Out of Control


8/43 99

Control Charts

SPC uses graphical displa"s .non as control

charts to monitor a production process' Control charts pro&ide a basis or deciding

hether the &ariation in the output is due tocommon causes in control or assignable

causes out o control'


9/43 ::

Control Charts

To important lines on a control chart are the

upper control limit ;CLand loer controllimit LCL.

These lines are chosen so that hen theprocess is in control* there ill be a high

probabilit" that the sample


10/43 1010

T"pes o Control Charts

#nxchart is used i the ,ualit" o the output is

measured in terms o a &ariable such aslength* eight* temperature* and so on'

xrepresents the mean &alue ound in a sampleo the output'

#n Rchart is used to monitor the range o themeasurements in the sample'

#pchart is used to monitor the proportiondeecti&e in the sample'

#n npchart is used to monitor the number odeecti&e items in the sample'


11/43 11

xChart Structure

;CL

LCL

Process MeanWhen in Control

Center Line

Time

x


12/43 1212

Control Limits or anxChart

Process Mean and Standard 7e&iation >non

;CL ? + $ x

LCL ? $ x


13/43 1$1$

8@ampleA Branite oc. Co'

Control Limits or anxChartA Process Mean

and Standard 7e&iation >non

The eight o bags o cement


14/43 1)


Control Limits or anxChartA Process Mean

and Standard 7e&iation >non

? 0* ? 1'* n? :

;CL ? 0 F $' ? 1'

LCL ? 0 - $' ? )9'

x n= = =1

:0' '


15/43 1

Control Limits or anxChart

Process Mean and Standard 7e&iation

;n.non

hereA

x? o&erall sample mean

R? a&erage range

A2? a constant that depends on nG ta.enrom

%Hactors or Control Charts( table

;CL ? x A R+ 2

?I

LCL ? x A R 2


16/43 1414

Hactors orxand RControl Charts

Hactors Table Partial

n d2 A 2 d3 D 3 D 4

2'$24 0'66 0'94) 0 2'11)

4 2'$) 0')9$ 0'9)9 0 2'00)

6 2'60) 0')1: 0'9$$ 0'064 1':2)9 2'9)6 0'$6$ 0'920 0'1$4 1'94)

: 2':60 0'$$6 0'909 0'19) 1'914

10 $'069 0'$09 0'6:6 0'22$ 1'666

. . . . . .

. . . . . .


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16

;CL ? RD)

LCL ? RD$

hereA R? a&erage range

D$* D)? constants that depend on nG

ound in %Hactors or Control

Charts( table

Control Limits or an RChart

I

II


18/43

1919

Hactors orxand RControl Charts

Hactors Table Partial

n d2 A 2 d3 D 3 D 4

2'$24 0'66 0'94) 0 2'11)

4 2'$) 0')9$ 0'9)9 0 2'00)

6 2'60) 0')1: 0'9$$ 0'064 1':2)9 2'9)6 0'$6$ 0'920 0'1$4 1'94)

: 2':60 0'$$6 0'909 0'19) 1'914

10 $'069 0'$09 0'6:6 0'22$ 1'666

. . . . . .

. . . . . .


19/43

1:1:


Control Limits or x and RChartsA Process

Meanand Standard 7e&iation ;n.non

Suppose Branite does not .no the truemean and standard de&iation or its bag


20/43

2020

Control Limits or RChartA Process Mean

and Standard 7e&iation ;n.non

x? 0'01* R? '$22* n?

;CL ? RD)? '$222'11) ? '491 LCL ? RD$? '$220 ? 0


_=

_

_


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2121


RChart

A B C D E FRChart 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 20Sa!"e #$%er

Sa!"eRan&e

R

'C'

(C'


22/43

2222

Control Limits or x ChartA Process Mean

and Standard 7e&iation ;n.non

x? 0'01* R? '$22* n?

;CL ?xFA2R? 0'01 F '66'$22 ?0'1:4

LCL ?x -A2R ? 0'01 - '66'$22 ?

):'92)


=

==


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2$2$


x Chart

xChart for Granite Rock Co.

4).7

4).8

4).)

50.0

50.1

50.2

50.3

0 5 10 15 20Sa!"e #$%er

Sa!"e

*ean

(C'

'C'


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2)2)

Control Limits or a pChart

hereA

assumingA

npJ

n1-p J

KoteA 3 computed LCL is negati&e* set LCL ?0

;CL ? p p

+ $

LCL ? p p $

p p pn

= 1


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8@ampleA Korest an.

8&er" chec. cashed or deposited at

Korest an. must be encoded ith theamount o the chec. beore it can begin theHederal eser&e clearing process' Theaccurac" o the chec. encoding process is o

upmost importance' 3 there is an"discrepanc" beteen the amount a chec. ismade out or and the encoded amount* thechec. is deecti&e'


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8@ampleA Korest an.

Tent" samples* each consisting o 20

chec.s* ere selected and e@amined hen theencoding process as .non to be operatingcorrectl"' The number o deecti&e chec.sound in the samples ollo'

) 1 0 $ 2 6 ) 0 2 $

2 9 0 $ 4 ) 2 0 $ 4


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8@ampleA Korest an.


Suppose Korest does not .no theproportion o deecti&e chec.s* p* or theencoding process hen it is in control'

We ill treat the data 20 samples

collected as one large sample and computethe a&erage number o deecti&e chec.s or allthe data' That &alue can then be used toestimate p'


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8@ampleA Korest an.


8stimated p? 90/2020 ? 90/000 ? '014

1 '0141 '014 '016))'006:$4

20 20p

p p

n

= = = =

;CL ? $ '014 $'006:$4 '0$:909pp + = + =

LCL ? $ '014 $'006:$4 -'006909 0pp = = =


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2:2:

8@ampleA Korest an.

pChart

p Chart for #or+e,t Bank

0.000

0.0050.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0 5 10 15 20

Sa!"e #$%er

S

a!"e-ro!ortionp

(C'

'C'


30/43

$0$0

Control Limits or an npChart

assumingA

npJ n1-p J

KoteA 3 computed LCL is negati&e* set LCL? 0

;CL ? np np p+ $ 1

LCL ? np np p $ 1


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$1$1

3nterpretation o Control Charts

The location and pattern o points in a control

chart enable us to determine* ith a smallprobabilit" o error* hether a process is instatistical control'

# primar" indication that a process ma" be out

o control is a data point outside the controllimits'

Certain patterns o points ithin the controllimits can be arning signals o ,ualit"

problemsA Large number o points on one side o

center line'

Si@ or se&en points in a ro that indicate

either an increasing or decreasing trend'


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$2$2

#cceptance Sampling

#cceptance sampling is a statistical method

that enables us to base the accept-reectdecision on the inspection o a sample o itemsrom the lot'

#cceptance sampling has ad&antages o&er

100 inspection includingA less e@pensi&e*less product damage* eer peoplein&ol&ed* ' ' ' and more'


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#cceptance Sampling Procedure

Lot recei&ed

Sample selected

Sampled itemsinspected or ,ualit"

esults compared ithspeci


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$)$)

#cceptance Sampling

#cceptance sampling is based on h"pothesis-

testing methodolog"' The h"pothesis areA

H0A Bood-,ualit" lot

HaA Poor-,ualit" lot


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The Nutcomes o #cceptance Sampling

T"pe 3 and T"pe 33 8rrors

State of the Lot

Decision

Correct7ecision

T"pe 33 8rrorConsumerDs is.

Correct

7ecision

T"pe 3 8rrorProducerDs is.

RejectH0eect the Lot

AccetH0#ccept the Lot

H0 True

Bood-!ualit" Lot

Ha True

Poor-!ualit" Lot


36/43

$4$4

inomial Probabilit" Hunction or #cceptance

Sampling

hereA

n? sample si+e

p? proportion o deecti&e items in lot

x? number o deecti&e items in sample

fx ? probabilit" oxdeecti&e items insample

Probabilit" o #ccepting a Lot

f x n

x n xp px n x

O

O O

=

1


37/43

$6$6

8@ampleA #cceptance Sampling

#n inspector ta.es a sample o 20 items rom a

lot'5er polic" is to accept a lot i no more than 2

deecti&e

items are ound in the sample'

#ssuming that percent o a lot is deecti&e*hat is

the probabilit" that she ill accept a lotE eecta lotE

n ? 20* c? 2* andp? '0

P#ccept Lot ? f0 F f1 F f2

? '$9 F '$66) F '1996

? ':2)4


38/43

$9$9

8@ampleA #cceptance Sampling

;sing the Tables o inomial Probabilities

p

n x .05 .10 .15 .20 .25 .30 .35 .40 .45 .50

20 0 .3585 .1216 .0388 .0115 .0032 .0008 .0002 .0000 .0000 .0000

1 .3774 .2702 .1368 .0576 .0211 .0068 .0020 .0005 .0001 .0000

2 .1887 .2852 .2293 .1369 .0669 .0278 .0100 .0031 .0008 .0002

3 .0596 .1901 .2428 .2054 .1339 .0716 .0323 .0123 .0040 .0011

4 .0133 .0898 .1821 .2182 .1897 .1304 .0738 .0350 .0139 .0046

5 .0022 .0319 .1028 .1746 .2023 .1789 .1272 .0746 .0365 .0148

6 .0003 .0089 .0454 .1091 .1686 .1916 .1712 .1244 .0746 .0370

7 .0000 .0020 .0160 .0545 .1124 .1643 .1844 .1659 .1221 .0739

8 .0000 .0004 .0046 .0222 .0609 .1144 .1614 .1797 .1623 .1201

9 .0000 .0001 .0011 .0074 .0271 .0654 .1158 .1597 .1771 .1602


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$:$:

Selecting an #cceptance Sampling Plan

3n ormulating a plan* managers must speci"

to &alues or the raction deecti&e in the lot' a? the probabilit" that a lot ithp0

deecti&es ill be reected'

b ? the probabilit" that a lot ithp1

deecti&es ill be accepted'

Then* the &alues o nand care selected thatresult in an acceptance sampling plan thatcomes closest to meeting both the aand b

re,uirements speci


40/43

)0)0

Nperating Characteristic Cur&e

'10'10'20'20

'$0'$0

')0')0

'0'0

'40'40

'60'60

'90'90

':0':0

Pro!

a!ilit"

ofAccetin#

the

Lot

Pro!

a!ilit"

ofAccetin#

th

e

Lot

0 10 1 20 2 0 10 1 20 2

1'001'00

Percent 7eecti&e in the Lot

p0 p1

$% & '

n? 1* c? 0

p0? '0$* p1? '1

? '$446* ? '096)

l i l S li l


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)1

Multiple Sampling Plans

# multiple sampling plan uses to or more

stages o sampling' #t each stage the decision possibilities areA

stop sampling and accept the lot*

stop sampling and reect the lot* or

continue sampling'

Multiple sampling plans oten result in asmaller total sample si+e than single-sampleplans ith the same T"pe 3 error and T"pe 33

error probabilities'

# T St # t S li Pl


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)2)2

# To-Stage #cceptance Sampling Plan

3nspect n1items

Hindx1deecti&e items in this sample

x1 c1E

x1J c2E

3nspect n2additional items

#cceptthe lot

eectthe lot

x1 Fx2 c$E

Hindx2deecti&e items in this sample

Qes

QesKo

Ko

KoQes

8 d Ch t 20


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8nd o Chapter 20

Statistics for Business and Economics: bab 20

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