statistics for business and economics: bab 20
TRANSCRIPT
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Slides Prepared by
JOHN S. LOUCKSSt. Edwards University
2002 South-Western College Publishing/Thomson Learning
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UCL
CL
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'(
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!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'
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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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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'
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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
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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'
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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
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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'
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xChart Structure
;CL
LCL
Process MeanWhen in Control
Center Line
Time
x
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Control Limits or anxChart
Process Mean and Standard 7e&iation >non
;CL ? + $ x
LCL ? $ x
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8@ampleA Branite oc. Co'
Control Limits or anxChartA Process Mean
and Standard 7e&iation >non
The eight o bags o cement
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8@ampleA Branite oc. Co'
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' '
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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
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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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;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
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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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1:1:
8@ampleA Branite oc. Co'
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
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2020
Control Limits or RChartA Process Mean
and Standard 7e&iation ;n.non
x? 0'01* R? '$22* n?
;CL ? RD)? '$222'11) ? '491 LCL ? RD$? '$220 ? 0
8@ampleA Branite oc. Co'
_=
_
_
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2121
8@ampleA Branite oc. Co'
RChart
A B C D E FRChart for Granite Rock Co.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0 5 10 15 20Sa!"e #$%er
Sa!"eRan&e
R
'C'
(C'
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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)
8@ampleA Branite oc. Co'
=
==
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2$2$
8@ampleA Branite oc. Co'
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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2424
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.
Control Limits or a pChart
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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2929
8@ampleA Korest an.
Control Limits or a pChart
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'
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$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
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$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
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$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
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$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
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)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