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Ch3 Inference About Process Quality. Sampling from a Normal distribution Sampling from a Bernoulli distribution Sampling from a Poisson distribution Estimation of process parameter 5. Hypothesis testing. ( a ) Point estimator ( b ) Interval estimation ( confidence interval ). - PowerPoint PPT Presentation
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1
Ch3 Inference About Process Quality
1. Sampling from a Normal distribution
2. Sampling from a Bernoulli distribution
3. Sampling from a Poisson distribution
4. Estimation of process parameter
5. Hypothesis testing
( a ) Point estimator( b ) Interval estimation ( confidence interval )
2
假設 ,則
nxx ,,1 ~ ),( 2N ~ ),(2
nN
nx
2
2
1
n
i ni xx ~ 21n
( with ,當 時, )kt 1,0 2
k
k k )1,0(Ntk
1. Sampling from a Normal distribution
, 其中
~221 nxxy
nS
x ~
1nt
0 2n
n
iin xx
nS
1
22 )(1
1
3
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,/
/F , vV
uUvu 其中 U and V indep. ~
and
2u
2v
e.g. ~
其中 is the sample var. of i.i.d.
22
22
21
21
/
/
S
S ,1,1F21 nn
21S ,,, 111 nxx ),( 2
11 N
is the sample var. of i.i.d. 22S ,,, 221 nxx ),( 2
22 N
4
,)1()()(][
0
an
k
knk ppk
nanxPaxP
pxE )(
n
ppxVar
)1()(
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假設 i.i.d. Bernoulli with success prob.= p
nxx ,,1
令 ~ B ( n , p )nxxx 1
a discrete r.v. with range space
n
i ixn
x1
1
}1,1
,2
,1
,0{n
n
nn
2. Sampling from a Bernoulli distribution
5
假設 i.i.d.
nxx ,,1 )(P ~ nxxx 1 )( nP
a discrete r.v. with taking values },2
,1
,0{ nn
n
i ixn
x1
1
,!
1)()(
][
0
an
k
kn nek
anxPaxP
,)( xEn
xVar
)(
3. Sampling from a Poisson distribution
6
令 indep. ,1
m
i ii xaLix )( iP
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( e.g. A unit of product can have m different types of defect, each modeled with a Poisson distribution with parameter )i
此稱為 demerit procedure, 若不全為 1, 則 L 一般未必為 Poisson 分佈。
ia
7
( a) Point estimation: Important properties of an estimation ( 1 ) Unbiased ( 2 ) Minimum variance
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4. Estimation of process parameter
In general, and are unbiased estimators of thepopulation mean and variance, respectively.但 S 則一般並非 population standard deviation 的unbiased estimator.
X2S
e.g. Poisson , Binomial Xˆ Xˆ p
8
( b) Interval estimation:
1)( ULP
[L,U] 稱為 的 two sided confidence interval. )%1(100
1)( UP
稱為 的 one sided confidence interval. )%1(100 ],( U
9
nzx
n
zx or
Lower C.I. Upper C.I.
nZx
nZx
22
Two sided C.I.
2
)1(,2
1
22
2
)1(,2
2 )1()1(
nn
SnSn
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e.g. nxx ,,1 i.i.d. ),( 2N
當 variance unknown, 則以 取代 , S 取代 。
1,2
nt
2
Z
)%1(100 two-sided C.I. On the variance
10
1. C.I. on the difference in two means( a ) Variance known
( b ) Variance unknown
2. C.I. on the ratio of the variance of two Normal distribution
1,1,2
22
21
22
21
1,1,2
122
21
1212
nnnnF
S
SF
S
S
2
22
1
21
2
21212
22
1
21
2
21 nnzxx
nnzxx
211,
2
21
2121
1,2
21
11
11
21
21
nnStxx
nnStxx
pnn
pnn
11
3. C.I. on Binomial parameter
( c ) If n is large, p is small, then use Poisson.
( b ) If n is small, then use Binomial distribution.
C.I. on the difference of two binomial parameter and .1p 2p
n
ppzpp
n
ppzp
)ˆ1(ˆˆ
)ˆ1(ˆˆ
22
2
22
1
11
2
21212
22
1
11
2
21
)ˆ1(ˆ)ˆ1(ˆˆˆ
)ˆ1(ˆ)ˆ1(ˆˆˆ
n
pp
n
ppzpppp
n
pp
n
ppzpp
( a ) If n is large, and , use Normal.9.01.0 p
12
Hypotheses Testing
1. Null hypotheses
2. Alternative hypotheses
3. Test statistic
4. Rejection region( or critical region)
13
=P(Type I error)=P(reject | is true)
0H 0H
(在 Q.C. work, 有時亦可稱為 produce’s risk. )
=P(Type II error)=P(fail to reject | is false)
0H 0H
( consumes’s risk )
Power=1- =P(Type II error)=P( reject | is false)0H 0H
Specify and design a test procedure maximize the power( minimize , a function of sample size. )
p-value = The smallest level of significance that would lead to rejection of the null hypotheses.
14
1. Test on means of normal distribution, variance known
Test statistic
|))(|1(2value 0Zp
00 : H0: aHv.s.
n
xZ
/0
2
22
1
21
0
nn
xZ
or
15
Tests on Means with Known Variance
16
2. Test Means of Normal Distribution, Variance Unknown
17
Test on Binomial Parameter
Test on Poisson parameter
n
xZ
/0
00
,
)1(
)5.0(
,)1(
)5.0(
00
0
00
0
0
pnp
npxpnp
npx
Zif 0npx
if 0npx
00 : ppH 0: ppH a v.s.
00 : H0: aHv.s.
18
Probability of Type II error
n
xZ
/0
0
00 : H0: aHv.s.
)()(2/2/
nn
ZZ
19
Operating characteristic (O.C.) curve
20
Tests Means of Normal Distribution, Variance Unknown
3. Paired Data
n
i id ddn
S1
22 )(1
1nS
dt
d /0 ~ 1nt
21
Test on variance of Normal distribution