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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

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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

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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

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（ 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

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（ 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

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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

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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

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Hypotheses Testing

1. Null hypotheses

2. Alternative hypotheses

3. Test statistic

4. Rejection region（ or critical region）

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=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.

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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

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2. Test Means of Normal Distribution, Variance Unknown

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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.

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Probability of Type II error

n

xZ

/0

0

00 : H0: aHv.s.

)()(2/2/

nn

ZZ

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Operating characteristic (O.C.) curve

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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