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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 ).

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ch3 inference about process quality
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)

slide2

假設 ,則

, 其中

( with ,當 時, )

1. Sampling from a Normal distribution

slide3

e.g.~

其中 is the sample var. of i.i.d.

is the sample var. of i.i.d.

其中U and V indep.~ and

回上頁

slide4

假設 i.i.d. Bernoulli with success prob.= p

令 ~ B(n , p)

a discrete r.v. with range space

2. Sampling from a Bernoulli distribution

回上頁

slide5

假設 i.i.d.

a discrete r.v. with taking values

3. Sampling from a Poisson distribution

slide6

令 indep.

(e.g. A unit of product can have m different types of

defect, each modeled with a Poisson distribution

with parameter )

此稱為 demerit procedure, 若不全為1,

則L一般未必為Poisson分佈。

回上頁

slide7

In general, and are unbiased estimators of the

population mean and variance, respectively.

但S則一般並非 population standard deviation 的

unbiased estimator.

e.g. Poisson , Binomial

4. Estimation of process parameter

(a)Point estimation:

Important properties of an estimation

(1)Unbiased

(2)Minimum variance

回上頁

slide8

[L,U]稱為 的 two sided confidence interval.

稱為 的 one sided confidence interval.

(b)Interval estimation:

slide9

Two sided C.I.

e.g.

i.i.d.

當variance unknown, 則以 取代 ,

S 取代 。

two-sided C.I. On the variance

or

Lower C.I.

Upper C.I.

回上頁

slide10

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

slide11

(a)If n is large, and , use Normal.

C.I. on the difference of two binomial parameter and .

3. C.I. on Binomial parameter

(b)If n is small, then use Binomial distribution.

(c)If n is large, p is small, then use Poisson.

slide12

Hypotheses Testing

  • Null hypotheses
  • Alternative hypotheses
  • Test statistic
  • Rejection region(or critical region)
slide13

=P(Type II error)=P(fail to reject | is false)

Power=1- =P(Type II error)=P( reject | is false)

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.

=P(Type I error)=P(reject | is true)

(在Q.C. work, 有時亦可稱為produce’s risk. )

(consumes’s risk)

slide14

v.s.

or

1. Test on means of normal distribution, variance known

Test statistic

slide17

v.s.

if

v.s.

if

Test on Binomial Parameter

Test on Poisson parameter

slide18

v.s.

Probability of Type II error

slide20

Tests Means of Normal Distribution, Variance Unknown

3. Paired Data