Lecture 4 (Inf Pro Q) Basic

8/6/2019 Lecture 4 (Inf Pro Q) Basic

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Lecture 4 Statistical Methods (II)

Inferences About Process Quality

Ming-Hung Shu ( ), Professor

Department of Industrial Engineering & ManagementNational Kaohsiung University of Applied Sciences


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Learning Outlines

H ypothesis TestingSampling Errors

Type I Error and Type II Error

Producers Risk and Consumers RiskConfidence level (1-Type I error) and Power (1-Type II error)

Inference on the Mean of a Populationwith Variance KnownConfidence Interval on the Mean withVariance Known

Please put more emphasis on Section 4.3 in Textbook p. 112-116.


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H ypothesis Testing


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B asic Concepts

In statistical methods (I), the use of probabilitydistributions in modeling the output of a process.The parameters like were assumed known.

It is unrealistic for most of practical cases.

In general, these parameters are unknown andneed to be estimated by S 2 , respectively.

Based on sample dataParameter Estimation

By doing so, sampling errors are raisedType I error and Type II error ; producers risk and consumers risk ; Confidence level (1-Type I error) and Power (1-Type 2 error)

Q 2W p

x p


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Hypothesis Testing Sam pling

Error

Statistical Inference ( H ypothesis Testing) : drawing conclusions about the information

contained in a sample and making a decisionfor the unknown population .

Sampling to DetermineParent Distribution


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An important part of any hypothesis testingproblem is determining the parameter valuespecified in the null and alternativehypotheses . How can we determine that?

Result from past knowledge or evidenceResult from contractual or design specificationsResult from some model of process

As to realize whether the parameter valuehas changed , then periodically test thehypothesis.


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Suppose we think that the mean of inside diameter of a bearing is 1.5 inches. We may express thisstatementH0 (Null hypothesis) :

H1 ( A lternative hypothesis) :

=1.5 Q

1.5 Q {

H ypothesis testing procedures are quiet

useful in many types (Part 3 and Part 4)of statistical quality control problem.


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Sampling Errors


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Inference for a Single Sample

Example : We think the average mpg of one type of cars is35 mpgH0 (Null hypothesis) : H1 ( A lternative hypothesis) :

From the sample of 25 cars, the sample average mpg wasfound to be 33 mpg. Assume the true standard deviationmpg is 5. Is our thinking right?

35!Q

35{Q

Sampling Errors need to betake into account

before making a decision


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7

R j ctR j ct

t r j ct

R j cti R i s


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Sampling Errors

Types of ErrorsType I error - re ecting the null hypothesiswhen it is true.

Pr(Type I error) = E, sometimes called the

producer

s ri sk .The level of significance is a probability . It isalso known as the probability of a Type I error

(want this to be small) - rejecting the null

hypothesis when it is true .

Note : The smaller the Type I error, the more confidence(the more evidence) has when the null hypothesis is re ected.


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Significance Level

The level of significance (confidence) ,E determines the s ize of the rejectionregion .

How small? Usually want inmodern technology for evaluating productsquality. (three standard deviations of mean)

0.0027E !

Remember number values : 3, 1.96, and 1.645


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P ower of a Test

Type II error - not re ecting the nullhypothesis when it is false.

Pr(Type II error) = F, sometimes called thecon s umer s ri sk .

The Power of a test of hypothesis isgiven by 1 - FThat is, 1 - F is the probability of

correctly rejecting the null hypothesis,or the probability of rejecting the nullhypothesis when the alternative istrue .


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Inference on the Mean of aPopulation, Variance Known


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Inference for a Single Sample

Example : H0 (Null hypothesis) : H1 ( A lternative hypothesis) : From the sample of 25 cars, the average mpg was found to

be 31.5. Assume the true standard deviation mpg is 10/3and . What is your conclusion?

35{35!

3 3 3 5 3 7

R j

R j

n o t re je c t

R e je c t io n R e g io n s

0.0027E !


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H ypothesis TestingH ypotheses : H0: H1:Test Statistic :

Significance Level , ERejection Region :

If Z0 falls into either of the two regionsabove, re ect H 0

o Q Q! o Q Q

0

0/

x

Z n

Q

W !

/2 0 /2o Z Z or Z Z

E E "


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Exercise

Significance Level, E=0.0027 (for Products Quality)Rejection Region :

Significance Level, E=0.05 (for General Engineering)Rejection Region :

Significance Level , E=0.10 (for Social Science)

Rejection Region :

/2 0 /2oor

E E"

/2 0 /2o Z Z or Z Z

E E "

/2 0 /2oor

E E"

Remind Numbers 3, 1.96, and 1.645


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F or E = 0.0027

Example 4-2H ypotheses : H0: H1:

Test Statistic :

Significance Level , E = 0.002Rejection Region : Since 3.50 > 3 , re ect H 0 and conclude thatthe lot mean pressure strength exceeds175 psi.

17 5!Q 1 7 5Q {

5 0.32 5/1017 5182

Z0 !!

0 / 23 Z Z

E" !


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F or An Advanced Study (See Textbook Ex4-1 p.114) for One Sided


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Confidence Interval on the Meanof a Population, Variance Known


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Confidence IntervalsA general 100(1- E)% two-sided confidenceinterval on the true population mean, Q is

L ower Upper

100(1- E)% One-sided confidence intervals are :

[ ] 1 P L U Q Ee e !

[ ] (1 ) [ ] (1 ) P U P L Q E Q Ee ! e !


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Continued Confidence Intervals

Two-Sided : (This is the basic idea for constructing the upper and lower controllimits in control charts)

L ower Upper

See the text for one-sided confidence intervals.

2 2

P r [ ] 1 x Z x Z n n

E EW W Q Ee e !


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F or E = 0.05

Example 4.2: (See Textbook Ex. 4-2 p. 116)Reconsider Example 3-1. Suppose a 95%two-sided confidence interval is specified.

Our estimate of the mean bursting strength is182 psi s 3.92 psi with 95% confidence

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10 1018 2 1 .96 18 2 1 .96

2 5 2 5

17 8 .0 8 18 5 .92

x Z x Z n n

E E

W W Q

Q

Q

e e

e e

e e


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F or E = 0.0027

Example 3.2: (See Textbook Ex. 4-2 p. 116)Reconsider Example 3-1. Suppose a 99. 3%two-sided confidence interval is specified.

Our estimate of the mean bursting strength is182 psi s 6 psi with 95% confidence

/ 2 / 2

10 1018 2 3 18 2 3

2 5 2 5

176 188

x Z x Z n n

E E

W W Q

Q

Q

e e

e e

e e


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F inal Words

Statistical methods are used to makedecisions about a processIs the process out of control?

Is the process average you were given the truevalue?What is the true process variability ?

Is the process performance/capabilityacceptable?


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H omework

P. 169 Ex. 4.1 (a) (c) and Ex. 4.4 (a) (c)P. 169 Ex. 4.5


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Exercises

P. 44 Ex. 1. ( Answer in P. 8 and 9) and Ex.1.9 ( Answer in P. 13)P. 99 Ex. 3.2 and Ex.3.8

P. 169 Ex. 4.1 (a) (c) and Ex. 4.4 (a) (c)P. 169 Ex. 4.5 ( Bonus )

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Lecture 4 (Inf Pro Q) Basic