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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
/ 2 / 2
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 )