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1、Revision:1.00Date:June 20016 6西格玛绿带培训西格玛绿带培训MaterialsMaterialsTWOTWO-6-4-20246标准偏差标准偏差标准偏差标准偏差第二天:Tests of Hypotheses-Week 1 recap of Statistics Terminology-Introduction to Student T distribution-Example in using Student T distribution-Summary of formula for Confidence Limits-Introduction to Hypothe
2、sis Testing-The elements of Hypothesis Testing-Break-Large sample Test of Hypothesis about a population mean-p-Values,the observed significance levels-Small sample Test of Hypothesis about a population mean-Measuring the power of hypothesis testing-Calculating Type II Error probabilities-Hypothesis
3、Exercise I-Lunch-Hypothesis Exercise I Presentation-Comparing 2 population Means:Independent Sampling-Comparing 2 population Means:Paired Difference Experiments-Comparing 2 population Proportions:F-Test-Break-Hypothesis Testing Exercise II(paper clip)-Hypothesis Testing Presentation-第一天wrap up第二天:An
4、alysis of variance 和simple linear regression-Chi-square:A test of independence-Chi-square:Inferences about a population variance-Chi-square exercise-ANOVA-Analysis of variance-ANOVA Analysis of variance case study-Break-Testing the fittness of a probability distribution-Chi-square:a goodness of fit
5、test-The Kolmogorov-Smirnov Test-Goodness of fit exercise using dice-Result 和discussion on exercise-Lunch-Probabilistic 关系hip of a regression model-Fitting model with least square approach-Assumptions 和variance estimator-Making inference about the slope-Coefficient of Correlation 和Determination-Exam
6、ple of simple linear regression-Simple linear regression exercise(using statapult)-Break-Simple linear regression exercise(cont)-Presentation of results-第二天wrap upDay 3:Multiple regression 和model building-Introduction to multiple regression model-Building a model-Fitting the model with least squares
7、 approach-Assumptions for model-Usefulness of a model-Analysis of variance-Using the model for estimation 和prediction-Pitfalls in prediction model-Break-Multiple regression exercise(statapult)-Presentation for multiple regression exercise-Lunch-Qualitative data 和dummy variables-Models with 2 or more
8、 quantitative independent variables-Testing the model-Models with one qualitative independent variable-Comparing slopes 和response curve-Break-Model building example-Stepwise regression an approach to screen out factors-Day 3 wrap upDay 4:设计of Experiment-Overview of Experimental Design-What is a desi
9、gned experiment-Objective of experimental 设计和its capability in identifying the effect of factors-One factor at a time(OFAT)versus 设计of experiment(DOE)for modelling-Orthogonality 和its importance to DOE-H和calculation for building simple linear model-Type 和uses of DOE,(i.e.linear screening,linear model
10、ling,和non-linear modelling)-OFAT versus DOE 和its impact in a screening experiment-Types of screening DOEs-Break-Points to note when conducting DOE-Screening DOE exercise using statapult-Interpretating the screening DOEs result-Lunch-Modelling DOE(Full factoria with interactions)-Interpreting interac
11、tion of factors-Pareto of factors significance-Graphical interpretation of DOE results-某些rules of thumb in DOE-实例of Modelling DOE 和its analysis-Break-Modelling DOE exercise with statapult-Target practice 和confirmation run-Day 4 wrap upDay 5:Statistical 流程Control-What is Statistical 流程Control-Control
12、 chart the voice of the 流程-流程control versus 流程capability-Types of control chart available 和its application-Observing trends for control chart-Out of Control reaction-Introduction to Xbar R Chart-Xbar R Chart example-Assignable 和Chance causes in SPC-Rule of thumb for SPC run test-Break-Xbar R Chart e
13、xercise(using Dice)-Introduction to Xbar S Chart-Implementing Xbar S Chart-为什么Xbar S Chart?-Introduction to Individual Moving Range Chart-Implementing Individual Moving Range Chart-为什么Xbar S Chart?-Lunch-Choosing the sub-group-Choosing the correct sample size-Sampling frequency-Introduction to contr
14、ol charts for attribute data-np Charts,p Charts,c Charts,u Charts-Break-Attribute control chart exercise(paper clip)-Out of control not necessarily is bad-Day 5 wrap upRecap of Statistical TerminologyDistributions differs in locationDistributions differs in spreadDistributions differs in shapeNormal
15、 Distribution-6-5-4-3-2-1 01 2 3 4 5 6 -99.9999998%-99.73%-95.45%-68.27%-3 variation is called natural tolerance Area under a Normal Distribution流程流程capability potential,CpBased on the assumptions that:1.流程is normalNormal Distribution-6 -5 -4 -3 -2 -1 01 2 3 4 5 6 Lower Spec LimitLSLUpper Spec Limit
16、USLSpecification Center2.It is a 2-sided specification3.流程mean is centered to the device specificationSpread in specificationNatural toleranceCP=USL-LSL6 8 6=1.33流程流程Capability Index,Cpk1.Based on the assumption that the 流程is normal 和in control2.An index that pare the 流程center with specification cen
17、terNormal Distribution-6 -5 -4 -3 -2 -1 01 2 3 4 5 6 Lower Spec LimitLSLUpper Spec LimitUSLSpecification CenterTherefore when,Cpk 20)Estimated 标准偏差标准偏差,R/d2 Population 标准偏差标准偏差,(when sample size,n 20)Probability TheoryProbability is the chance for an event to occur.Statistical dependence/independenc
18、e Posterior probability Relative frequency Make decision through probability distributions(i.e.Binomial,Poisson,Normal)Central Limit TheoremRegardless the actual distribution of the population,the distribution of the mean for sub-groups of sample from that distribution,will be normally distributed w
19、ith sample mean approximately equal to the population mean.Set confidence interval for sample based on normal distribution.A basis to compare samples using normal distribution,hence making statistical comparison of the actual populations.It does not implies that the population is always normally dis
20、tributed.(Cp,Cpk must always based on the assumption that 流程流程is normal)Inferential StatisticsThe 流程流程of interpreting the sample data to draw conclusions about the population from which the sample was taken.Confidence Interval(Determine confidence level for a sampling mean to fluctuate)T-Test 和和F-Te
21、st(Determine if the underlying populations is significantly different in terms of the means 和和variations)Chi-Square Test of Independence(Test if the sample proportions are significantly different)Correlation 和和Regression(Determine if 关系关系hip between variables exists,和和generate model equation to pred
22、ict the out e of a single output variable)Central Limit Theorem1.The mean x of the sampling distribution will approximately equal to the population mean regardless of the sample size.The larger the sample size,the closer the sample mean is towards the population mean.2.The sampling distribution of t
23、he mean will approach normality regardless of the actual population distribution.3.It assures us that the sampling distribution of the mean approaches normal as the sample size increases.=150Population distributionx =150Sampling distribution(n=5)x =150Sampling distribution(n=20)x =150Sampling distri
24、bution(n=30)=150Population distributionx =150Sampling distribution(n=5)某些某些take aways for sample size 和和sampling distribution For large sample size(i.e.n 30),the sampling distribution of x will approach normality regardless the actual distribution of the sampled population.For small sample size(i.e.
25、n 30),the sampling distribution of x is exactly normal if the sampled population is normal,和will be approximately normal if the sampled population is also approximately normally distributed.The point estimate of population 标准偏差 using S equation may 提供a poor estimation if the sample size is small.Int
26、roduction to Student t Distrbution Discovered in 1908 by W.S.Gosset from Guinness Brewery in Ireland.To compensate for 标准偏差 dependence on small sample size.Contain two random quantities(x 和S),whereas normal distribution contains only one random quantity(x only)As sample size increases,the t distribu
27、tion will become closer to that of standard normal distribution(or z distribution).Percentiles of the t DistributionWhereby,df =Degree of freedom =n(sample size)1Shaded area =one-tailed probability of occurence =1 Shaded areaApplicable when:Sample size 30 标准偏差 is unknown Population distribution is a
28、t least approximately normally distributedt(a a,u u)a aArea under the curvePercentiles of the Normal Distribution/Z DistributionZa aArea under the curveWhereby,Shaded area =one-tailed probability of occurence =1 Shaded areaStudent t Distrbution exampleFDA requires pharmaceutical panies to perform ex
29、tensive tests on all new drugs before they can be marketed to the public.The first phase of testing will be on animals,while the second phase will be on human on a limited basis.PWD is a pharmaceutical pany currently in the second phase of testing on a new antibiotic project.The chemists are interes
30、ted to know the effect of the new antibiotic on the human blood pressure,和they are only allowed to test on 6 patients.The result of the increase in blood pressure of the 6 tested patients are as below:(1.7 ,3.0 ,0.8 ,3.4 ,2.7 ,2.1 )Construct a 95%confidence interval for the average increase in blood
31、 pressure for patients taking the new antibiotic,using both normal 和t distributions.Student t Distrbution example(cont)Using normal or z distributionUsing student t distributionAlthough the confidence level is the same,using t distribution will result in a larger interval value,because:标准偏差标准偏差,S fo
32、r small sample size is probably not accurate 标准偏差标准偏差,S for small sample size is probably too optimistic Wider interval is therefore necessary to achieve the required confidence level Summary of formula for confidence limit6 Sigma 流程和流程和1.5 Sigma Shift in MeanStatistically,a 流程that is 6 Sigma with r
33、espect to its specifications is:Normal Distribution-6-5-4-3-2-1 01 2 3 4 5 6 -99.9999999998%-LSLUSLDPM=0.002Cp=2Cpk=2But Motorola defines 6 Sigma with a scenario of 1.5 Sigma shift in meanDPM=3.4Cp=2Cpk=1.51.5 某些某些Explanations on 1.5 Sigma Mean Shift 1.Motorla has conducted a lot of experiments,和fou
34、nd that in long term,the 流程mean will shift within 1.5 sigma if the 流程is under control.2.1.5 sigma mean shift in a 3 Sigma 流程control plan will be translated to approximately 14%of the time a data point will be out of control,和this is deem acceptable in statistical 流程control(SPC)practices.Normal Distr
35、ibution-3-2-1 01 2 3-99.74%-LCLUCLDistribution with 1.5 Sigma Shift-3-2-1 01 2 3-86.64%-LCLUCLOut of control data pointsOur Explanation Most frequently used sample size for SPC in industry is 3 to 5 units per sampling.Take the middle value of 4 as an average sample size used in the sampling.Assuming
36、 the 流程is of 6 sigma capability,is in control,和is normally distributed.Under the confidence interval for sampling distribution,we expect the average value of the samples to fluctuate within 3 standard errors(i.e.natural tolerance),giving confidence interval of:Introduction to Hypothesis Testing?What
37、 is hypothesis testing in statistic?A hypothesis is“a tentative assumption made in order to draw out or test its logical or empirical consequences.A statistical hypothesis is a statement about the value of one of the characteristics for one or more populations.The purpose of the hypothesis is to est
38、ablish a basis,so that one can gather evidence to either disprove the statement or accept it as true.lExample of statistical hypothesis The average mute time using Highway 92 is shorter than using France Avenue.This 流程流程change will not cause any effect on the downstream 流程流程es.The variation of Vendo
39、r Bs parts are 40%wider than those of Vendor A.Elements of Hypothesis TestingPossible out es for hypothesis testing on two tested populations:No Significant DifferenceSignificant Difference in VariationSignificant Difference in MeanSignificant Difference in both Mean 和和Variationm m1 m m2 1=2m m1 m m
40、2 1 2m m1=m m2 1 2m m1=m m2 1=2为什么为什么Hypothesis Testing?Many problems require a decision to accept or reject a statement about a parameter.That statement is a Hypothesis.It represents the translation of a practical question into a statistical question.Statistical testing 提供提供s an objective solution,
41、with known risks,to questions which are traditionally answered subjectively.It is a stepping stone to 设计设计of Experiment,DOE.Hypothesis Testing Descriptions Hypothesis Testing answers the practical question:“Is there a real difference between A 和和B?In hypothesis testing,relatively small samples are u
42、sed to answer questions about population parameters.There is always a chance that a sample that is not representative of the population being selected 和和results in drawing a wrong conclusion.Elements of Hypothesis Testing(cont)The Null Hypothesis Statement generally assumed to be true unless suffici
43、ent evidence is found to be contrary Often assumed to be the status quo,or the preferred out e.However,it sometimes represents a state you strongly want to disprove.Designated as H0 In hypothesis testing,we always bias toward null hypothesisThe Alternative Hypothesis(or Research Hypothesis)Statement
44、 that will be accepted only if data 提供convincing evidence of its truth(i.e.by rejecting the null hypothesis).Instead of paring two populations,it can also be based on a specific engineering difference in a characteristic value that one desires to detect(i.e.instead of asking is 1=2,we ask is 1 450).
45、Designated as H1Elements of Hypothesis Testing(cont)Example if we want to test whether a population mean is equal to 500,we would translate it to:Null Hypothesis,H0:m mp=500和consider alternate hypothesis as:Alternate Hypothesis,H1:m mp 500;(2 tails test)Remember confidence interval,at 95%confidence
46、level states that:95%of the time the mean value will fluctuate within the confidence interval(limit)5%chance that the mean is natural fluctuation,but we think it is not alpha()probability-Confidence limit-m mH0=5000.025 of area0.025 of area(a a/2)reject area(a a/2)reject area1.96std error1.96std err
47、orType II ErrorAccepting a null hypothesis(H0),when it is false.Probability of this error equals b bType I ErrorRejecting the null hypothesis(H0),when it is true.Probability of this error equals a aIf m mp is within confidence limit,accept the null hypothesis H0.If m mp is in reject area,reject the
48、null hypothesis H0.Use the std error observed from the sample to set confidence limit on 500(H0).The assumption is H0 has the same variance as p.Elements of Hypothesis Testing(cont)Other possible alternate hypothesis are:1.Alternate Hypothesis,H1:m mp 500;(1 tail test)2.Alternate Hypothesis,H1:m mp
49、500 For 95%confidence level,=0.05.Since H1 is one tail test,reject area does not need to be divided by 2.From standard normal distribution table:Z-value of 1.645 will give 0.95 area,leaving to be 0.05.Therefore if p is more than 500 by 1.645 std error,it will be in the reject area,和we will reject th
50、e null hypothesis H0,concluding on alternate hypothesis H1 that p is 500.某些某些hypothesis testings that are applicable to engineers:The impact on response measurement with new 和和old 流程流程parameters.Comparison of a new vendors parts(which are slightly more expensive)to the present vendor,when variation