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1、硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Revision:1.00Date:June 20016 西格玛绿带培训MaterialsTWO-6-4-2 0 2 4 6标准偏差 标准偏差硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 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-In
2、troduction to Hypothesis 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 prob
3、abilities-Hypothesis 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 Presenta
4、tion-第一天wrap up硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 第二天:Analysis 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
5、 distribution-Chi-square:a goodness of fit 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-Coe
6、fficient of Correlation 和Determination-Example of simple linear regression-Simple linear regression exercise(using statapult)-Break-Simple linear regression exercise(cont)-Presentation of results-第二天wrap up硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Day 3:Multiple regression 和model building-Introduction to multiple regre
7、ssion model-Building a model-Fitting the model with least squares 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 exerc
8、ise-Lunch-Qualitative data 和dummy variables-Models with 2 or more 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 up硬商品
9、买卖在阿里巴巴 软商品交易在阿里巧巧 Day 4:设计of Experiment-Overview of Experimental Design-What is a designed 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calculatio
10、n for building simple linear model-Type 和uses of DOE,(i.e.linear screening,linear modelling,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 screenin
11、g DOEs result-Lunch-Modelling DOE(Full factoria with interactions)-Interpreting interaction 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confirma
12、tion run-Day 4 wrap up硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Day 5:Statistical 流程Control-What is Statistical 流程Control-Control 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 Ch
13、art-Xbar R Chart example-Assignable 和Chance causes in SPC-Rule of thumb for SPC run test-Break-Xbar R Chart exercise(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
14、Chart?-Lunch-Choosing the sub-group-Choosing the correct sample size-Sampling frequency-Introduction to control 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 up硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 R
15、ecap of Statistical TerminologyDistributions differs in locationDistributions differs in spreadDistributions differs in shapeNormal Distribution-6-5-4-3-2-1 0 1 2 3 4 5 6-99.9999998%-99.73%-95.45%-68.27%-3 variation is called natural tolerance Area under a Normal Distribution硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 流程
16、capability potential,CpBased on the assumptions that:1.流程is normalNormal Distribution-6-5-4-3-2-1 0 1 2 3 4 5 6Lower Spec LimitLSLUpper Spec LimitUSLSpecification Center2.It is a 2-sided specification3.流程mean is centered to the device specificationSpread in specificationNatural toleranceCP=USL-LSL68
17、6=1.33硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 流程Capability Index,Cpk1.Based on the assumption that the 流程is normal 和in control2.An index that compare the 流程center with specification centerNormal Distribution-6-5-4-3-2-1 0 1 2 3 4 5 6Lower Spec LimitLSLUpper Spec LimitUSLSpecification CenterTherefore when,Cpk Cp;then
18、流程is not centeredCpk=Cp;then 流程is centeredUSL-Y3Y-LSL3Cpk=min,硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 The 流程of collecting,presenting 和describing sample data,using graphical 工具和numbers.Pareto Chart Population mean Histogram Population 标准偏差Descriptive StatisticsEstimates for Descriptive StatisticsThe 流程of estimating th
19、e population parameters from sample(s)that was taken from the population.Sample mean,X Population mean,m Sample 标准偏差,S Population 标准偏差,(when sample size,n 20)Estimated 标准偏差,R/d2 Population 标准偏差,(when sample size,n 20)硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Probability TheoryProbability is the chance for an event to o
20、ccur.Statistical dependence/independence 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 dist
21、ribution,will be normally distributed with 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 tha
22、t the population is always normally distributed.(Cp,Cpk must always based on the assumption that 流程is normal)硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Inferential StatisticsThe 流程of interpreting the sample data to draw conclusions about the population from which the sample was taken.Confidence Interval(Determine confid
23、ence level for a sampling mean to fluctuate)T-Test 和F-Test(Determine if the underlying populations is significantly different in terms of the means 和variations)Chi-Square Test of Independence(Test if the sample proportions are significantly different)Correlation 和Regression(Determine if 关系hip betwee
24、n variables exists,和generate model equation to predict the outcome of a single output variable)硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 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
25、sample mean is towards the population mean.2.The sampling distribution of the 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=150Sampli
26、ng distribution(n=5)x=150Sampling distribution(n=20)x=150Sampling distribution(n=30)=150Population distributionx=150Sampling distribution(n=5)硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 某些take aways for sample size 和sampling distribution For large sample size(i.e.n 30),the sampling distribution of x will approach normali
27、ty regardless the actual distribution of the sampled population.For small sample size(i.e.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 o
28、f population 标准偏差 using S equation may 提供a poor estimation if the sample size is small.Introduction 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 dis
29、tribution contains only one random quantity(x only)As sample size increases,the t distribution will become closer to that of standard normal distribution(or z distribution).硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Percentiles of the t DistributionWhereby,df=Degree of freedom=n(sample size)1Shaded area=one-tailed proba
30、bility of occurence=1 Shaded areaApplicable when:Sample size 30 标准偏差 is unknown Population distribution is at least approximately normally distributedt(a,u)aArea under the curve硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Percentiles of the Normal Distribution/Z DistributionZaArea under the curveWhereby,Shaded area=one-ta
31、iled probability of occurence=1 Shaded area硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Student t Distrbution exampleFDA requires pharmaceutical companies to perform extensive 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 hu
32、man on a limited basis.PWD is a pharmaceutical company currently in the second phase of testing on a new antibiotic project.The chemists are interested 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 bloo
33、d 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 pressure for patients taking the new antibiotic,using both normal 和t distributions.硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Student t Distrbution example(cont)Using normal or z
34、 distribution Using student t distributionAlthough the confidence level is the same,using t distribution will result in a larger interval value,because:标准偏差,S for small sample size is probably not accurate 标准偏差,S for small sample size is probably too optimistic Wider interval is therefore necessary
35、to achieve the required confidence level 硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Summary of formula for confidence limit硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 6 Sigma 流程和1.5 Sigma Shift in MeanStatistically,a 流程that is 6 Sigma with respect to its specifications is:Normal Distribution-6-5-4-3-2-1 0 1 2 3 4 5 6-99.9999999998%-LSL USLDP
36、M=0.002Cp=2Cpk=2But Motorola defines 6 Sigma with a scenario of 1.5 Sigma shift in meanDPM=3.4Cp=2Cpk=1.51.5硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 某些Explanations on 1.5 Sigma Mean Shift 1.Motorla has conducted a lot of experiments,和found that in long term,the 流程mean will shift within 1.5 sigma if the 流程is under cont
37、rol.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 Distribution-3-2-1 0 1 2 3-99.74%-LCL UCLDistribution with 1.5 Sigma Shift-3-2-1 0 1 2
38、3-86.64%-LCL UCLOut of control data points硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Our 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 the 流程is of 6 sigma capability,is in control,和is normal
39、ly 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:硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Introduction to Hypothesis Testing?What is hypothesis testing in statisti
40、c?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 establish a basis,so that one can ga
41、ther evidence to either disprove the statement or accept it as true.lExample of statistical hypothesis The average commute time using Highway 92 is shorter than using France Avenue.This 流程change will not cause any effect on the downstream 流程es.The variation of Vendor Bs parts are 40%wider than those
42、 of Vendor A.硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Elements of Hypothesis TestingPossible outcomes for hypothesis testing on two tested populations:No Significant DifferenceSignificant Difference in VariationSignificant Difference in MeanSignificant Difference in both Mean 和Variationm1 m21=2m1 m21 2m1=m21 2m1=m21=2
43、硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 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s an objective solution,with known ris
44、ks,to questions which are traditionally answered subjectively.It is a stepping stone to 设计of Experiment,DOE.Hypothesis Testing Descriptions Hypothesis Testing answers the practical question:“Is there a real difference between A 和B?”In hypothesis testing,relatively small samples are used to answer qu
45、estions about population parameters.There is always a chance that a sample that is not representative of the population being selected 和results in drawing a wrong conclusion.硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Elements of Hypothesis Testing(cont)The Null Hypothesis Statement generally assumed to be true unless su
46、fficient evidence is found to be contrary Often assumed to be the status quo,or the preferred outcome.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)St
47、atement that will be accepted only if data 提供convincing evidence of its truth(i.e.by rejecting the null hypothesis).Instead of comparing 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
48、is 1 450).Designated as H1硬商品买卖在阿里巴巴 软商品交易在阿里巧巧 Elements 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:mp=500和consider alternate hypothesis as:Alternate Hypothesis,H1:mp 500;(2 tails test)Remember confidenc
49、e interval,at 95%confidence 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-mH0=5000.025 of area 0.025 of area(a/2)reject area(a/2)reject area1
50、.96std error1.96std errorType II ErrorAccepting a null hypothesis(H0),when it is false.Probability of this error equals bType I ErrorRejecting the null hypothesis(H0),when it is true.Probability of this error equals aIf mp is within confidence limit,accept the null hypothesis H0.If mp is in reject a