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1、硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 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硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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 Stu
2、dent T distribution-Summary of formula for Confidence Limits-Introduction 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
3、 the power of hypothesis testing-Calculating Type II Error probabilities-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-Hypothe
4、sis Testing Exercise II(paper clip)-Hypothesis Testing Presentation-第一天wrap up硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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 varianc
5、e-ANOVA Analysis of variance case study-Break-Testing the fittness of a probability 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 s
6、quare approach-Assumptions 和variance estimator-Making inference about the slope-Coefficient 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硬
7、商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Day 3:Multiple regression 和model building-Introduction to multiple regression 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 pr
8、ediction model-Break-Multiple regression exercise(statapult)-Presentation for multiple regression exercise-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 cu
9、rve-Break-Model building example-Stepwise regression an approach to screen out factors-Day 3 wrap up硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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 f
10、actors-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 modelling,和non-linear modelling)-OFAT versus DOE 和its impact in a screening experiment-Types of
11、 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 interaction of factors-Pareto of factors significance-Graphical interpretation of DOE results-某些r
12、ules 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 up硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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ca
13、pability-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 exercise(using Dice)-Introduction to Xbar S Chart
14、-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 control charts for attribute data-np Charts,p Charts,
15、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硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Recap of Statistical TerminologyDistributions differs in locationDistributions differs in spreadDistributions differs in shapeNormal Distr
16、ibution-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硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 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
17、 Lower Spec LimitLSLUpper Spec LimitUSLSpecification 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硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 流程流程Capability Index,Cpk1.Based on the assumption that the 流
18、程is normal 和in control2.An index that compare the 流程center with specification centerNormal 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)硬商品买卖在阿里巴巴硬商品买卖在阿里巴
19、巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Probability TheoryProbability is the chance for an event to occur.Statistical dependence/independence Posterior probability Relative frequency Make decision through probability distributions(i.e.Binomial,Poisson,Normal)Central Limit TheoremRegardless the actual distribution of
20、 the population,the distribution of the mean for sub-groups of sample from that distribution,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,h
21、ence making statistical comparison of the actual populations.It does not implies that the population is always normally distributed.(Cp,Cpk must always based on the assumption that 流程流程is normal)硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Inferential StatisticsThe 流程流程of interpreting the sample data t
22、o 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-Test(Determine if the underlying populations is significantly different in terms of the means 和和variations)Chi-Square Test of Independenc
23、e(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 predict the outcome of a single output variable)硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Central Limit Theorem1.The mean x of the sampling
24、 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 the mean will approach normality regardless of the actual population distribution.3.It assur
25、es 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 distribution(n=30)=150Population distributionx =150Sampling distribution(n=5)硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴
26、 软商品交易在阿里巧巧软商品交易在阿里巧巧 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.n 30),the sampling distribution of x is exactly n
27、ormal 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.Introduction to Student t Distrbution Discovered in
28、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 distribution will become closer to that of standard norma
29、l distribution(or z distribution).硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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 probability of occurence =1 Shaded areaApplicable when:Sample size 30 标准偏差 is unknown Population distribution is at least
30、 approximately normally distributedt(a a,u u)a aArea under the curve硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Percentiles of the Normal Distribution/Z DistributionZa aArea under the curveWhereby,Shaded area =one-tailed probability of occurence =1 Shaded area硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧
31、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 human on a limited basis.PWD is a pharmaceutical company currently
32、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 blood pressure of the 6 tested patients are as below:(1.7 ,3.0 ,0.8 ,
33、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硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Student t Distrbution example(cont)Using normal or z distributionUsing student t distributi
34、onAlthough 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 for 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 confiden
35、ce level 硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 6 Sigma 流程和流程和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 01 2 3 4 5 6 -99.9999999998
36、%-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 硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 某些某些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
37、 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 Distribution-3-2-1 01 2 3-99.74%-LCLUCLDistribution wi
38、th 1.5 Sigma Shift-3-2-1 01 2 3-86.64%-LCLUCLOut of control data points硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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
39、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:硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Intr
40、oduction to Hypothesis Testing?What 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
41、 purpose of the hypothesis is to establish a basis,so that one can gather 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 流程流程change will not cause any effect on the d
42、ownstream 流程流程es.The variation of Vendor Bs parts are 40%wider than those of Vendor A.硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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 Dif
43、ference in MeanSignificant Difference in both Mean 和和Variationm m1 m m2 1=2m m1 m m2 1 2m m1=m m2 1 2m m1=m m2 1=2硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 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 re
44、presents the translation of a practical question into a statistical question.Statistical testing 提供提供s an objective solution,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
45、answers the practical question:“Is there a real difference between A 和和B?”In hypothesis testing,relatively small samples are used 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
46、wrong conclusion.硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧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 sufficient evidence is found to be contrary Often assumed to be the status quo,or the preferred outcome.However,it sometimes represents a s
47、tate you strongly want to disprove.Designated as H0 In hypothesis testing,we always bias toward null hypothesisThe Alternative Hypothesis(or Research Hypothesis)Statement that will be accepted only if data 提供convincing evidence of its truth(i.e.by rejecting the null hypothesis).Instead of comparing
48、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).Designated as H1硬商品买卖在阿里巴巴硬商品买卖在阿里巴巴 软商品交易在阿里巧巧软商品交易在阿里巧巧 Elements of Hypothesis Testing(cont)Example if we want to test whethe
49、r 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 level states that:95%of the time the mean value will fluctuate within the confidence
50、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 errorType II ErrorAccepting a null hypothesis(H0),when it is false.Probability of this e