六西格玛绿带培训2.pptx
Revision: 1.00Date: June 20016 6西格玛绿带培训西格玛绿带培训MaterialsMaterialsTWOTWO-6-4-20246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 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 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- Hypothesis Testing Exercise II (paper clip)- Hypothesis Testing Presentation- 第一天wrap up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 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- 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 upDay 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 prediction 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 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 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 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 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 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 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 upRecap of Statistical TerminologyDistributions differs in locationDistributions differs in spreadDistributions differs in shapeNormal 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 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流程流程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 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 / 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 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, hence 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)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-Test(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 predict the outcome 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 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.m = 150Population distributionx = 150Sampling distribution(n = 5)x = 150Sampling distribution(n = 20)x = 150Sampling distribution(n = 30)m = 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. 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.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 distribution 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).Percentiles of the t DistributionWhereby,df = Degree of freedom = n (sample size) 1Shaded area = one-tailed probability of occurencea = 1 Shaded areaApplicable when: Sample size 30 标准偏差 is unknown Population distribution is at least approximately normally distributed0.750.900.950.9750.990.9950.999511.00003.07776.313712.706231.821063.6559636.577620.81651.88562.92004.30276.96459.925031.599830.76491.63772.35343.18244.54075.840812.924440.74071.53322.13182.77653.74694.60418.610150.72671.47592.01502.57063.36494.03216.868560.71761.43981.94322.44693.14273.70745.958770.71111.41491.89462.36462.99793.49955.408180.70641.39681.85952.30602.89653.35545.041490.70271.38301.83312.26222.82143.24984.7809100.69981.37221.81252.22812.76383.16934.5868110.69741.36341.79592.20102.71813.10584.4369120.69551.35621.78232.17882.68103.05454.3178130.69381.35021.77092.16042.65033.01234.2209140.69241.34501.76132.14482.62452.97684.1403150.69121.34061.75312.13152.60252.94674.0728160.69011.33681.74592.11992.58352.92084.0149170.68921.33341.73962.10982.56692.89823.9651180.68841.33041.73412.10092.55242.87843.9217190.68761.32771.72912.09302.53952.86093.8833200.68701.32531.72472.08602.52802.84533.8496210.68641.32321.72072.07962.51762.83143.8193220.68581.32121.71712.07392.50832.81883.7922230.68531.31951.71392.06872.49992.80733.7676240.68481.31781.71092.06392.49222.79703.7454250.68441.31631.70812.05952.48512.78743.7251260.68401.31501.70562.05552.47862.77873.7067270.68371.31371.70332.05182.47272.77073.6895280.68341.31251.70112.04842.46712.76333.6739290.68301.31141.69912.04522.46202.75643.6595300.68281.31041.69732.04232.45732.75003.6460 0.6741.2821.6451.962.3262.5763.291Area under the curvedf, u ut ( a a, u u )a aArea under the curvePercentiles of the Normal Distribution / Z Distribution00.010.020.030.040.050.060.070.080.0900.50000.50400.50800.51200.51600.51990.52390.52790.53190.53590.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.57530.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61410.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65170.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68790.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72240.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75490.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78520.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81330.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.838910.84130.84380.84610.84850.85080.85310.85540.85770.85990.86211.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88301.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90151.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91771.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93191.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94411.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95451.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96331.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97061.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.976720.97720.97780.97830.97880.97930.97980.98030.98080.98120.98172.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98572.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.98902.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.99162.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.99362.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.99522.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.99642.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.99742.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.99812.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.998630.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990Area under the curveZZa aArea under the curveWhereby,Shaded area = one-tailed probability of occurencea = 1 Shaded areaStudent 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 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 , 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.Student t Distrbution example (cont)Using normal or z distribution0.76 2.283 (0.388) 1.96 2.283 nS Z X interval confidence 95% S Deviation, Std 2.283 613.7 2.1) 2.7 3.4 0.8 3 (1.7 X Mean,0.05 level, Confidence6 n size, Sample2) / (aa95. 0 6Using student t distribution0.997 2.283 (0.388) 2.571 2.283 nS t X interval confidence 95% S Deviation, Std 2.283 613.7 2.1) 2.7 3.4 0.8 3 (1.7 X Mean,1) - 6 (i.e. 5 freedom, of Degree0.05 level, Confidence6 n size, Sample2) / (aua95. 0 6Although 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 confidence level Summary of formula for confidence limitdata attribute for npq2) / Z( p limit Confidence data continuous for nS2) / Z( X limit Confidenceknown is deviation standard population whenor ,30) (n size sample large Foraadata attribute for npq2) / t( p limit Confidence data continuous for nS2) / t( X limit Confidencedeviation standard population unknown with30) (n size sample small Foraa6 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% -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found 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 Distribution-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 the 流程is of 6 sigma capability, is in control, 和is normall
