6Sigma綠帶培訓(xùn)教材-2

1、Revision: 1.00Date: June 20016西格瑪綠帶培訓(xùn)MaterialsTWO-6-4-20246規(guī)范偏向.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 H

2、ypothesis 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- Hypothesi

3、s 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

4、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 Ko

5、lmogorov-Smirnov Test Goodness of fit exercise using dice Result 和discussion on exercise-Lunch- Probabilistic 關(guān)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 sim

6、ple linear regression Simple linear regression exercise (using statapult)-Break- Simple linear regression exercise (cont) Presentation of results 第二天wrap up.Day 3: Multiple regression 和model building Introduction to multiple regression model Building a model Fitting the model with least squares appr

7、oach 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 q

8、uantitative 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.Day 4: 設(shè)計of Experiment Overview of Experimental Design What is a des

9、igned experiment Objective of experimental 設(shè)計和its capability in identifying the effect of factors One factor at a time (OFAT) versus 設(shè)計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, line

10、ar 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) Interpre

11、ting interaction of factors Pareto of factors significance Graphical interpretation of DOE results 某些rules of thumb in DOE 實例of Modelling DOE 和its analysis-Break- Modelling DOE exercise with statapult Target practice 和confirmation run Day 4 wrap up.Day 5: Statistical 流程Control What is Statistical 流程

12、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

13、- Xbar R Chart exercise (using Dice) Introduction to Xbar S Chart Implementing Xbar S Chart 為什么Xbar S Chart ? Introduction to Individual Moving Range Chart Implementing Individual Moving Range Chart 為什么Xbar S Chart ?-Lunch- Choosing the sub-group Choosing the correct sample size Sampling frequency I

14、ntroduction 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.Recap of Statistical TerminologyDistributions differs in locationDistributions differs in spreadDistributi

15、ons differs in shapeNormal Distribution-6-5-4-3-2-10123456 - 99.9999998% - 99.73% - 95.45% -68.27%- 3 variation is called natural tolerance Area under a Normal Distribution.流程capability potential, CpBased on the assumptions that :流程is normalNormal Distribution-6-5-4-3-2-10123456Lower Spec LimitLSLUp

16、per Spec LimitUSLSpecification CenterIt is a 2-sided specification流程mean is centered to the device specificationSpread in specificationNatural toleranceCP =USL - LSL686= 1.33.流程Capability Index, CpkBased on the assumption that the 流程is normal 和in control2. An index that compare the 流程center with spe

17、cification centerNormal Distribution-6-5-4-3-2-10123456Lower Spec LimitLSLUpper Spec LimitUSLSpecification CenterTherefore when ,Cpk 20) Estimated 規(guī)范偏向, R/d2 Population 規(guī)范偏向, (when sample size, n 20) .Probability TheoryProbability is the chance for an event to occur. Statistical dependence / indepen

18、dence 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 dist

19、ributed 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

20、 normally distributed.(Cp, Cpk must always based on the assumption that 流程is normal).Inferential StatisticsThe 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)

21、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 關(guān)系hip between variables exists, 和generate model equati

22、on to predict the outcome of a single output variable).Central Limit TheoremThe 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 dis

23、tribution 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 = 2

24、0)x = 150Sampling distribution(n = 30)m = 150Population distributionx = 150Sampling distribution(n = 5).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 normality regardless the actual distribution of the sampled popu

25、lation. 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 規(guī)范偏向 using S equation may 提供a poor est

26、imation 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 規(guī)范偏向 dependence on small sample size. Contain two random quantities (x 和S), whereas normal distribution contains only one random quantity (x

27、 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: Sam

28、ple size 30 規(guī)范偏向 is unknown Population distribution is at least approximately normally distributedt ( a, u )aArea under the curve.Percentiles of the Normal Distribution / Z DistributionZaArea under the curveWhereby,Shaded area = one-tailed probability of occurencea = 1 Shaded area.Student t Distrbut

29、ion 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 p

30、hase 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

31、.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 distributionUsing student t distributionAlthough the confidence level is the same,

32、 using t distribution will result in a larger interval value, because: 規(guī)范偏向, S for small sample size is probably not accurate 規(guī)范偏向, 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 lim

33、it.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-6-5-4-3-2-10123456 - 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Ex

34、planations on 1.5 Sigma Mean Shift 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.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 c

35、ontrol, 和this is deem acceptable in statistical 流程control (SPC) practices.Normal Distribution-3-2-10123- 99.74% -LCLUCLDistribution with 1.5 Sigma Shift-3-2-10123- 86.64% -LCLUCLOut of control data points.Our Explanation Most frequently used sample size for SPC in industry is 3 to 5 units per sampli

36、ng. 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 normally distributed. Under the confidence interval for sampling distribution, we expect the average value of the samples to fluctuate within 3 standard errors (

37、i.e. natural tolerance), giving confidence interval of:.Introduction 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 val

38、ue of one of the characteristics for one or more populations. The 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. Example of statistical hypothesis The average commute time using Highway 92 is shorter than usin

39、g 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 of Vendor A.Elements of Hypothesis TestingPossible outcomes for hypothesis testing on two tested populations:No Significant DifferenceSignificant Difference in V

40、ariationSignificant Difference in MeanSignificant Difference in both Mean 和Variationm1 m21 = 2m1 m21 2m1 = m21 2m1 = m21 = 2.為什么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

41、practical question into a statistical question. Statistical testing 提供s an objective solution, with known risks, to questions which are traditionally answered subjectively. It is a stepping stone to 設(shè)計of Experiment, DOE.Hypothesis Testing Descriptions Hypothesis Testing answers the practical questio

42、n: “Is there a real difference between A 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results in drawing a wrong conclusion.Elements o

43、f 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 state you strongly want to disprove. Designated as H0 In hypothesi

44、s 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 two populations, it can also be based on a specific engine

45、ering difference in a characteristic value that one desires to detect (i.e. instead of asking is m1 = m2, we ask is m1 450). Designated as H1.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 =

46、 500和consider alternate hypothesis as:Alternate Hypothesis, H1 : 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 interval (limit) 5% chance that the mean is natural fluctuation, but we think

47、 it is not alpha (a) probability- Confidence limit -mH0 = 5000.025 of area0.025 of area(a/2)reject area(a/2)reject area1.96std 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

48、is true. Probability of this error equals aIf mp is within confidence limit, accept the null hypothesis H0.If mp is in reject area, reject the null hypothesis H0.Use the std error observed from the sample to set confidence limit on 500 (mH0). The assumption is mH0 has the same variance as mp.Element

49、s of Hypothesis Testing (cont)Other possible alternate hypothesis are:Alternate Hypothesis, H1 : mp 500 ; (1 tail test)Alternate Hypothesis, H1 : mp 500 For 95% confidence level, a = 0.05.Since H1 is one tail test, reject area does not need to be divided by 2.From standard normal distribution table:

50、Z-value of 1.645 will give 0.95 area, leaving a to be 0.05.Therefore if mp is more than 500 by 1.645 std error, it will be in the reject area, 和we will reject the null hypothesis H0, concluding on alternate hypothesis H1 that mp is 500.某些hypothesis testings that are applicable to engineers: The impa

51、ct on response measurement with new 和old 流程parameters. Comparison of a new vendors parts (which are slightly more expensive) to the present vendor, when variation is a major issue. Is the yield on Tester ECTZ21 the same as the yield on Tester ECTZ33 ?流程Situations Comparison of one population from a

52、single 流程to a desirable standard Comparison of two populations from two different 流程esor Single sided: comparison considers a difference only if it is greater or only if it is less, but not both. Two sided: comparison considers any difference of ine質(zhì)量 important. Inferences based on a single sample“L

53、arge sample test of hypothesis about a population meanExample:An automotive manufacturer wants to evaluate if their new throttle 設(shè)計on all the latest car model is able to give an adequate response time, resulting in an predictable pick-up of the vehicle speed when the fuel pedal is being depressed. B

54、ased on finite element modelling, the 設(shè)計team committed that the throttle response time is 1.2 msec, 和this is the recommended value that will give the driver the best control over the vehicle acceleration.The test engineer of this project has tested on 100 vehicles with the new throttle 設(shè)計和obtain an

55、average throttle response time of 1.05 msec with a 規(guī)范偏向 S of 0.5 msec. Based on 99% confidence level, can he concluded that the new throttle 設(shè)計will give an average response time of 1.2 msec ?.“Large sample test of hypothesis about a population mean (cont)Solution:Since the sample size is relatively

56、large (i.e. 30), we should use z statistic.m X = 1.05 msec ;s S = 0.5 msec ;n = 100 ;Null hypothesis H0 : m = mH0 (1.2 msec)Alternate hypothesis H1 : m mH0 (1.2 msec)- Acceptance Area -mH0 = 1.20.005 of area0.005 of area(a/2)Reject Area(a/2)Reject Area2.58std error2.58std errorFrom standard normal d

57、istribution table,The Z value corresponding to 0.005 tail area is 2.58.a = 0.01 (2 tails), since 2 tails test, therefore tail area = a/2 = 0.005 ;How many std error is X away from 1.2 msec ?X = 1.02Therefore X is 3 std errors away from 1.2 msec.- Acceptance Area -mH0 = 1.20.005 of area0.005 of area(

58、a/2)Reject Area(a/2)Reject Area2.58std error2.58std errorX = 1.02“Large sample test of hypothesis about a population mean (cont)Baed on 99% confidence level, since X is at the negative reject area, we will reject the null hypothesis 和conclude on the alternate hypothesis that the average response tim

59、e is significantly different than 1.2 msecThe average response time appears to be lower than 1.2 msec.What does 99% confidence level means in the above example ?It defines the limits whereby 99% of the average sampling value should fall within, given the desirable (hypothesised) mean as mH0. Any val

60、ue fall outside this confidence limit indicates the sample mean is significantly different from mH0. In other words, we will only conclude the alternate hypothesis H1 (that the means are different) if we are more than 99% sure.The Observed Significance level, p-value p-value is the probability for c