【正文】 x) – A Few ThoughtsPg 8 ?March 01, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. March 16, 2001
Make sure the process settings cover the likely productionrange (but not too far).
Too great a range
points outside the normal range mayhave too great an effect on the model.
Too small a range
Error term may dominate the fit.
Take several replicates at each input setting (x).
Replicate runs help increase the model accuracy.
Randomize runs whenever practical.
Run order is often significant factor.
The output (y) at different inputs (x抯) is not alwaysindependent of previous settings.A good spread in the data is required for agood model. Consider two examples:All of the data is collected at the normalprocess settings. In this case, regression willtry to fit a linear model to a bination ofrandom process variation and randommeasurement variation. The results will be ofno value.The second case is when most of the datais clustered around the standard settingsexcept for a couple of points at the extremeranges. In this case, the extreme pointscontrol the fit of the model. If one of theextreme points is a flyer, then the model willbe in error due to the flyer.The ideal case is for the Black Belt tocollect a range of data throughout the processspace. 置信區(qū)間:Confidence Intervals
A population is the set of all measurements of interest to the experimenter
A sample is a subset of measurements selected from the population
An inference is a statement about a population parameter based oninformation contained in a sample Two types of inference
Estimation A poll has been devised to determine the public’s reaction to anew political scandal. The purpose is to estimate the reactionof all Americans by polling a representative sample
Hypothesis testing A vaccine for Lyme disease has been developed but the rateof negative side effects is %. A new vaccine has beendeveloped and it is desired to know if the rate of negative sideeffects is lower than %.The other branch of statistics isdescriptive. Its purpose is merely todescribe a set of measurements.Inferential statistics is used to guess whatGod knows about a population from a sample.Within inferential statistics, there are twotypes: estimation and hypothesis testing.Estimation is trying to guess the populationstatistics from a sample. Hypothesis testingconcerns evaluating a sample statistic andparing it to some hypothetical value.Estimates and the CLT
What is the best estimate of the population mean using sample data? The sample mean!
How good of an estimate is the sample mean? What factors influence the accuracy of the estimate of the meanfrom sample data? Recall that:
The variation in the distribution of sample means is a function of thevariance of the Population and the sample size!n Pop X /σ σ =What About Small Samples?
If the population standard deviation is known (it almost never is) usethe previous formula for small samples, too
If the population sigma is unknown (it usually is): The estimate for standard deviation (s) is used The tdistribution is used instead of the normal (Z) distribution
Q: What is a tdistribution? The tdistribution is a family of bellshaped (normallike)distributions that are dependent on sample size The smaller the sample size n, the wider and flatter thedistributionns t X μ ns t X n n 1 , 2 / 1 , 2 / +≤≤α αThe tdistribution is the general case forany sample where the population standarddeviation is unknown. However, with largesamples, the t and zdistributions are nearlyidentical, so either can be used.You can verify this in Minitab bygenerating a large sample of normal data andthen analyzing it with both the z and tdistribution routines.Proportions and Binomial Experiments Pg 35 . April 01, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. April 3, 2001
Proportion data is usually the result of a binomialtypeexperiment Binomial experiments (or Bernoulli trials) are those thathave only one of two outes, either a “success” or a“failure” The probability of this type of experiment is described by abinomial distribution, a plicated distribution In many cases the normal distribution can be used toapproximate the binomial distribution When nxp 5 and nx(1p)5
μ= nxp and σ 2 = nxpx(1p)Binomial distributions are discussed inalmost every statistics textbook. Calculationswith them is not necessarily difficult, but it istedious if it must be done manually. Minitabhas routines, however, that greatly simplifiesthe calculations.If the binomial approximation applies andthe data can be estimated with a normaldistribution other statistical tests and controlcharts can be used that would not be availableotherwise.Try to construct your experiments suchthat the binomial approximation is valid.A general rule of thumb: for the normalapproximation to apply, have a sample size ofat least 30 and large enough to guarantees atleast 5 successes. 假設(shè)測試:Introduction to Hypothesis TestingA Bright IdeaNotes:Pg 5 11 Nov 2000 ?April 01, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved.
A light bulb pany is trying to produce a brighter light bulb for thesame energy. It is hoped that a change in the filament coatingprocess will produce a brighter light.
The engineer collected the last ten light bulbs made before theprocess change and the first ten after the change. The mean lightoutput of the old process bulbs is 1251 lumens and the new processis 1273 lumens.
Does the increase of 22 in the means of the two groups represent areal improvement?
Could the difference between