Basic One and Two Sample Inference FormulasSome More Applications of Simple One-Sample Methods of InferenceApplication to a Single Sample Consisting of Single Measurements of a Single Measurand Made Using Multiple Devices (From a Large Population of Such Devices)Application to a Single Sample Consisting of Differences in Measurements on Multiple Measurands Made Using Two Devices (Assuming Device Linearity)Applications of Simple Two-Sample Methods of InferenceApplication to Two Samples Consisting of Repeat Measurements of a Single Measurand Made Using Two Different DevicesApplication to Two Samples Consisting of Single Measurements Made With Two (Linear) Devices On Multiple Measurands From a Stable Process (Only One Device Used for a Given Measurand)Application to Two Samples Consisting of Repeat Measurements Made With One Linear Device On Two MeasurandsApplication to Two Samples Consisting of Single Measurements Made Using a Single Linear Device on Multiple Measurands Produced by Two Stable ProcessesSummaryIE 361 Module 3More on Elementary Statistics and MetrologyReading: Section 2.2 Statistical Quality Assurance for Engineers(Section 2.2 of Revised SQAME )Prof. Steve Vardeman and Prof. Max MorrisIow a State UniversityAugust 2008Vardeman and Morris (I o wa State University) IE 361 Module 3 A ugust 2008 1 / 28Basic One and Two Sample Inference FormulasIn Module 2 we reminded you that based on a model for y1, y2, . . . , ynofsampling from a normal distribution with mean µ and standard deviationσ, elementary con…dence limits arey  tspnfor estimating µ (1)andssn 1χ2upperand ssn 1χ2lowerfor estimating σ (2)(where degrees of freedom for t, χ2upper, and χ2lowerare n 1) and thenmade some simple applications of these limits in contexts that recognizemeasurement error.Vardeman and Morris (I o wa State University) IE 361 Module 3 August 2008 2 / 28Basic One and Two Sample Inference FormulasParallel to the one sample formulas are the two sample formulas ofelementary statistics. These are based on a model that says thaty11, y12, . . . , y1n1and y21, y22, . . . , y2n2are independent samples from normal distributions with respective meansµ1and µ2and respective standard deviations σ1and σ2. In this context,the so-called "Satterthwaite" approximation gives limitsy1y2ˆtss21n1+s22n2for estimating µ1 µ2(3)where appropriate "approximate degrees of freedom" forˆt areˆν =s21n1+s22n22s41(n11)n21+s42(n21)n22Vardeman and Morris (I o wa State University) IE 361 Module 3 August 2008 3 / 28Basic One and Two Sample Inference Formulasˆν =s21n1+s22n22s41(n11)n21+s42(n21)n22(This is a formula that you may not have seen in an elementary statisticscourse, where only methods valid when one assumes that σ1= σ2aretypically presented.) It turns out that above min((n11), (n21))ˆν,so that a simple conservative version of this method uses degrees offreedomˆν = min((n11), (n21))Further, in the two-sample context, standard elementary con…dence limitsfor comparing standard deviations ares1s21qF(n11) ,(n21) ,upperands1s21qF(n11) ,(n21) ,lowerforσ1σ2(4)(and be reminded that F(n11) ,(n21) ,lower= 1/F(n21) ,(n11) ,upperso thatstandard F tables giving only upper percentage points can be employed).Vardeman and Morris (I o wa State University) IE 361 Module 3 August 2008 4 / 28Basic One and Two Sample Inference FormulasIf these formulas do not look familiar, you should immediately stop andreview them. Their use can, for example, be seen in Chapter 6 ofVardeman and Jobe’s Basic Engineering Data Collection and Analysis or inthe Stat 231 text. Here we will consider a variety of applications of themto problems that arise in metrological studies for quality assurance. Ourbasic objective is to amply illustrate (and help you develop the thoughtprocess necessary to successfully employ) the basic insight thatHow sources of physical variation interact with a data collectionplan governs what of practical importance can be learned from adata set, and in particular, how measurement error is re‡ected inthe data set.Vardeman and Morris (I o wa State University) IE 361 Module 3 August 2008 5 / 28Some More Applications of Simple One-Sample Methodsof InferenceIn Module 2, we considered inference based on a "single sample" made upof either1repeat measurements on a single measurand made using the samedevice2single measurements made on multiple measurands coming from astable process made using the same linear deviceHere we present two more applications of one sample formulas.Vardeman and Morris (I o wa State University) IE 361 Module 3 August 2008 6 / 28Application to a Single Sample Consisting of SingleMeasurements of a Single Measurand Made Using MultipleDevices (From a Large Population of Such Devices)There are quality assurance contexts in which an organization has many"similar" measurement "devices" that could potentially be used to domeasuring. In particular, a given piece of equipment might well be usedby any of a large number of operators. (Recall that we are using the word"device" to describe a particular combination of equipment, people,procedures, etc. used to produce a measurement. So, in this language,di¤erent operators with a …xed piece of equipment are di¤erent "devices.")A way to try to compare these devices would be to use some (say n ofthem) to measure a single measurand. This is illustrated in the next…gure.Vardeman and Morris (I o wa State University) IE 361 Module 3 August 2008 7 / 28Single Measurements of a Measurand From MultipleDevicesFigure: Cartoon Illustrating the Measurement of One Item Using Multiple Devices(From a Large Population of Such) As suming a Common σdevi ceVardeman and Morris (I o wa State University) IE 361 Module 3 August 2008 8 / 28Single Measurements of a Measurand From MultipleDevicesIn this context, a measurement is of the formy = x + ewhere e = δ + e , for δ the (randomly selected) bias of the device usedand e a measurement error with mean 0 and standard deviation σd evi ce(representing a repeat measurement variability for the particular device).So one might writey = x + δ + e Thinking of x as …xed and δ and e as independent random variables (δwith mean µδ, the average device bias, and standard deviation σδmeasuring variability in device biases) the laws of mean and variance fromelementary probability then imply thatµy= x + µδ+ 0 = x + µδandσy=q0 + σ2δ+