Graduate Lectures and Problems in QualityControl and Engineering Statistics:Theory and MethodsTo AccompanyStatistical Quality Assurance Methods for EngineersbyVardeman and JobeStephen B. VardemanV2.0: January 2001c° Stephen Vardeman 2001. Permission to copy for educationalpurposes granted by the author, subject to the requirement thatthis title page be a¢xed to each copy (full or partial) produced.2Contents1 Measurement and Statistics 11.1 Theory for Range-Based Estimation of Variances . . . . . . . . . 11.2 Theory for Sample-Variance-Based Estimation of Variances . . . 31.3 Sample Variances and Gage R&R . . . . . . . . . . . . . . . . . . 41.4 ANOVA and Gage R&R . . . . . . . . . . . . . . . . . . . . . . . 51.5 Con…dence Intervals for Gage R&R Studies . . . . . . . . . . . . 71.6 Calibration and Regression Analysis . . . . . . . . . . . . . . . . 101.7 Crude Gaging and Statistics . . . . . . . . . . . . . . . . . . . . . 111.7.1 Distributions of Sample Means and Ranges from IntegerObservations . . . . . . . . . . . . . . . . . . . . . . . . . 121.7.2 Estimation Based on Integer-Rounded Normal Data . . . 132 Process Monitoring 212.1 Some Theory for Stationary Discrete Time Finite State MarkovChains With a Single Absorbing State . . . . . . . . . . . . . . . 212.2 Some Applications of Markov Chains to the Analysis of ProcessMonitoring Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Integral Equations and Run Length Properties of Process Moni-toring Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 An Introduction to Discrete Stochastic Control Theory/MinimumVariance Control 373.1 General Exposition . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Process Characterization and Capability Analysis 454.1 General Comments on Assessing and Dissecting “Overall Variation” 454.2 More on Analysis Under the Hierarchical Random E¤ects Model 474.3 Finite Population Sampling and Balanced Hierarchical Structures 505 Sampling Inspection 535.1 More on Fraction Nonconforming Acceptance Sampling . . . . . 535.2 Imperfect Inspection and Acceptance Sampling . . . . . . . . . . 5834 CONTENTS5.3 Some Details Concerning the Economic Analysis of Sampling In-spection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 Problems 691 Measurement and Statistics . . . . . . . . . . . . . . . . . . . . . 692 Process Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . 743 Engineering Control and Stochastic Control Theory . . . . . . . 934 Process Characterization . . . . . . . . . . . . . . . . . . . . . . . 1015 Sampling Inspection . . . . . . . . . . . . . . . . . . . . . . . . . 115A Useful Probabilistic Approximation 127Chapter 1Measurement and StatisticsV&J §2.2 presents an introduction to the topic of measurement and the relevanceof the subject of statistics to the measurement enterprise. This chapter expandssomewhat on the topics presented in V&J and raises some additional issues.Note that V&J equation (2.1) and the discussion on page 19 of V&J arecentral to the role of statistics in describing measurements in engineering andquality assurance. Much of Stat 531 concerns “process variation.” The discus-sion on and around page 19 points out that variation in measurements from aprocess will include both components of “real” process variation and measure-ment variation.1.1 Theory for Range-Based Estimation of Vari-ancesSuppose that X1; X2; : : : ; Xnare iid Normal (¹,¾2) random variables and letR = max Xi¡ min Xi= max(Xi¡ ¹) ¡ min(Xi¡ ¹)= ¾µmaxµXi¡ ¹¾¶¡ minµXi¡ ¹¾¶¶= ¾ (max Zi¡ min Zi)where Zi= (Xi¡ ¹)=¾. Then Z1; Z2; : : : ; Znare iid standard normal randomvariables. So for purposes of studying the distribution of the range of iid normalvariables, it su¢ces to study the standard normal case. (One can derive “general¾” facts from the “¾ = 1” facts by multiplying by ¾.)Consider …rst the matter of the …nding the mean of the range of n iid stan-dard normal variables, Z1; : : : ; Zn. LetU = min Zi; V = max Ziand W = V ¡ U :12 CHAPTER 1. MEASUREMENT AND STATISTICSThenEW = EV ¡EUand¡EU = ¡E min Zi= E(¡min Zi) = E max(¡Zi) ;where the n variables ¡Z1; ¡Z2; : : : ; ¡Znare iid standard normal. ThusEW = EV ¡EU = 2EV :Then, (as is standard in the theory of order statistics) note thatV · t , all n values Ziare · t :So with © the standard normal cdf,P[V · t] = ©n(t)and thus a pdf for V isf(v) = nÁ(v)©n¡1(v) :SoEV =Z1¡1v¡nÁ(v)©n¡1(v)¢dv ;and the evaluation of this integral becomes a (very small) problem in numericalanalysis. The value of this integral clearly depends upon n. It is standard toinvent a constant (whose dependence upon n we will display explicitly)d2(n):= EW = 2EVthat is tabled in Table A.1 of V&J. With this notation, clearlyER = ¾d2(n) ;(and the range-based formulas in Section 2.2 of V&J are based on this simplefact).To …nd more properties of W (and hence R) requires appeal to a well-knownorder statistics result giving the joint density of two order statistics. The jointdensity of U and V isf(u; v) =½n(n ¡ 1)Á(u)Á(v) (©(v) ¡©(u))n¡2for v > u0 otherwise :A transformation then easily shows that the joint density of U and W = V ¡Uisg(u; w) =½n(n ¡ 1)Á(u)Á(u + w) (©(u + w) ¡©(u))n¡2for w > 00 otherwise :1.2. THEORY FOR SAMPLE-VARIANCE-BASED ESTIMATION OF VARIANCES3Then, for example, the cdf of W isP[W · t] =Zt0Z1¡1g(u; w)dudw ;and the mean of W2isEW2=Z10Z1¡1w2g(u; w)dudw :Note that upon computing EW and EW2, one can compute both the varianceof WVar W = EW2¡ (EW )2and the standard deviation of W,pVar W . It is common to give this standarddeviation the name d3(n) (where we continue to make the dependence on nexplicit and again this constant is tabled in Table A.1 of V&J). Clearly, havingcomputed d3(n):=pVar W , one then haspVar R = ¾d3(n) :1.2 Theory for Sample-Variance-Based Estima-tion of VariancesContinue to suppose that X1; X2; : …