Analysis of Variance and Design of Experiments-II

Size: px
Start display at page:

Download "Analysis of Variance and Design of Experiments-II"

Transcription

1 Anly of Vrne Degn of Experment-II MODULE VI LECTURE - 8 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr Shlbh Deprtment of Mthemt & Sttt Indn Inttute of Tehnology Knpur

2 Tretment ontrt: Mn effet The uefulne of hvng ovrte n the model provde more urte more pree etmte of tretment ontrt Now dut for the ovrte to remove be ttrbutble to dfferene mong expermentl unt Now redue the b by dutng tretment men ontrt Conder the whole-plot prt of the degn Conder the model gven n equton ( whh follow: w w w w y = µ + β x + r + β ( x x + w + β ( x x + β ( x x x + x + ε(1 + h ooo h hoo ooo oo ooo ho hoo oo ooo h + β ( x x + ( w + β ( x x x + x + β ( x x x + x + ε( ( oo ooo o oo oo ooo h ho o oo h = µ + r + w + β ( x x w h ho hoo x + x w + x x x + x + oo ooo ε(1 h ( β ( h ho o oo ε( h, where w µ = µ + β x ooo r = r + β ( x x w h h hoo ooo w = w + β ( x x w oo ooo = + β ( x x oo ooo ( w k = ( w + β ( xo xoo xoo + xooo

3 3 µ w The unbed etmte of re obtned from th model follow: y oo = ˆ µ + wˆ y = wˆ oo wth = 0 But the unbed etmte of ontrt of the form w re needed nted of thee ontrt Ung the defnton mpled n equton (, the unbed etmte re gven ˆw ˆ ˆ ˆw y = µ + β x + w + β ( x x oo ooo oo ooo ˆ ˆ w y = w + β x oo ooo Sne xh ' re the oberved ontnt, o ˆ ˆ ˆ w µ + w = y β x oo oo ˆ ˆ w w = y β x oo oo Thee re the requred ontrt the duted mn effet ontrt

4 4 Further, ( x oo ( 1 Vr ˆ w σ + σ = + r E(1 x w x w E MSE(1 = σ + σ1 MSE (1 h ( r 1( t 1 1 degree of freedom Compron mong the plt-plot tretment level re mlr Then The unbed etmte of the ontrt of the form unbed etmte re gven by y oo = ˆ µ + ˆ ˆ oo = y ˆw ˆ ˆ ˆ y = µ + β x + + β ( x x oo ooo oo ooo ˆ ˆ y = + β x oo oo Thee expreon n be re-expreed re needed Ung the defnton n equton (, the ˆ ˆ ˆ ˆw µ + = y ( β x x β xˆ oo oo ooo ooo ˆ ˆ = y β x oo oo Alo x oo Vr ˆ = + σ rt E( E MSE( = σ MSE ( h ( r 1 t ( 1 1 degree of freedom

5 5 Interton ontrt Interton ontrt re of the form y o wth = 0 The trd error of uh ontrt depend on the nture of the ontrt Conder the ontrt mong plt-plot level for fxed whole-plot level Thee re of the form y o wth = 0 wth pef vlue Then The duted ontrt t vrne E y = + w o ( ( ( ( β ( o oo = + w + x x ( ˆ ( ˆ + w = y β ( x x o o oo ( x x 1 o oo σ r E ( Next onder the ontrt mong whole-plot tretment level, ether t the me plt-plot tretment level or t the dfferent plt-plot tretment level Thee ontrt tke the generl form y o, where 0 Then E y = w + + w o ( ( w = ( w + β ( x x + + ( w + β ( x x oo ooo o oo =

6 6 The duted ontrt ˆw ( ˆ ˆ ( ( ( ˆ w + + w = y β x x β ( x x o oo ooo o oo h vrne ( σ1 + σ ( x ( oo x x o x ooo oo + ( σ1 + σ + σ r E(1 x ( wx E w There no ne etmte of th vrne A moderte mplfton obtned by plttng th n two e Conder frt the whole-plot ontrt t one plt-plot level Th mple tht the ' re zero for ll ' the generl duted ontrt then beome t vrne beome ˆ β ( An unbed etmte of th vrne gven by ( ( wˆ + ˆ + ( w = y x x, ' ' ' ' o ' o ' oo ' o ' oo ( 1 ( x o ' xoo ( 1 σ + σ ' σ + + σ = σ + σ r E( r E( ( x x 1 o ' oo 1 ' + MSE + ' MSE r E r ( ( (1 It dffult to fnd t ext degree of freedom The Stterwte formul ued to pproxmte the degree of freedom whh ued for the tet of hypothe onfdene ntervl etmton For the more generl e, the unbed etmte of the vrne gven by The ext degree of freedom re dffult to obtn The pproxmte degree of freedom n be obtned ung Stterthwte pproxmton ( x x ( x x ( (1 w w w w 1 1( xoo xooo o oo 1 ( xoo xooo + + MSE + + MSE r E(1 x x E( x x r E(1 x x

7 7 Anly of ovrne wth one whole-plot ovrte n RBD We onder llutrte the nly of ovrne for the plt-plot experment wth whole-plot n n RBD one ovrte oted wth the whole-plot Developng the model We begn wth the model y = µ + r + w + βx + ε(1 + + ( w + ε(, h= 1,, r, = 1,, t, = 1,,, h h h h h where h = 1,,,r; = 1,,,t = 1,,,, the ovrte Aume tht both the whole- plt-plot tretment re fxed effet, mplyng tht wo = o = ( w o = ( w o = 0 The xh ' re oberved ontnt The ε (1 h ε ( h re dentlly ndependently normlly dtrbuted eh wth men 0 vrne σ σ repetvely Moreover they re mutully ndependent Rewrte the model to olte the ovrte ontrbuton to b vrne Ung x = x + ( x x + ( x x + ( x x x + x, ho oo ho oo o oo h ho o oo the model rewrtten x h 1 y = µ + βx + r + β( x x + w + β( x x + β( x x x + x + ε(1 + + ( w + ε( h oo h ho oo o oo h ho o oo h h = µ + r + w + β( x x x + x + ε(1 + + ( w + ε( h h ho o oo h h where µ = µ + βx, oo r = r + β ( x x, h h ho oo w = w + β ( x x o oo

8 8 The term n β ( x x x + x h ho o remove ll of thee µ, r, w h repreent the ontrbuton to b from the expermentl unt v the ovrte, repreent the ontrbuton to the vrne The nly of ovrne provde dutment to Computton n the nly of ovrne One the model ontruted, the next tep to ontrut the ompt nly of ovrne tble follow: Anly of ovrne tble for the plt-plot wth only whole-plot ovrte Soure y vrble Covrte Men M M M Blok B B B W W W W Error (1 E (1 E (1 E (1 S S W S W S Error ( E ( Totl T T T

9 9 There re no ovrne dutment n the plt-plot trtum In the whole-plot trtum, we hve ˆ β = E(1 E(1 ˆ ( σ + σ1 Vr( β = E(1 1 E (1 MSE(1 = E(1 [( r 1( t 1 1] E(1 MSW 1 [ W + E(1 ] E(1 = W + ( t 1 W + E(1 E(1 Contrt Sne there only whole-plot ovrte, o only the whole-plot tretment ontrt re duted Conder the ontrt wth = 0 Rewrte th Th ontrt h vrne The etmte of y = wˆ + ˆ β ( x x oo o oo wˆ = y ˆ β x oo oo x o r E(1 ( σ σ1 ( σ + σ wth ( r 1( t 1 1 degree of freedom gven by MSE(1 1

Analysis of Variance and Design of Experiments-II

Analysis of Variance and Design of Experiments-II Anlyi of Vrince nd Deign of Experiment-II MODULE VI LECTURE - 7 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr. Shlbh Deprtment of Mthemtic & Sttitic Indin Intitute of Technology Knpur Anlyi of covrince ith one

More information

8. INVERSE Z-TRANSFORM

8. INVERSE Z-TRANSFORM 8. INVERSE Z-TRANSFORM The proce by whch Z-trnform of tme ere, nmely X(), returned to the tme domn clled the nvere Z-trnform. The nvere Z-trnform defned by: Computer tudy Z X M-fle trn.m ued to fnd nvere

More information

v v at 1 2 d vit at v v 2a d

v v at 1 2 d vit at v v 2a d SPH3UW Unt. Accelerton n One Denon Pge o 9 Note Phyc Inventory Accelerton the rte o chnge o velocty. Averge ccelerton, ve the chnge n velocty dvded by the te ntervl, v v v ve. t t v dv Intntneou ccelerton

More information

Lecture 4: Piecewise Cubic Interpolation

Lecture 4: Piecewise Cubic Interpolation Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml

More information

6 Random Errors in Chemical Analysis

6 Random Errors in Chemical Analysis 6 Rndom Error n Cheml Anl 6A The ture of Rndom Error 6A- Rndom Error Soure? Fg. 6- Three-dmenonl plot howng olute error n Kjeldhl ntrogen determnton for four dfferent nlt. Anlt Pree Aurte 4 Tle 6- Pole

More information

Jens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers

Jens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers Jens Sebel (Unversty of Appled Scences Kserslutern) An Interctve Introducton to Complex Numbers 1. Introducton We know tht some polynoml equtons do not hve ny solutons on R/. Exmple 1.1: Solve x + 1= for

More information

Learning Enhancement Team

Learning Enhancement Team Lernng Enhnement Tem Worsheet: The Cross Produt These re the model nswers for the worsheet tht hs questons on the ross produt etween vetors. The Cross Produt study gude. z x y. Loong t mge, you n see tht

More information

Simple Linear Regression Analysis

Simple Linear Regression Analysis LINEAR REGREION ANALYSIS MODULE II Lecture - 5 Smple Lear Regreo Aaly Dr Shalabh Departmet of Mathematc Stattc Ida Ittute of Techology Kapur Jot cofdece rego for A jot cofdece rego for ca alo be foud Such

More information

Concept of Activity. Concept of Activity. Thermodynamic Equilibrium Constants [ C] [ D] [ A] [ B]

Concept of Activity. Concept of Activity. Thermodynamic Equilibrium Constants [ C] [ D] [ A] [ B] Conept of Atvty Equlbrum onstnt s thermodynm property of n equlbrum system. For heml reton t equlbrum; Conept of Atvty Thermodynm Equlbrum Constnts A + bb = C + dd d [C] [D] [A] [B] b Conentrton equlbrum

More information

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Analyss of Varance and Desgn of Experments- MODULE LECTURE - 6 EXPERMENTAL DESGN MODELS Dr. Shalabh Department of Mathematcs and Statstcs ndan nsttute of Technology Kanpur Two-way classfcaton wth nteractons

More information

Let us look at a linear equation for a one-port network, for example some load with a reflection coefficient s, Figure L6.

Let us look at a linear equation for a one-port network, for example some load with a reflection coefficient s, Figure L6. ECEN 5004, prng 08 Actve Mcrowve Crcut Zoy Popovc, Unverty of Colordo, Boulder LECURE 5 IGNAL FLOW GRAPH FOR MICROWAVE CIRCUI ANALYI In mny text on mcrowve mplfer (e.g. the clc one by Gonzlez), gnl flow-grph

More information

2. Work each problem on the exam booklet in the space provided.

2. Work each problem on the exam booklet in the space provided. ECE470 EXAM # SOLUTIONS SPRING 08 Intructon:. Cloed-book, cloed-note, open-mnd exm.. Work ech problem on the exm booklet n the pce provded.. Wrte netly nd clerly for prtl credt. Cro out ny mterl you do

More information

MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER

MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER I One M Queston Fnd the unt veto n the deton of Let ˆ ˆ 9 Let & If Ae the vetos & equl? But vetos e not equl sne the oespondng omponents e dstnt e detons

More information

ECONOMETRIC THEORY. MODULE IV Lecture - 16 Predictions in Linear Regression Model

ECONOMETRIC THEORY. MODULE IV Lecture - 16 Predictions in Linear Regression Model ECONOMETRIC THEORY MODULE IV Lecture - 16 Predictions in Liner Regression Model Dr. Shlbh Deprtent of Mthetics nd Sttistics Indin Institute of Technology Knpur Prediction of vlues of study vrible An iportnt

More information

Variable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia

Variable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia Vrble tme mpltude mplfcton nd quntum lgorthms for lner lgebr Andrs Ambns Unversty of Ltv Tlk outlne. ew verson of mpltude mplfcton;. Quntum lgorthm for testng f A s sngulr; 3. Quntum lgorthm for solvng

More information

Abhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no

Abhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no hlsh Clsses Clss- XII Dte: 0- - SOLUTION Chp - 9,0, MM 50 Mo no-996 If nd re poston vets of nd B respetvel, fnd the poston vet of pont C n B produed suh tht C B vet r C B = where = hs length nd dreton

More information

KULLBACK-LEIBLER DISTANCE BETWEEN COMPLEX GENERALIZED GAUSSIAN DISTRIBUTIONS

KULLBACK-LEIBLER DISTANCE BETWEEN COMPLEX GENERALIZED GAUSSIAN DISTRIBUTIONS 0th Europen Sgnl Proessng Conferene (EUSIPCO 0) uhrest, Romn, August 7-3, 0 KULLACK-LEILER DISTANCE ETWEEN COMPLEX GENERALIZED GAUSSIAN DISTRIUTIONS Corn Nfornt, Ynnk erthoumeu, Ion Nfornt, Alexndru Isr

More information

UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II

UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )

More information

Quiz: Experimental Physics Lab-I

Quiz: Experimental Physics Lab-I Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled

More information

Predictors Using Partially Conditional 2 Stage Response Error Ed Stanek

Predictors Using Partially Conditional 2 Stage Response Error Ed Stanek Predctor ng Partally Condtonal Stage Repone Error Ed Stane TRODCTO We explore the predctor that wll relt n a mple random ample wth repone error when a dfferent model potlated The model we decrbe here cloely

More information

7.2 Volume. A cross section is the shape we get when cutting straight through an object.

7.2 Volume. A cross section is the shape we get when cutting straight through an object. 7. Volume Let s revew the volume of smple sold, cylnder frst. Cylnder s volume=se re heght. As llustrted n Fgure (). Fgure ( nd (c) re specl cylnders. Fgure () s rght crculr cylnder. Fgure (c) s ox. A

More information

Lecture 5 Single factor design and analysis

Lecture 5 Single factor design and analysis Lectue 5 Sngle fcto desgn nd nlss Completel ndomzed desgn (CRD Completel ndomzed desgn In the desgn of expements, completel ndomzed desgns e fo studng the effects of one pm fcto wthout the need to tke

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Analyss of Varance and Desgn of Experment-I MODULE VIII LECTURE - 34 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS EFFECTS MODEL Dr Shalabh Department of Mathematcs and Statstcs Indan

More information

Statistics 423 Midterm Examination Winter 2009

Statistics 423 Midterm Examination Winter 2009 Sttstcs 43 Mdterm Exmnton Wnter 009 Nme: e-ml: 1. Plese prnt your nme nd e-ml ddress n the bove spces.. Do not turn ths pge untl nstructed to do so. 3. Ths s closed book exmnton. You my hve your hnd clcultor

More information

Section 11.5 Estimation of difference of two proportions

Section 11.5 Estimation of difference of two proportions ection.5 Estimtion of difference of two proportions As seen in estimtion of difference of two mens for nonnorml popultion bsed on lrge smple sizes, one cn use CLT in the pproximtion of the distribution

More information

Dennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1

Dennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1 Denns Brcker, 2001 Dept of Industrl Engneerng The Unversty of Iow MDP: Tx pge 1 A tx serves three djcent towns: A, B, nd C. Ech tme the tx dschrges pssenger, the drver must choose from three possble ctons:

More information

Principle Component Analysis

Principle Component Analysis Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors

More information

Math 497C Sep 17, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 17, Curves and Surfaces Fall 2004, PSU Mth 497C Sep 17, 004 1 Curves nd Surfces Fll 004, PSU Lecture Notes 3 1.8 The generl defnton of curvture; Fox-Mlnor s Theorem Let α: [, b] R n be curve nd P = {t 0,...,t n } be prtton of [, b], then the

More information

Small signal analysis

Small signal analysis Small gnal analy. ntroducton Let u conder the crcut hown n Fg., where the nonlnear retor decrbed by the equaton g v havng graphcal repreentaton hown n Fg.. ( G (t G v(t v Fg. Fg. a D current ource wherea

More information

Dynamics of Linked Hierarchies. Constrained dynamics The Featherstone equations

Dynamics of Linked Hierarchies. Constrained dynamics The Featherstone equations Dynm o Lnke Herrhe Contrne ynm The Fethertone equton Contrne ynm pply ore to one omponent, other omponent repotone, rom ner to r, to ty tne ontrnt F Contrne Boy Dynm Chpter 4 n: Mrth mpule-be Dynm Smulton

More information

Lecture 9-3/8/10-14 Spatial Description and Transformation

Lecture 9-3/8/10-14 Spatial Description and Transformation Letue 9-8- tl Deton nd nfomton Homewo No. Due 9. Fme ngement onl. Do not lulte...8..7.8 Otonl et edt hot oof tht = - Homewo No. egned due 9 tud eton.-.. olve oblem:.....7.8. ee lde 6 7. e Mtlb on. f oble.

More information

Additional File 1 - Detailed explanation of the expression level CPD

Additional File 1 - Detailed explanation of the expression level CPD Addtonal Fle - Detaled explanaton of the expreon level CPD A mentoned n the man text, the man CPD for the uterng model cont of two ndvdual factor: P( level gen P( level gen P ( level gen 2 (.).. CPD factor

More information

4. UNBALANCED 3 FAULTS

4. UNBALANCED 3 FAULTS 4. UNBALANCED AULTS So fr: we hve tudied lned fult ut unlned fult re more ommon. Need: to nlye unlned ytem. Could: nlye three-wire ytem V n V n V n Mot ommon fult type = ingle-phe to ground i.e. write

More information

LINEAR REGRESSION ANALYSIS. MODULE VIII Lecture Indicator Variables

LINEAR REGRESSION ANALYSIS. MODULE VIII Lecture Indicator Variables LINEAR REGRESSION ANALYSIS MODULE VIII Lecture - 7 Indcator Varables Dr. Shalabh Department of Maematcs and Statstcs Indan Insttute of Technology Kanpur Indcator varables versus quanttatve explanatory

More information

Chapter Newton-Raphson Method of Solving a Nonlinear Equation

Chapter Newton-Raphson Method of Solving a Nonlinear Equation Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson

More information

PROBABILISTIC RESTORATION DESIGN FOR RC SLAB BRIDGES

PROBABILISTIC RESTORATION DESIGN FOR RC SLAB BRIDGES - Tehnl Pper - PROBABILISTIC RSTORATION DSIGN FOR RC SLAB BRIDGS Abrhm Gebre TARKGN *, Ttu TSUBAKI * ABSTRACT In e o oler brge, eng muh egn normton poble ult. Retorton egn n mportnt metho to etmte the

More information

Rank One Update And the Google Matrix by Al Bernstein Signal Science, LLC

Rank One Update And the Google Matrix by Al Bernstein Signal Science, LLC Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses

More information

International Journal of Pure and Applied Sciences and Technology

International Journal of Pure and Applied Sciences and Technology Int. J. Pure Appl. Sc. Technol., () (), pp. 44-49 Interntonl Journl of Pure nd Appled Scences nd Technolog ISSN 9-67 Avlle onlne t www.jopst.n Reserch Pper Numercl Soluton for Non-Lner Fredholm Integrl

More information

Harmonic oscillator approximation

Harmonic oscillator approximation armonc ocllator approxmaton armonc ocllator approxmaton Euaton to be olved We are fndng a mnmum of the functon under the retrcton where W P, P,..., P, Q, Q,..., Q P, P,..., P, Q, Q,..., Q lnwgner functon

More information

Team. Outline. Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference

Team. Outline. Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference Team Stattc and Art: Samplng, Repone Error, Mxed Model, Mng Data, and nference Ed Stanek Unverty of Maachuett- Amhert, USA 9/5/8 9/5/8 Outlne. Example: Doe-repone Model n Toxcology. ow to Predct Realzed

More information

Applied Statistics Qualifier Examination

Applied Statistics Qualifier Examination Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng

More information

Introduction to Robotics (Fag 3480)

Introduction to Robotics (Fag 3480) Intouton to Robot (Fg 8) Vå Robet Woo (Hw Engneeeng n pple Sene-B) Ole Jkob Elle PhD (Mofe fo IFI/UIO) Føtemnuen II Inttutt fo Infomtkk Unvetetet Olo Sekjonlee Teknolog Intevenjonenteet Olo Unvetetkehu

More information

Many-Body Calculations of the Isotope Shift

Many-Body Calculations of the Isotope Shift Mny-Body Clcultons of the Isotope Shft W. R. Johnson Mrch 11, 1 1 Introducton Atomc energy levels re commonly evluted ssumng tht the nucler mss s nfnte. In ths report, we consder correctons to tomc levels

More information

TELCOM 2130 Time Varying Queues. David Tipper Associate Professor Graduate Telecommunications and Networking Program University of Pittsburgh Slides 7

TELCOM 2130 Time Varying Queues. David Tipper Associate Professor Graduate Telecommunications and Networking Program University of Pittsburgh Slides 7 TELOM 3 Tme Vryng Queues Dvd Tpper Assote Professor Grdute Teleommuntons nd Networkng Progrm Unversty of Pttsburgh ldes 7 Tme Vryng Behvor Teletrff typlly hs lrge tme of dy vrtons Men number of lls per

More information

Two Approaches to Proving. Goldbach s Conjecture

Two Approaches to Proving. Goldbach s Conjecture Two Approache to Provng Goldbach Conecture By Bernard Farley Adved By Charle Parry May 3 rd 5 A Bref Introducton to Goldbach Conecture In 74 Goldbach made h mot famou contrbuton n mathematc wth the conecture

More information

2.3 Least-Square regressions

2.3 Least-Square regressions .3 Leat-Square regreon Eample.10 How do chldren grow? The pattern of growth vare from chld to chld, o we can bet undertandng the general pattern b followng the average heght of a number of chldren. Here

More information

x = , so that calculated

x = , so that calculated Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to

More information

Shuai Dong. Using Math and Science to improve your game

Shuai Dong. Using Math and Science to improve your game Computtonl phscs Shu Dong Usng Mth nd Sene to mprove our gme Appromton of funtons Lner nterpolton Lgrnge nterpolton Newton nterpolton Lner sstem method Lest-squres ppromton Mllkn eperment Wht s nterpolton?

More information

INTERPOLATION(1) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek

INTERPOLATION(1) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek ELM Numercl Anlss Dr Muhrrem Mercmek INTEPOLATION ELM Numercl Anlss Some of the contents re dopted from Lurene V. Fusett, Appled Numercl Anlss usng MATLAB. Prentce Hll Inc., 999 ELM Numercl Anlss Dr Muhrrem

More information

E-Companion: Mathematical Proofs

E-Companion: Mathematical Proofs E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth

More information

Chapter 2 Organizing and Summarizing Data. Chapter 3 Numerically Summarizing Data. Chapter 4 Describing the Relation between Two Variables

Chapter 2 Organizing and Summarizing Data. Chapter 3 Numerically Summarizing Data. Chapter 4 Describing the Relation between Two Variables Copyright 013 Peron Eduction, Inc. Tble nd Formul for Sullivn, Sttitic: Informed Deciion Uing Dt 013 Peron Eduction, Inc Chpter Orgnizing nd Summrizing Dt Reltive frequency = frequency um of ll frequencie

More information

Problem #1. Known: All required parameters. Schematic: Find: Depth of freezing as function of time. Strategy:

Problem #1. Known: All required parameters. Schematic: Find: Depth of freezing as function of time. Strategy: BEE 3500 013 Prelm Soluton Problem #1 Known: All requred parameter. Schematc: Fnd: Depth of freezng a functon of tme. Strategy: In thee mplfed analy for freezng tme, a wa done n cla for a lab geometry,

More information

Quick Visit to Bernoulli Land

Quick Visit to Bernoulli Land Although we have een the Bernoull equaton and een t derved before, th next note how t dervaton for an uncopreble & nvcd flow. The dervaton follow that of Kuethe &Chow ot cloely (I lke t better than Anderon).

More information

17 Nested and Higher Order Designs

17 Nested and Higher Order Designs 54 17 Nested and Hgher Order Desgns 17.1 Two-Way Analyss of Varance Consder an experment n whch the treatments are combnatons of two or more nfluences on the response. The ndvdual nfluences wll be called

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

523 P a g e. is measured through p. should be slower for lesser values of p and faster for greater values of p. If we set p*

523 P a g e. is measured through p. should be slower for lesser values of p and faster for greater values of p. If we set p* R. Smpth Kumr, R. Kruthk, R. Rdhkrshnn / Interntonl Journl of Engneerng Reserch nd Applctons (IJERA) ISSN: 48-96 www.jer.com Vol., Issue 4, July-August 0, pp.5-58 Constructon Of Mxed Smplng Plns Indexed

More information

Al-Zangana Iraqi Journal of Science, 2016, Vol. 57, No.2A, pp:

Al-Zangana Iraqi Journal of Science, 2016, Vol. 57, No.2A, pp: Results n Projetve Geometry PG( r,), r, Emd Bkr Al-Zngn* Deprtment of Mthemts, College of Sene, Al-Mustnsryh Unversty, Bghdd, Ir Abstrt In projetve plne over fnte feld F, on s the unue omplete ( ) r nd

More information

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I lear regreo, we coder the frequecy dtrbuto of oe varable (Y) at each of everal level of a ecod varable (X). Y kow a the depedet varable. The

More information

ψ ij has the eigenvalue

ψ ij has the eigenvalue Moller Plesset Perturbton Theory In Moller-Plesset (MP) perturbton theory one tes the unperturbed Hmltonn for n tom or molecule s the sum of the one prtcle Foc opertors H F() where the egenfunctons of

More information

Physical Optics Driven Method of Moments Based on Adaptive Grouping Technique

Physical Optics Driven Method of Moments Based on Adaptive Grouping Technique December, 0 Mcrowve evew Phycl Optc Drven Method of Moment Bed on Adptve Groupng Technque Modrg S. Tć, Brno M. Kolundž Abtrct ecently we ntroduced new tertve method for nlyng lrge perfectly conductng ctterer,

More information

Chemical Reaction Engineering

Chemical Reaction Engineering Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F 0 E 0 F E Q W

More information

a = Acceleration Linear Motion Acceleration Changing Velocity All these Velocities? Acceleration and Freefall Physics 114

a = Acceleration Linear Motion Acceleration Changing Velocity All these Velocities? Acceleration and Freefall Physics 114 Lner Accelerton nd Freell Phyc 4 Eyre Denton o ccelerton Both de o equton re equl Mgntude Unt Drecton (t ector!) Accelerton Mgntude Mgntude Unt Unt Drecton Drecton 4/3/07 Module 3-Phy4-Eyre 4/3/07 Module

More information

ˆ A = A 0 e i (k r ωt) + c.c. ( ωt) e ikr. + c.c. k,j

ˆ A = A 0 e i (k r ωt) + c.c. ( ωt) e ikr. + c.c. k,j p. Supp. 9- Suppleent to Rate of Absorpton and Stulated Esson Here are a ouple of ore detaled dervatons: Let s look a lttle ore arefully at the rate of absorpton w k ndued by an sotrop, broadband lght

More information

Not at Steady State! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Yes! Class 15. Is the following possible?

Not at Steady State! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Yes! Class 15. Is the following possible? Chapter 5-6 (where we are gong) Ideal gae and lqud (today) Dente Partal preure Non-deal gae (next tme) Eqn. of tate Reduced preure and temperature Compreblty chart (z) Vapor-lqud ytem (Ch. 6) Vapor preure

More information

Chemical Reaction Engineering

Chemical Reaction Engineering Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F E F E + Q! 0

More information

3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4

3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4 // Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,,.... - - - Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples

More information

No! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Survey Results. Class 15. Is the following possible?

No! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Survey Results. Class 15. Is the following possible? Survey Reult Chapter 5-6 (where we are gong) % of Student 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% Hour Spent on ChE 273 1-2 3-4 5-6 7-8 9-10 11+ Hour/Week 2008 2009 2010 2011 2012 2013 2014 2015 2017 F17

More information

Electrochemical Thermodynamics. Interfaces and Energy Conversion

Electrochemical Thermodynamics. Interfaces and Energy Conversion CHE465/865, 2006-3, Lecture 6, 18 th Sep., 2006 Electrochemcl Thermodynmcs Interfces nd Energy Converson Where does the energy contrbuton F zϕ dn come from? Frst lw of thermodynmcs (conservton of energy):

More information

A Study on Root Properties of Super Hyperbolic GKM algebra

A Study on Root Properties of Super Hyperbolic GKM algebra Stuy on Root Popetes o Supe Hypebol GKM lgeb G.Uth n M.Pyn Deptment o Mthemts Phypp s College Chenn Tmlnu In. bstt: In ths ppe the Supe hypebol genelze K-Mooy lgebs o nente type s ene n the mly s lso elte.

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

PHYS 2421 Fields and Waves

PHYS 2421 Fields and Waves PHYS 242 Felds nd Wves Instucto: Joge A. López Offce: PSCI 29 A, Phone: 747-7528 Textook: Unvesty Physcs e, Young nd Feedmn 23. Electc potentl enegy 23.2 Electc potentl 23.3 Clcultng electc potentl 23.4

More information

Haddow s Experiment:

Haddow s Experiment: schemtc drwng of Hddow's expermentl set-up movng pston non-contctng moton sensor bems of sprng steel poston vres to djust frequences blocks of sold steel shker Hddow s Experment: terr frm Theoretcl nd

More information

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n 0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke

More information

Iowa Training Systems Trial Snus Hill Winery Madrid, IA

Iowa Training Systems Trial Snus Hill Winery Madrid, IA Iow Trining Systems Tril Snus Hill Winery Mdrid, IA Din R. Cohrn nd Gil R. Nonneke Deprtment of Hortiulture, Iow Stte University Bkground nd Rtionle: Over the lst severl yers, five sttes hve een evluting

More information

Trigonometry. Trigonometry. Solutions. Curriculum Ready ACMMG: 223, 224, 245.

Trigonometry. Trigonometry. Solutions. Curriculum Ready ACMMG: 223, 224, 245. Trgonometry Trgonometry Solutons Currulum Redy CMMG:, 4, 4 www.mthlets.om Trgonometry Solutons Bss Pge questons. Identfy f the followng trngles re rght ngled or not. Trngles,, d, e re rght ngled ndted

More information

x=0 x=0 Positive Negative Positions Positions x=0 Positive Negative Positions Positions

x=0 x=0 Positive Negative Positions Positions x=0 Positive Negative Positions Positions Knemtc Quntte Lner Moton Phyc 101 Eyre Tme Intnt t Fundmentl Tme Interl Defned Poton x Fundmentl Dplcement Defned Aerge Velocty g Defned Aerge Accelerton g Defned Knemtc Quntte Scler: Mgntude Tme Intnt,

More information

and decompose in cycles of length two

and decompose in cycles of length two Permutaton of Proceedng of the Natona Conference On Undergraduate Reearch (NCUR) 006 Domncan Unverty of Caforna San Rafae, Caforna Apr - 4, 007 that are gven by bnoma and decompoe n cyce of ength two Yeena

More information

STK4900/ Lecture 4 Program. Counterfactuals and causal effects. Example (cf. practical exercise 10)

STK4900/ Lecture 4 Program. Counterfactuals and causal effects. Example (cf. practical exercise 10) STK4900/9900 - Leture 4 Program 1. Counterfatuals and ausal effets 2. Confoundng 3. Interaton 4. More on ANOVA Setons 4.1, 4.4, 4.6 Supplementary materal on ANOVA Example (f. pratal exerse 10) How does

More information

Online Appendix to. Mandating Behavioral Conformity in Social Groups with Conformist Members

Online Appendix to. Mandating Behavioral Conformity in Social Groups with Conformist Members Onlne Appendx to Mndtng Behvorl Conformty n Socl Groups wth Conformst Members Peter Grzl Andrze Bnk (Correspondng uthor) Deprtment of Economcs, The Wllms School, Wshngton nd Lee Unversty, Lexngton, 4450

More information

Chapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters

Chapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters Chapter 6 The Effect of the GPS Sytematc Error on Deformaton Parameter 6.. General Beutler et al., (988) dd the frt comprehenve tudy on the GPS ytematc error. Baed on a geometrc approach and aumng a unform

More information

Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212. Chapters 14, 15 & 16. Professor Ahmadi, Ph.D. Department of Management

Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212. Chapters 14, 15 & 16. Professor Ahmadi, Ph.D. Department of Management Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 1 Chapters 14, 15 & 16 Professor Ahmad, Ph.D. Department of Management Revsed August 005 Chapter 14 Formulas Smple Lnear Regresson Model: y =

More information

Kinematics Quantities. Linear Motion. Coordinate System. Kinematics Quantities. Velocity. Position. Don t Forget Units!

Kinematics Quantities. Linear Motion. Coordinate System. Kinematics Quantities. Velocity. Position. Don t Forget Units! Knemtc Quntte Lner Phyc 11 Eyre Tme Intnt t Fundmentl Tme Interl t Dened Poton Fundmentl Dplcement Dened Aerge g Dened Aerge Accelerton g Dened Knemtc Quntte Scler: Mgntude Tme Intnt, Tme Interl nd Speed

More information

VECTORS VECTORS VECTORS VECTORS. 2. Vector Representation. 1. Definition. 3. Types of Vectors. 5. Vector Operations I. 4. Equal and Opposite Vectors

VECTORS VECTORS VECTORS VECTORS. 2. Vector Representation. 1. Definition. 3. Types of Vectors. 5. Vector Operations I. 4. Equal and Opposite Vectors 1. Defnton A vetor s n entt tht m represent phsl quntt tht hs mgntude nd dreton s opposed to slr tht ls dreton.. Vetor Representton A vetor n e represented grphll n rrow. The length of the rrow s the mgntude

More information

Multivariate Time Series Analysis

Multivariate Time Series Analysis Mulvre me Sere Anl Le { : } be Mulvre me ere. Denon: () = men vlue uncon o { : } = E[ ] or. (,) = Lgged covrnce mr o { : } = E{[ - ()][ - ()]'} or, Denon: e me ere { : } onr e jon drbuon o,,, e me e jon

More information

Modeling uncertainty using probabilities

Modeling uncertainty using probabilities S 1571 Itroduto to I Leture 23 Modelg uertty usg probbltes Mlos Huskreht mlos@s.ptt.edu 5329 Seott Squre dmstrto Fl exm: Deember 11 2006 12:00-1:50pm 5129 Seott Squre Uertty To mke dgost feree possble

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed

More information

The corresponding link function is the complementary log-log link The logistic model is comparable with the probit model if

The corresponding link function is the complementary log-log link The logistic model is comparable with the probit model if SK300 and SK400 Lnk funtons for bnomal GLMs Autumn 08 We motvate the dsusson by the beetle eample GLMs for bnomal and multnomal data Covers the followng materal from hapters 5 and 6: Seton 5.6., 5.6.3,

More information

4. Eccentric axial loading, cross-section core

4. Eccentric axial loading, cross-section core . Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we

More information

Polynomial Regression Models

Polynomial Regression Models LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance

More information

" = #N d$ B. Electromagnetic Induction. v ) $ d v % l. Electromagnetic Induction and Faraday s Law. Faraday s Law of Induction

 = #N d$ B. Electromagnetic Induction. v ) $ d v % l. Electromagnetic Induction and Faraday s Law. Faraday s Law of Induction Eletromgnet Induton nd Frdy s w Eletromgnet Induton Mhel Frdy (1791-1867) dsoered tht hngng mgnet feld ould produe n eletr urrent n ondutor pled n the mgnet feld. uh urrent s lled n ndued urrent. The phenomenon

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10 University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

More information

DCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)

DCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x) DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng

More information

Systems of Equations (SUR, GMM, and 3SLS)

Systems of Equations (SUR, GMM, and 3SLS) Lecture otes on Advanced Econometrcs Takash Yamano Fall Semester 4 Lecture 4: Sstems of Equatons (SUR, MM, and 3SLS) Seemngl Unrelated Regresson (SUR) Model Consder a set of lnear equatons: $ + ɛ $ + ɛ

More information

Electromagnetism Notes, NYU Spring 2018

Electromagnetism Notes, NYU Spring 2018 Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system

More information

GRADE 4. Division WORKSHEETS

GRADE 4. Division WORKSHEETS GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.

More information

Model Fitting and Robust Regression Methods

Model Fitting and Robust Regression Methods Dertment o Comuter Engneerng Unverst o Clorn t Snt Cruz Model Fttng nd Robust Regresson Methods CMPE 64: Imge Anlss nd Comuter Vson H o Fttng lnes nd ellses to mge dt Dertment o Comuter Engneerng Unverst

More information

The multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted

The multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted Appendx Proof of heorem he multvarate Gauan probablty denty functon for random vector X (X,,X ) px exp / / x x mean and varance equal to the th dagonal term of, denoted he margnal dtrbuton of X Gauan wth

More information

M/G/1/GD/ / System. ! Pollaczek-Khinchin (PK) Equation. ! Steady-state probabilities. ! Finding L, W q, W. ! π 0 = 1 ρ

M/G/1/GD/ / System. ! Pollaczek-Khinchin (PK) Equation. ! Steady-state probabilities. ! Finding L, W q, W. ! π 0 = 1 ρ M/G//GD/ / System! Pollcze-Khnchn (PK) Equton L q 2 2 λ σ s 2( + ρ ρ! Stedy-stte probbltes! π 0 ρ! Fndng L, q, ) 2 2 M/M/R/GD/K/K System! Drw the trnston dgrm! Derve the stedy-stte probbltes:! Fnd L,L

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

More information