# Student Activity 3: Single Factor ANOVA

Size: px
Start display at page:

Transcription

1 MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether more thn one tretment is pplied to n experimentl unit (ie if tretments re given in combintion) nd if so how mny tretments re given in combintion, whether ll possible tretment combintions re used or not decides the nture of experiment Such informtion pertining to tretments is clled the TREATMENT STRUCTURE of the experiment To minimize systemtic bis, the tretments (or tretment combintions) re usully pplied ccording to rndomiztion scheme The rndomiztion used nd how tretments or tretment combintions re pplied to experimentl units will determine the nture of the design Informtion regrding the rndomiztion scheme used nd how tretments or tretment combintions re pplied to experimentl units is clled the DESIGN STRUCTURE of the experiment ONE-WAY TREATMENT STRUCTURE WITH COMPLETELY RANDOMIZED DESIGN STRUCTURE In short One-wy nlysis in CR design eg: Suppose we hve 4 blood pressure medicines plus plcebo They re to be given on one-drug-to-one ptient mnner So we hve one wy tretment structure In order to hve n estimte of ptient to ptient vrition, ech drug will be given to ten ptients We sy tht ech tretment will be replicted 10 times Suppose we select 50 ptients from the trget popultion (sy hypertensive mles between ges 45-55) rndomly Now rndomly select 10 of the 50 nd give them drug 1, chose nother 10 from the remining 40 nd give the drug, etc Here the ptients nd ssigned to the drugs in completely rndom fshion So we hve CR design structure

2 MATH 40 The Generl One-Wy Model Equl number of replictions Equl subclss numbers Blnced cse Suppose we hve "" tretments nd ech tretment is pplied to n experimentl units in CR design We now mesure the responses Y ij of experimentl units to tretments Suppose n pproprite model to describe the Y ij is Y ij i ij i 1,,,, j 1,,n, where ij ~ iid N0, Hence, the model implies tht Y ij independentn ~ i, i is clled the popultion men for the i th tretment, i 1,, (1) given bove is clled the mens model In contrst, the clssicl model is Y ij i ij where i j ~ iid N0, The i 's re clled the tretment effects nd is considered fixed effect (ie the tretment levels re specificlly chosen by the experimenter nd cn be replicted exctly ny number of times) Note tht i i One hypothesis the experimenter my wish to test is This is equivlent to testing : 1 3 vs H : t lest one i differs from the rest : 1 vs H : not The model () bove is over prmeterized (ie hs too mny prmeters so tht no single prmeter cn be estimted uniquely)

3 MATH 40 Thus, we enforce the restriction i i 1 0 With this restriction the hypotheses for the clssicl model becomes : 1 0 vs H : t lest one i is non-zero In order to test the bove hypotheses, we compute the following sums of squres n SS totl Y ij Y i 1 j 1 where Y Y / N, Y nd N n n Y ij i 1 j 1 Also compute nd compute SS tretment n Y i Y where Y i 1 n SS Error It cn be shown tht n i 1 j1 i 1 n Y ij i1 Y ij Y i, i 1,,,n SS Totl SS Tretment SS Error Thus, usully only SS Totl nd SS Tretment re computed nd then SS Error is obtined by subtrction Computtion is speeded up by using the following formuls insted of the ones given previously: n SST y NY i1 j1 ij i1 SStretment n Y NY i

4 MATH 40 Liner Model Theory cn be used to show tht SS Tretment ~ x 1 with non-centrlity prmeter, which becomes zero if nd only if is true Further, SS Error ~ x N with zero non-centrlity prmeter Moreover, SS Tretment nd SS Error re independent nd thus, under, F MS Tretment / MS Error ~ F 1,N, where MS Tretment SS Tretment / 1 nd MS Error SS Error / N If is flse, the bove F sttistic will hve non-centrl F distribution with the sme degrees of freedom nd hence will tend to be lrger thn centrl F rndom vrible Thus, n pproprite test for testing : 1 vs H : not is: Reject Ho t significnce level if F F, 1, N The results re usully summrized in n nlysis of vrince tble s follows: ANOVA Source of Vrition df SS MS F Tretments 1 SS Tretment MS Tretment MS Tretment MS Error Error N SS Error MS Error Totl (djusted for the men) N 1 SS Totl MS Totl Note: MS Totl SS Totl / N 1 Remrk: A computer output would lwys give the p -vlue of the F -test

5 MATH 40 An Exmple of Anlysis of Vrince of Dt from Blnced One-wy Experiment conducted in Completely Rndomized Design The wether resistnce of four pints ws tested by crrying out the following experiment Twenty metl pnels were rndomly selected from stockpile nd their surfces were prepred for pinting From these, four pnels were rndomly selected nd primer ws pplied to them The sme primer ws pplied to the other sixteen pnels, but unlike the first four, ech of these were then pinted with one of the four pints Specificlly, ech pint ws pplied to four rndomly selected pnels The primer coted but unpinted pnels s well s the pinted pnels were then rndomly lid out in n re open to the elements After two yers, the pnels were gthered nd the cumultive re of rust in ech pnel ws mesured (in squre mm) The resulting dt is s follows: Primer only Ltex A Oil Bsed I Oil bsed II Oil bsed III We will now construct the ANOVA tble for the bove experiment Note tht Y 11 =0, Y 1 =40,, Y 14 =40,, Y 54 =10 Y1 = 5, Y = 145, Y3 = 135, Y4 = 10, Y5 = 1175, Y = Yij =475,100 i1 j1 So, SST= 475,100 N(1485) = 475,100 0(1485) = 34,0550 Note tht we used formul tht looks different from wht is given in notes, but this is equivlent to tht formul 5 Also, SSTretment = n( Yi ) N(1485) = 4(118,0815) - 0(1485) i1 = 31,800 Agin, note tht we used formul tht looks different from wht is given in notes, but this is equivlent to tht formul Now, by subtrction, SSE = SST SSTretment = 34, ,800 = 7750

6 MATH 40 ANOVA Source of Vrition DF Sum of Squres Men Squres F rtio Tretment 4 31,800 7, Error Totl 19 34,0550 1,7937 Now let us test the hypothesis of no tretment differences ginst the lterntive tht t lest one tretment men is different from the others t 005 significnce level From F-tbles, F005, 4, Since 4707 > 30556, we reject the null hypothesis of no tretment difference nd conclude tht t lest one tretment men is different from the others Plese note tht in computing SSTotl we cn use either of two formuls n (, ) n n Yi j i1 j1 i, j i, j i1 j1 i1 j1 N SST Y NY Y Note tht NY = n ( Y ) i1 j1 N i, j Similrly, we hve two choices in computing SSTretment n (, ) Yi j i1 j1 i n i i1 i1 N SSTretment n Y NY Y

7 MATH 40 Running the Single Fctor Anlysis of Vrince on SAS Suppose tht you wnt to determine whether the men of the vrible rust vries by different type of pint You cn use PROC GLM (generl liner model) to conduct one-wy ANOVA In ddition, you cn request summry nd dignostic plots using ODS Grphics Here is the progrm: options ls=78; dt pint; input pint \$ rust; dtlines; primer 0 primer 40 primer 00 primer 40 ltex 150 ltex 130 ltex 140 ltex 160 oil1 150 oil1 140 oil1 130 oil1 10 oil 130 oil 10 oil 10 oil 110 oil3 10 oil3 130 oil3 100 oil3 10 ; proc print dt= pint; run; ods grphics on; title "Running one-wy ANOVA"; proc glm dt=pint plots=dignostics; clss pint; model rust = pint/solution ss3 ss1; mens pint/hovtest; run; Remrk: 1 you specified the independent vrible on CLASS sttement you use MODEL sttement to specify your dependent vrible s well s your design

8 MATH 40 3 The MEANS sttement tht follows the MODEL sttement requests tht PROC GLM print the men nd stndrd devition for your dependent vrible for ech vlue of your independent vrible The HOVTEST option on the MEANS sttement requests tht SAS perform Levene s test for homogeneity of vrince 4 The order of sttements within procedure usully does not mtter However, in PROC GLM, you must specify your CLASS sttement before your MODEL sttement nd your MODEL sttement before the MEANS sttement (or before other sttements such s lest squres mens or contrsts) 5 By defult, PROC GLM produces both Type I nd Type III sum of squres (SS) s output Type I SS shows the effect of ech fctor s it is entered into the design; type III SS shows the effect of ech fctor, controlling for ll the other fctors They re the sme in the one fctor cse You cn specify the SS3 in the option if you only wnt to see Type III SS Plese interpret the bove SAS output

9 MATH 40 Assignment: 1 Redo problem 33 on textbook Plese mke sure to include the SAS code nd output Four different designs for digitl computer circuit re being studied in order to compre the mount of noise present The following dt hve been obtined () Is the sme mount of noise present for ll four designs? (b) Anlyze the resuduls from this experiment Are the nlysis of vrince ssumptions stisfied? (c) Use Tukey s test to compre pirs of tretment mens Wht is your conclusion? (d) Use LSD method to compre pirs of tretment mens Wht is your conclusion? (e) Construct set of orthogonl contrsts (f) Compre the tretments mens of circuit design 1 to design, 3, nd 4 (g) Compre the tretments mens of circuit design 4 to design 1 nd 3 A phrmceuticl mnufcturer wnts to investigte the bioctivity of new drug A completely rndomized single-fctor experiment ws conducted with three dosge levels, nd the following results were obtined (Plese do this problem without SAS) () Construct the ANOVA tble (b) Is there evidence to indicte tht dosge level ffects bioctivity? Why? (c) Set the µ1 s the popultion men for dosge 0g, µ s the popultion men for dosge 30g, nd µ3 s the popultion men for dosge 40g Use t-test to compre the 0g nd 30g tretment mens Tht is, to test H0: µ1= µ (d) Use either t-test or F-test for testing dosge 40g versus dosge 0g nd 30g Tht is, to test H0: µ1 + µ - *µ3=0 (e) Construct set of orthogonl contrsts, nd compute the sums of squres for ech tretment (f) Use Scheffe s method to compre the tretment mens between 30g nd 40g dosge (g) Use Tukey s method to compre the tretment mens between 30g nd 40g dosge (h) Use LSD method to compre the tretment mens between 30g nd 40g dosge (i) Assume tht dosge 0g is control tretment, pply Dunnett s test to compre tretment mens with the control

### Tests for the Ratio of Two Poisson Rates

Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson

### Chapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses

Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of

### For the percentage of full time students at RCC the symbols would be:

Mth 17/171 Chpter 7- ypothesis Testing with One Smple This chpter is s simple s the previous one, except it is more interesting In this chpter we will test clims concerning the sme prmeters tht we worked

### MIXED MODELS (Sections ) I) In the unrestricted model, interactions are treated as in the random effects model:

1 2 MIXED MODELS (Sections 17.7 17.8) Exmple: Suppose tht in the fiber breking strength exmple, the four mchines used were the only ones of interest, but the interest ws over wide rnge of opertors, nd

### Comparison Procedures

Comprison Procedures Single Fctor, Between-Subects Cse /8/ Comprison Procedures, One-Fctor ANOVA, Between Subects Two Comprison Strtegies post hoc (fter-the-fct) pproch You re interested in discovering

### The steps of the hypothesis test

ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of

### Non-Linear & Logistic Regression

Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### Lecture INF4350 October 12008

Biosttistics ti ti Lecture INF4350 October 12008 Anj Bråthen Kristoffersen Biomedicl Reserch Group Deprtment of informtics, UiO Gol Presenttion of dt descriptive tbles nd grphs Sensitivity, specificity,

### 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

### 14.3 comparing two populations: based on independent samples

Chpter4 Nonprmetric Sttistics Introduction: : methods for mking inferences bout popultion prmeters (confidence intervl nd hypothesis testing) rely on the ssumptions bout probbility distribution of smpled

### Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

### Chapter 6 Continuous Random Variables and Distributions

Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete

### Design and Analysis of Single-Factor Experiments: The Analysis of Variance

13 CHAPTER OUTLINE Design nd Anlysis of Single-Fctor Experiments: The Anlysis of Vrince 13-1 DESIGNING ENGINEERING EXPERIMENTS 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 13-2.1 An Exmple 13-2.2

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

### Multivariate problems and matrix algebra

University of Ferrr Stefno Bonnini Multivrite problems nd mtrix lgebr Multivrite problems Multivrite sttisticl nlysis dels with dt contining observtions on two or more chrcteristics (vribles) ech mesured

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### Lecture 7 notes Nodal Analysis

Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions

### Vyacheslav Telnin. Search for New Numbers.

Vycheslv Telnin Serch for New Numbers. 1 CHAPTER I 2 I.1 Introduction. In 1984, in the first issue for tht yer of the Science nd Life mgzine, I red the rticle "Non-Stndrd Anlysis" by V. Uspensky, in which

### Chapter 1: Fundamentals

Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,

### Chapter 5 : Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll

### Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of

### List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

### Conservation Law. Chapter Goal. 5.2 Theory

Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very

### 4.1. Probability Density Functions

STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

### Math 1B, lecture 4: Error bounds for numerical methods

Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

### New Expansion and Infinite Series

Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

### Response aodels and aini.al designs for mixtures of n of m iteas. useful for intercropping and other investigations

Response odels nd ini.l designs for mixtures of n of m ites useful for intercropping nd oer investigtions BY W. T. FEDERER B.iomet:r.ics lln.it:, Cornell lln.ivers.it:y, It:hc, NY 14853, ll.s.a. AND D.

### Theoretical foundations of Gaussian quadrature

Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

### SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

### 7.2 The Definite Integral

7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

### CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

### 13: Diffusion in 2 Energy Groups

3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

### Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading

Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method

### Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

### 1 Probability Density Functions

Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### 8 Laplace s Method and Local Limit Theorems

8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

### Arithmetic & Algebra. NCTM National Conference, 2017

NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

### Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.

Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we

### Lecture 3 Gaussian Probability Distribution

Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil

### Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 17

CS 70 Discrete Mthemtics nd Proility Theory Summer 2014 Jmes Cook Note 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion, y tking

### Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17

EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,

### Genetic Programming. Outline. Evolutionary Strategies. Evolutionary strategies Genetic programming Summary

Outline Genetic Progrmming Evolutionry strtegies Genetic progrmming Summry Bsed on the mteril provided y Professor Michel Negnevitsky Evolutionry Strtegies An pproch simulting nturl evolution ws proposed

### Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

### 1. Extend QR downwards to meet the x-axis at U(6, 0). y

In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

### NUMERICAL INTEGRATION

NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls

### 3.1 Review of Sine, Cosine and Tangent for Right Angles

Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

### Consolidation Worksheet

Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is

### Credibility Hypothesis Testing of Fuzzy Triangular Distributions

666663 Journl of Uncertin Systems Vol.9, No., pp.6-74, 5 Online t: www.jus.org.uk Credibility Hypothesis Testing of Fuzzy Tringulr Distributions S. Smpth, B. Rmy Received April 3; Revised 4 April 4 Abstrct

### How do you know you have SLE?

Simultneous Liner Equtions Simultneous Liner Equtions nd Liner Algebr Simultneous liner equtions (SLE s) occur frequently in Sttics, Dynmics, Circuits nd other engineering clsses Need to be ble to, nd

### Solution for Assignment 1 : Intro to Probability and Statistics, PAC learning

Solution for Assignment 1 : Intro to Probbility nd Sttistics, PAC lerning 10-701/15-781: Mchine Lerning (Fll 004) Due: Sept. 30th 004, Thursdy, Strt of clss Question 1. Bsic Probbility ( 18 pts) 1.1 (

### Improper Integrals, and Differential Equations

Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

### AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

### Orthogonal Polynomials

Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils

### Bases for Vector Spaces

Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

### A Matrix Algebra Primer

A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,

### Problem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume

Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom r vrible. Define new rndom vrible Y = g. Find the pdf of Y. Method: Step : Step : Step 3: Plot Y = g( ). Find F ( y) by mpping

### I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

### Physics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018

Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.

### 1 Online Learning and Regret Minimization

2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in

### Acceptance Sampling by Attributes

Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

### Numerical integration

2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

### Experiments with a Single Factor: The Analysis of Variance (ANOVA) Dr. Mohammad Abuhaiba 1

Experiments with Single Fctor: The Anlsis of Vrince (ANOVA) Dr. Mohmmd Abuhib 1 Wht If There Are More Thn Two Fctor Levels? The t-test does not directl ppl There re lots of prcticl situtions where there

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

### Equations and Inequalities

Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

### Elementary Linear Algebra

Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร

### Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

### Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

### UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3

UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,

### Best Approximation. Chapter The General Case

Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

### 8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims

### Recitation 3: More Applications of the Derivative

Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

### Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

### 5 Probability densities

5 Probbility densities 5. Continuous rndom vribles 5. The norml distribution 5.3 The norml pproimtion to the binomil distribution 5.5 The uniorm distribution 5. Joint distribution discrete nd continuous

### Chapter 14. Matrix Representations of Linear Transformations

Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

### The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F

### Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014

Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method

### Session Trimester 2. Module Code: MATH08001 MATHEMATICS FOR DESIGN

School of Science & Sport Pisley Cmpus Session 05-6 Trimester Module Code: MATH0800 MATHEMATICS FOR DESIGN Dte: 0 th My 06 Time: 0.00.00 Instructions to Cndidtes:. Answer ALL questions in Section A. Section

### DIRECT CURRENT CIRCUITS

DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

### Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

### W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

### CS 188 Introduction to Artificial Intelligence Fall 2018 Note 7

CS 188 Introduction to Artificil Intelligence Fll 2018 Note 7 These lecture notes re hevily bsed on notes originlly written by Nikhil Shrm. Decision Networks In the third note, we lerned bout gme trees

### ( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

### The Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases.

Section 1 Section 2 The module clcultes nd plots isotherml 1-, 2- nd 3-metl predominnce re digrms. ccesses only compound dtbses. Tble of Contents Tble of Contents Opening the module Section 3 Stoichiometric

### 1.9 C 2 inner variations

46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for

### Chapter 2. Determinants

Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is

### Method: Step 1: Step 2: Find f. Step 3: = Y dy. Solution: 0, ( ) 0, y. Assume

Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom vrible. Define new rndom vrible Y g( ) ). Find the pdf of Y. Method: Step : Step : Step 3: Plot Y g( ). Find F ( ) b mpping

### MAA 4212 Improper Integrals

Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

### Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.

Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number

### Administrivia CSE 190: Reinforcement Learning: An Introduction

Administrivi CSE 190: Reinforcement Lerning: An Introduction Any emil sent to me bout the course should hve CSE 190 in the subject line! Chpter 4: Dynmic Progrmming Acknowledgment: A good number of these

### THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES

THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,

### Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order: