14.3 comparing two populations: based on independent samples

Size: px
Start display at page:

Download "14.3 comparing two populations: based on independent samples"

Transcription

1 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 popultion. (t-test, ANOVA) : sttisticl tests tht do not rely on ny underlying ssumptions bout the probbility distribution of the smpled popultion. : the brnch of inferentil sttistics devoted to distribution-free tests. : nonprmetric tests (or sttistics) bsed on the rnks of mesurements. 4.3 compring two popultions: bsed on independent smples When t-test is not pproprite, we cn use nonprmetric methods to compre two popultions. : compring the loction of center for two or more non-norml popultions. It tests the hypothesis tht the probbility distributions ssocited with the two popultions re equivlent. Wilcoxon rnk sum test: Define : the for popultion, D : the for popultion. Three reltionship between D nd D :. ypothesis: : : ( is shifted to the of D ) ( is shifted to the of D )

2 . Test sttistic: Step. ll the dt vlues (both smples) from to N, ( N = n+ n, n < n) (Note: ssign the of the rnks to ech of the tied observtions) Step. Test sttistict = Sum the rnks for the ( n ). 3. Rejection region: if nd D re not identicl; if is shifted to the left of D ; if is shifted to the right of D. Criticl vlues TL nd T U re from Tble XII, p87. Condition required for vlid rnk sum test:. the two smples re ;. the two probbility distributions from which the smples re drwn re.

3 ExmpleRUGS: At α =.5, Do the dt provide sufficient evidence to indicte shift in the probbility distributions for drug A nd B? Drug A Drug B Rection time(seconds) Rection time(seconds) Define : the probbility distribution for rection times under drug A( ) D : the probbility distribution for rection time under drug B; : : 3

4 Exmple : VENDERS: Rw mteril for n industril process is provided by two different venders. Four btches of rw mteril re purchsed from ech vender, nd the strength of the finished product (in p.s.i) is mesured for ech btch. At α =.5, do the dt provide sufficient evidence to indicte the probbility distributions of product strength for vender is shifted to left of the vender? Vender Vender

5 The Wilcoxon Rnk sum test for lrge smple ( ): ypothesis: : D nd D re identicl, nd D re not identicl, R.R: ( is shifted to the left of D, R.R: ) ( is shifted to the right of D, R.R: ) Test sttistic: n( n+ n + ) T z = nn ( n+ n+ ) Exmple: Verbl SAT scores for students rndomly selected from two different schools re listed below. Use the Wilcoxon rnk sum procedure to test the clim tht there is no difference in the scores from ech school. Use α =.5. School 55, 5, 77, 48, 75, 53, 58, 78, 6, 59, 73, 75 School 49, 44, 68, 43, 7, 59, 69, 55, 53, 63, 64, 54 5

6 4.5 compring three or more popultions: completely rndomized design Nonprmetric method lso cn used to nlyze the dt from CRD. : no ssumptions concerning the popultion probbility distribution; Rnk procedure:. Rnk observtions from smllest () to lrgest (n),. Add up the rnks for, R, j =,..., k j ypothesis: : : Test sttistic: Rejection region:, with degree of freedom. (p798) Conditions required for the vlid Kruskl-wllis test:. The k smples re,. there re mesurements in ech smple; 3. the k probbility distributions from which the smples re drwn re. 6

7 Exmple: OZONE EFFECT: To study the effects of ozone exposure on irwy resistnce, 5 helthy mle subjects were rndomly ssigned to ech of 3 exposure levels:.ppm,.6ppm,.ppm for one hour. The differences of irwy resistnce fter exposure nd before exposure for ech subject were recorded. At α =., is there evidence to conclude there exists difference in the probbility distribution mong three ozone exposure levels?. ppm ().6 ppm (). ppm (3) : : 7

8 Exmple: Vehicle miles: The U.S federl highwy dministrtion conducts nnul survey on motor vehicle trvel by type of vehicle nd publishes in highwy sttistics. Independent rndom smples of crs, buses, nd trucks provided the dt on number of miles driven lst yer, in thousnds. At α =.5, do the dt provide sufficient evidence to conclude tht difference exists in the probbility distribution of lst yer s miles mong crs, buses, nd trucks? crs buses trucks : : 8

9 4.6 Compring three or more popultions: rndomized Block design : no ssumptions concerning the popultion probbility distribution; Rnk procedure:. Rnk observtions within ;. Add up the rnks for ; R, j =,..., k ypothesis: : : j Test sttistic: : # of blocks; : # of tretments. Rejection region:, with degrees of freedom.(p798) Conditions required for vlid Friedmn test:. the tretments re ssigned to experimentl units within the blocks;. the mesurements cn be rnked within ; 3. the k probbility distributions from which the smples within ech block re drwn re. Exmple: Rection, (when drug effect is short-lived,no crryover effect; nd the drug effect vries gretly from person to person, it my be useful to employ RBD to find the effect of drug.) At α =.5, do the dt provide sufficient evidence to conclude the probbility distributions of the rection times for the three drugs differ in loction? Rection time for three drugs: subject Drug A Drug B Drug C

10 Exmple. In study compring the effects of four energy drinks on running speed, seven runners were timed (in seconds) running four miles. On ech dy, they were given single energy drink. The dt re listed below. Is there evidence of difference in the probbility distributions of the running times mong the four drinks? Use α =.. drink runner

11 4.7 Rnk Correltion In chpter, we hve introduced correltion coefficient r bsed on the observtions of two numericl vribles. SSxy r = SS SS xx yy x y x y x y... x n y n ere we introduce correltion coefficient bsed on of observtions: : it provides mesure of correltion between rnks. Where r s = SSuv SS SS u i = of the ith observtion in smple v i = of the ith observtion in smple n = uu SS ( u u )( v v ) u v uv i i i i vv ( ui)( vi) n = = ( u ) i SSuu = ( ui u ) = ui n ( v ) i SSvv = ( vi v ) = vi n Shortcut formul for r s : (no ties or less ties reltive to n ) r s =

12 Where (rnk difference of ith observtions between smple nd smple ) Note:. The vlue of r s lwys flls between,. indictes perfect positive correltion nd indictes perfect negtive correltion, 3. The closer r s flls to + or -, the the correltion between the rnks; the closer r s is to, the the correltion. Spermn s rnk correltion test: Define ρ : : ρ = ( ) : ρ ( ) Test sttistic: Rejection region: if : ρ ; if : ρ < if : ρ > Criticl vlue: rs, α from Tble XIV, p89. Note: The α vlues correspond to one-tiled test of : ρ =.

13 Condition required for vlid Spermn s Test:. The smple of experimentl units on which the two vribles re mesured is selected,. The probbility distributions of the two vribles re. Exmple: The number of bsences nd the finl grdes of 9 rndomly selected students from sttistics clss re given below. Cn you conclude tht there is correltion between the finl grde nd the number of bsences? Use α =.5. student bsence Finl grde

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

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

More information

Student Activity 3: Single Factor ANOVA

Student Activity 3: Single Factor ANOVA 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 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

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

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

More information

Tests for the Ratio of Two Poisson Rates

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

More information

Chapter 5 : Continuous Random Variables

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

More information

Comparison Procedures

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

More information

The steps of the hypothesis test

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

More information

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

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 (

More information

( ) 1. Algebra 2: Final Exam Review. y e + e e ) 4 x 10 = 10,000 = 9) Name

( ) 1. Algebra 2: Final Exam Review. y e + e e ) 4 x 10 = 10,000 = 9) Name Algebr : Finl Exm Review Nme Chpter 6 Grph ech function. Determine if the function represents exponentil growth or decy. Determine the eqution of the symptote, the domin nd the rnge of the function. )

More information

Continuous Random Variables

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

More information

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

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

More information

Lecture 3 Gaussian Probability Distribution

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

More information

Lecture INF4350 October 12008

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,

More information

4.1. Probability Density Functions

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

More information

Multi-Armed Bandits: Non-adaptive and Adaptive Sampling

Multi-Armed Bandits: Non-adaptive and Adaptive Sampling CSE 547/Stt 548: Mchine Lerning for Big Dt Lecture Multi-Armed Bndits: Non-dptive nd Adptive Smpling Instructor: Shm Kkde 1 The (stochstic) multi-rmed bndit problem The bsic prdigm is s follows: K Independent

More information

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

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,

More information

Normal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution

Normal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution Norml Distribution Lecture 6: More Binomil Distribution If X is rndom vrible with norml distribution with men µ nd vrince σ 2, X N (µ, σ 2, then P(X = x = f (x = 1 e 1 (x µ 2 2 σ 2 σ Sttistics 104 Colin

More information

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

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

More information

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

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

More information

Credibility Hypothesis Testing of Fuzzy Triangular Distributions

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

More information

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?

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

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

Joint Distribution of any Record Value and an Order Statistics

Joint Distribution of any Record Value and an Order Statistics Interntionl Mthemticl Forum, 4, 2009, no. 22, 09-03 Joint Distribution of ny Record Vlue nd n Order Sttistics Cihn Aksop Gzi University, Deprtment of Sttistics 06500 Teknikokullr, Ankr, Turkey entelpi@yhoo.com

More information

Multivariate problems and matrix algebra

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

More information

CS 188 Introduction to Artificial Intelligence Fall 2018 Note 7

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

More information

: : 8.2. Test About a Population Mean. STT 351 Hypotheses Testing Case I: A Normal Population with Known. - null hypothesis states 0

: : 8.2. Test About a Population Mean. STT 351 Hypotheses Testing Case I: A Normal Population with Known. - null hypothesis states 0 8.2. Test About Popultio Me. Cse I: A Norml Popultio with Kow. H - ull hypothesis sttes. X1, X 2,..., X - rdom smple of size from the orml popultio. The the smple me X N, / X X Whe H is true. X 8.2.1.

More information

5 Probability densities

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

More information

Non-Linear & Logistic Regression

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

More information

Lecture 21: Order statistics

Lecture 21: Order statistics Lecture : Order sttistics Suppose we hve N mesurements of sclr, x i =, N Tke ll mesurements nd sort them into scending order x x x 3 x N Define the mesured running integrl S N (x) = 0 for x < x = i/n for

More information

Acceptance Sampling by Attributes

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

More information

A New Statistic Feature of the Short-Time Amplitude Spectrum Values for Human s Unvoiced Pronunciation

A New Statistic Feature of the Short-Time Amplitude Spectrum Values for Human s Unvoiced Pronunciation Xiodong Zhung A ew Sttistic Feture of the Short-Time Amplitude Spectrum Vlues for Humn s Unvoiced Pronuncition IAODOG ZHUAG 1 1. Qingdo University, Electronics & Informtion College, Qingdo, 6671 CHIA Abstrct:

More information

New data structures to reduce data size and search time

New data structures to reduce data size and search time New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute

More information

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8 Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite

More information

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

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

More information

Chapter 3. Extensions

Chapter 3. Extensions Chpter 3 Extensions EXTENSION 3E: TESTS OF REPLICATION In Chpter 3, we ssumed tht the only comprison of interest in the two-group cse is tht between cell men model nd grnd men model. Tht is, we hve compred

More information

Monte Carlo method in solving numerical integration and differential equation

Monte Carlo method in solving numerical integration and differential equation Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The

More information

Continuous Random Variables

Continuous Random Variables CPSC 53 Systems Modeling nd Simultion Continuous Rndom Vriles Dr. Anirn Mhnti Deprtment of Computer Science University of Clgry mhnti@cpsc.uclgry.c Definitions A rndom vrile is sid to e continuous if there

More information

Math 426: Probability Final Exam Practice

Math 426: Probability Final Exam Practice Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by

More information

A signalling model of school grades: centralized versus decentralized examinations

A signalling model of school grades: centralized versus decentralized examinations A signlling model of school grdes: centrlized versus decentrlized exmintions Mri De Pol nd Vincenzo Scopp Diprtimento di Economi e Sttistic, Università dell Clbri m.depol@unicl.it; v.scopp@unicl.it 1 The

More information

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

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

More information

2.4 Linear Inequalities and Problem Solving

2.4 Linear Inequalities and Problem Solving Section.4 Liner Inequlities nd Problem Solving 77.4 Liner Inequlities nd Problem Solving S 1 Use Intervl Nottion. Solve Liner Inequlities Using the Addition Property of Inequlity. 3 Solve Liner Inequlities

More information

Predict Global Earth Temperature using Linier Regression

Predict Global Earth Temperature using Linier Regression Predict Globl Erth Temperture using Linier Regression Edwin Swndi Sijbt (23516012) Progrm Studi Mgister Informtik Sekolh Teknik Elektro dn Informtik ITB Jl. Gnesh 10 Bndung 40132, Indonesi 23516012@std.stei.itb.c.id

More information

Introduction to Group Theory

Introduction to Group Theory Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements

More information

CHM Physical Chemistry I Chapter 1 - Supplementary Material

CHM Physical Chemistry I Chapter 1 - Supplementary Material CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion

More information

Chapter 1: Fundamentals

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,

More information

1 Probability Density Functions

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

More information

Terminal Velocity and Raindrop Growth

Terminal Velocity and Raindrop Growth Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,

More information

MATH362 Fundamentals of Mathematical Finance

MATH362 Fundamentals of Mathematical Finance MATH362 Fundmentls of Mthemticl Finnce Solution to Homework Three Fll, 2007 Course Instructor: Prof. Y.K. Kwok. If outcome j occurs, then the gin is given by G j = g ij α i, + d where α i = i + d i We

More information

Hybrid Group Acceptance Sampling Plan Based on Size Biased Lomax Model

Hybrid Group Acceptance Sampling Plan Based on Size Biased Lomax Model Mthemtics nd Sttistics 2(3): 137-141, 2014 DOI: 10.13189/ms.2014.020305 http://www.hrpub.org Hybrid Group Acceptnce Smpling Pln Bsed on Size Bised Lomx Model R. Subb Ro 1,*, A. Ng Durgmmb 2, R.R.L. Kntm

More information

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VII

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VII Tutoril VII: Liner Regression Lst updted 5/8/06 b G.G. Botte Deprtment of Chemicl Engineering ChE-0: Approches to Chemicl Engineering Problem Solving MATLAB Tutoril VII Liner Regression Using Lest Squre

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

Driving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d

Driving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,

More information

Physics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15

Physics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15 Physics H - Introductory Quntum Physics I Homework #8 - Solutions Fll 4 Due 5:1 PM, Mondy 4/11/15 [55 points totl] Journl questions. Briefly shre your thoughts on the following questions: Of the mteril

More information

( 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

( 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

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

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

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

More information

Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)

Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar) Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of

More information

FBR Neutronics: Breeding potential, Breeding Ratio, Breeding Gain and Doubling time

FBR Neutronics: Breeding potential, Breeding Ratio, Breeding Gain and Doubling time FBR eutronics: Breeding potentil, Breeding Rtio, Breeding Gin nd Doubling time K.S. Rjn Proessor, School o Chemicl & Biotechnology SASTRA University Joint Inititive o IITs nd IISc Funded by MHRD Pge 1

More information

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 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.

More information

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present? University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the

More information

UNIT 3 Indices and Standard Form Activities

UNIT 3 Indices and Standard Form Activities UNIT 3 Indices nd Stndrd Form Activities Activities 3.1 Towers 3.2 Bode's Lw 3.3 Mesuring nd Stndrd Form 3.4 Stndrd Inde Form Notes nd Solutions (1 pge) ACTIVITY 3.1 Towers How mny cubes re needed to build

More information

Recitation 3: More Applications of the Derivative

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

More information

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from

More information

Joint distribution. Joint distribution. Marginal distributions. Joint distribution

Joint distribution. Joint distribution. Marginal distributions. Joint distribution Joint distribution To specify the joint distribution of n rndom vribles X 1,...,X n tht tke vlues in the smple spces E 1,...,E n we need probbility mesure, P, on E 1... E n = {(x 1,...,x n ) x i E i, i

More information

8 Laplace s Method and Local Limit Theorems

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

More information

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

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

More information

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?

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 out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

Math 135, Spring 2012: HW 7

Math 135, Spring 2012: HW 7 Mth 3, Spring : HW 7 Problem (p. 34 #). SOLUTION. Let N the number of risins per cookie. If N is Poisson rndom vrible with prmeter λ, then nd for this to be t lest.99, we need P (N ) P (N ) ep( λ) λ ln(.)

More information

Calculus of Variations

Calculus of Variations Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function

More information

Math 31S. Rumbos Fall Solutions to Assignment #16

Math 31S. Rumbos Fall Solutions to Assignment #16 Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)

More information

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es

More information

Session Trimester 2. Module Code: MATH08001 MATHEMATICS FOR DESIGN

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

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

Continuous probability distributions

Continuous probability distributions Chpter 1 Continuous probbility distributions 1.1 Introduction We cll x continuous rndom vrible in x b if x cn tke on ny vlue in this intervl. An exmple of rndom vrible is the height of dult humn mle, selected

More information

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses

More information

Vyacheslav Telnin. Search for New Numbers.

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

More information

PHYS Summer Professor Caillault Homework Solutions. Chapter 2

PHYS Summer Professor Caillault Homework Solutions. Chapter 2 PHYS 1111 - Summer 2007 - Professor Cillult Homework Solutions Chpter 2 5. Picture the Problem: The runner moves long the ovl trck. Strtegy: The distnce is the totl length of trvel, nd the displcement

More information

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Hypothesis Test We assume, calculate and conclude.

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Hypothesis Test We assume, calculate and conclude. Hypothesis Test We ssume, clculte d coclude. VIII. HYPOTHESIS TEST 8.. P-Vlue, Test Sttistic d Hypothesis Test [MATH] Terms for the mkig hypothesis test. Assume tht the uderlyig distributios re either

More information

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

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

More information

#6A&B Magnetic Field Mapping

#6A&B Magnetic Field Mapping #6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by

More information

Heat flux and total heat

Heat flux and total heat Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce

More information

Designing Information Devices and Systems I Spring 2018 Homework 7

Designing Information Devices and Systems I Spring 2018 Homework 7 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should

More information

Testing categorized bivariate normality with two-stage. polychoric correlation estimates

Testing categorized bivariate normality with two-stage. polychoric correlation estimates Testing ctegorized bivrite normlity with two-stge polychoric correltion estimtes Albert Mydeu-Olivres Dept. of Psychology University of Brcelon Address correspondence to: Albert Mydeu-Olivres. Fculty of

More information

13.4 Work done by Constant Forces

13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

Line and Surface Integrals: An Intuitive Understanding

Line and Surface Integrals: An Intuitive Understanding Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

Written as per the revised syllabus prescribed by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune.

Written as per the revised syllabus prescribed by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune. Written s per the revised syllbus prescribed by the Mhrshtr Stte Bord of Secondry nd Higher Secondry Eduction, Pune. Slient Fetures Written s per the new textbook. Exhustive coverge of entire syllbus.

More information

Chapter 4. Additional Variational Concepts

Chapter 4. Additional Variational Concepts Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.

More information

The use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.

The use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed. ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion

More information

Math 116 Final Exam April 26, 2013

Math 116 Final Exam April 26, 2013 Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems

More information

Quantitative Genomics and Genetics BTRY 4830/6830; PBSB

Quantitative Genomics and Genetics BTRY 4830/6830; PBSB Quntittive Genomics nd Genetics BTRY 4830/6830; PBSB.501.01 Lecture 1: (Brief) Introduction to Byesin Inference Json Mezey jgm45@cornell.edu My, 017 (T) 8:40-9:55 Announcements I will hve office hours

More information

ROB EBY Blinn College Mathematics Department

ROB EBY Blinn College Mathematics Department ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob Eby-Fll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous

More information

Mathematics 19A; Fall 2001; V. Ginzburg Practice Final Solutions

Mathematics 19A; Fall 2001; V. Ginzburg Practice Final Solutions Mthemtics 9A; Fll 200; V Ginzburg Prctice Finl Solutions For ech of the ten questions below, stte whether the ssertion is true or flse ) Let fx) be continuous t x Then x fx) f) Answer: T b) Let f be differentible

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

5 Accumulated Change: The Definite Integral

5 Accumulated Change: The Definite Integral 5 Accumulted Chnge: The Definite Integrl 5.1 Distnce nd Accumulted Chnge * How To Mesure Distnce Trveled nd Visulize Distnce on the Velocity Grph Distnce = Velocity Time Exmple 1 Suppose tht you trvel

More information

Numerical integration

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

More information

Bayesian Networks: Approximate Inference

Bayesian Networks: Approximate Inference pproches to inference yesin Networks: pproximte Inference xct inference Vrillimintion Join tree lgorithm pproximte inference Simplify the structure of the network to mkxct inferencfficient (vritionl methods,

More information