First Year Examination Department of Statistics, University of Florida

Size: px
Start display at page:

Download "First Year Examination Department of Statistics, University of Florida"

Transcription

1 Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve credt. 3. Questons 1 through 5 are the theory questons and questons 6 through 10 are the appled questons. You must do exactly four of the theory questons and exactly four of the appled questons 4. Wrte your answers to the theory questons on the blank paper provded. Wrte only on one sde of the paper, and start each queston on a new page. 5. Wrte your answers to the appled questons on the exam tself. 6. Whle the 10 questons are equally weghted, some questons are more dffcult than others. 7. The parts wthn a gven queston are not necessarly equally weghted. 8. You are allowed to use a calculator. 1

2 The followng abbrevatons and termnology are used throughout: ANOVA = analyss of varance CRD = completely randomzed desgn d = ndependent and dentcally dstrbuted mgf = moment generatng functon ML = maxmum lkelhood MOM = method of moments pdf = probablty densty functon pmf = probablty mass functon UMP = unformly most powerful Z + = {0, 1,, 3,... } N = {1,, 3,... } R + = (0, ) You may use the followng facts/formulas wthout proof: Beta densty: X Beta(α, β) means X has pdf where α > 0 and β > 0. f(x; α, β) = Gamma densty: X Gamma(α, β) means X has pdf where α > 0 and β > 0. f(x; α, β) = Γ(α + β) Γ(α)Γ(β) xα 1 (1 x) β 1 I (0,1) (x) 1 Γ(α) β α xα 1 e x/β I (0, ) (x) Negatve Bnomal mass functon: X NB(r, p) means X has pmf ( ) r + x 1 P (X = x) = p r (1 p) x I x Z +(x), where p (0, 1) and r N. Also, E(X) = r(1 p)/p and Var(X) = r(1 p)/p. Iterated Expectaton Formula: E(X) = E [E(X Y )]. Iterated Varance Formula: Var(X) = E [Var(X Y )] + Var [E(X Y )].

3 1. Suppose we have two urns. Urn I contans three yellow balls, four red balls, and two green balls. Urn II contans one yellow ball, two green balls, and four purple balls. Consder a two-stage experment n whch we randomly draw three balls from Urn I and move them to Urn II, and then we randomly draw one ball from (the new) Urn II. (a) Defne two events as follows: and A = { Two yellow balls and one green ball are moved from Urn I to Urn II } B = { A green ball s drawn from Urn II }. Fnd the probabltes of these two events. (Express all numercal answers n ths problem as ratos of ntegers or sums of ratos of ntegers - do not use any decmals.) (b) Are A and B ndependent? (c) Fnd the probablty that at least two of the balls moved from Urn I to Urn II were yellow, gven that the ball drawn from Urn II was yellow.. Let (X, Y ) be a bvarate random vector wth jont pdf [ ] f X,Y (x, y) = c I (0,1) (x)i ( 1,0) (y) xi ( 1,0) (x)i (0,1) (y), where c s an unknown normalzng constant. Let S R denote the support of the densty f X,Y (x, y). (a) Fnd the value of c. (b) Let A R denote the trangle whose vertces are at the ponts ( 1/, 0), (1/, 1/3) and (1/, ). Fnd P ( (X, Y ) A ). (c) Calculate the margnal pdf of X. (d) Calculate the condtonal pdf of Y gven X = x. (e) Fnd the condtonal expectaton of Y gven that X = 1/4. (f) Fnd a second jont pdf, call t f X,Y (x, y), that has exactly the same margnals as f X,Y (x, y), but whose support, call t S, has twce as large an area as S. (g) Fnd a thrd jont pdf, call t f X,Y (x, y), that has exactly the same x-margnal as f X,Y (x, y), but whose support, call t S, satsfes S S =. 3

4 3. Suppose that the condtonal pmf of Z gven θ s and that, margnally, θ Beta(α, β). (a) Fnd the margnal pmf of Z. P (Z = z θ) = θ (1 θ) z I Z +(z), (b) Derve a formula for E [ θ a (1 θ) b] that nvolves only α, β, a, b and the gamma functon. Does ths formula hold for all (a, b) R? (c) Calculate the margnal mean and varance of Z, assumng that α >. (Hnt: Use terated expectaton and varance.) For the remander of ths problem, assume that Z 1,..., Z n are d wth common pmf gven by where α >. P (Z = z; α) = α (α + 1) Γ(α) (z + 1)! Γ(α + z + 3) (d) Fnd the MOM estmator of α and call t α n = α n (z). (e) Show that and dentfy the functon v( ). n ( αn α ) d N ( 0, v(α) ), I Z +(z), 4. Let X 1,..., X n be d random varables from the pdf where θ > 0. f(x; θ) = x θ e x /θ I R +(x) (a) What knd of parameter s θ: locaton, scale or nether? (Explan your answer.) (b) Derve a formula for EX p that holds for any p >. (c) Fnd a functon of n, call t g(n), such that g(n) n =1 X4 s unbased for θ. (d) Fnd the ML estmator of θ. (e) Fnd the Cramér-Rao Lower Bound for the varance of an unbased estmator of θ. (f) Fnd the best unbased estmator of θ. 4

5 5. Suppose that Y 1,..., Y n are d NB(r, q). (a) Derve the mgf of Y 1. (b) Derve the dstrbuton of n =1 Y. (c) Suppose we have four dentcal quarters and we would lke to test H 0 : p 1/ vs. H 1 : p > 1/, where p s the unknown probablty that any one of the quarters comes up heads when t s flpped. We decde to perform an experment. Each quarter wll be pcked up and flpped repeatedly untl the frst head occurs. Let X denote the number of tals that occur before the frst head occurs when the th quarter s flpped, = 1,, 3, 4. Use the X s to construct a UMP sze test of H 0 vs. H 1. 5

6 Q.6. Consder models: Model 1: y u u ~ NID0, 1,...,4 j 1,..., 3 Model: y v NID0 p.6.a. In matrx form, we can wrte the models as follows: u 0 1 ~, v Model 1: 1 0 Y Wα U α Model : Y Xβ V β Fll n all empty values below: y y y y y y Y y y y y y y W X W' W _ X'X _ W' W 1 1 X'X X'Y β W' Y α p.6.b. For ths data, compute: Y Wα T Y Wα _ Y Xβ T Y Xβ _ Gve the test statstc and rejecton regon for testng: H 0 : = H A : Test Statstc: Rejecton Regon (based on 0.05 sgnfcance level): Do you reject the null hypothess that the mean s lnearly related to treatment level ()? Yes / No

7 Q.7. Derve the expected mean squares for treatments and error for the balanced 1-Way ANOVA wth g g treatments and n replcates per treatment: y 0 ~ NID0, 1

8 Q.8. A multple regresson model s ft based on n=30 ndvduals, relatng Y to X 1 and X (the model also contans an ntercept). The coeffcent of determnaton s R = Note: Y Y. p.8.a. Complete the Analyss of Varance and test H 0 : at the 0.05 sgnfcance level Source df SS MS F F(.05) Regresson Resdual Total p.8.b. Based on the least squares estmate of the parameter vector, and (X X) -1 gven below, test H 0 : at the 0.05 sgnfcance level (X'X)(-1) Beta-hat

9 Q.9. An experment s conducted as a Randomzed Complete Block Desgn wth 3 treatments appled to 8 blocks. The model ft s gven below, as well as the data: y j 4 0, ~ NID0, COV, 0, j, j' 1,...,3 j 1,...,8 0 j ~ NID j' 1 Trt1 Trt Trt3 Mean Block Block Block Block4 0 4 Block Block Block Block Mean y y 688 p.9.a. Complete the followng ANOVA table, and test whether treatment effects exst at the 0.05 sgnfcance level. H 0 : Source df SS MS F F(.05) Trts Blocks Error Total p.9.b. Use Bonferron s method to obtan smultaneous 95% confdence ntervals for all pars of dfferences among treatment means. 1 j1

10 Ths queston s based on the followng regresson model, and X s of full column rank (no lnear dependences among predctor varables). Y Xβ ε X n p' β p' 1 ε ~ N0, I Gven: a' x dx'ax d dx a dx. For a symmetrc matrx A: Ax E Y' AY travy μ Y ' Aμ Y Cochran s Theorem: Suppose Y s dstrbuted as follows wth nonsngular matrx V: Y ~ N μ, V rv n then f AV s dempotent : 1 1 Y' A Y sdstrbuted non - central wth : (a) df ( A) and (b) Noncentralty parameter : r Q.10.a. Derve the least squares estmator for μ' Aμ n n SS(Model) 1 SS(Resdua l) 1 Q.10.b. Obtan Y and e n matrx form (quadratc forms n Y). 1 1 Obtan the dstrbutons of the two sum of squares (be specfc wth regard to ther famly of dstrbutons, degrees of freedom, and non-centralty parameters).

Economics 130. Lecture 4 Simple Linear Regression Continued

Economics 130. Lecture 4 Simple Linear Regression Continued Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do

More information

STAT 3008 Applied Regression Analysis

STAT 3008 Applied Regression Analysis STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

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 Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models

More information

Here is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)

Here is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y) Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,

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

Statistics II Final Exam 26/6/18

Statistics II Final Exam 26/6/18 Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the

More information

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

STAT 511 FINAL EXAM NAME Spring 2001

STAT 511 FINAL EXAM NAME Spring 2001 STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte

More information

UNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours

UNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x

More information

ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics

ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott

More information

Stat 543 Exam 2 Spring 2016

Stat 543 Exam 2 Spring 2016 Stat 543 Exam 2 Sprng 2016 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of 11 questons. Do at least 10 of the 11 parts of the man exam.

More information

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9 Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,

More information

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1 Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons

More information

Stat 543 Exam 2 Spring 2016

Stat 543 Exam 2 Spring 2016 Stat 543 Exam 2 Sprng 206 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of questons. Do at least 0 of the parts of the man exam. I wll score

More information

Composite Hypotheses testing

Composite Hypotheses testing Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter

More information

ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)

ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) June 7, 016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston A B C Blank Queston

More information

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values Fall 007 Soluton to Mdterm Examnaton STAT 7 Dr. Goel. [0 ponts] For the general lnear model = X + ε, wth uncorrelated errors havng mean zero and varance σ, suppose that the desgn matrx X s not necessarly

More information

Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore

Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.

More information

Exam. Econometrics - Exam 1

Exam. Econometrics - Exam 1 Econometrcs - Exam 1 Exam Problem 1: (15 ponts) Suppose that the classcal regresson model apples but that the true value of the constant s zero. In order to answer the followng questons assume just one

More information

Chapter 13: Multiple Regression

Chapter 13: Multiple Regression Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to

More information

Lecture 3: Probability Distributions

Lecture 3: Probability Distributions Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the

More information

Chapter 12 Analysis of Covariance

Chapter 12 Analysis of Covariance Chapter Analyss of Covarance Any scentfc experment s performed to know somethng that s unknown about a group of treatments and to test certan hypothess about the correspondng treatment effect When varablty

More information

Department of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution

Department of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable

More information

Statistics MINITAB - Lab 2

Statistics MINITAB - Lab 2 Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that

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

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA 4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected

More information

ISQS 6348 Final Open notes, no books. Points out of 100 in parentheses. Y 1 ε 2

ISQS 6348 Final Open notes, no books. Points out of 100 in parentheses. Y 1 ε 2 ISQS 6348 Fnal Open notes, no books. Ponts out of 100 n parentheses. 1. The followng path dagram s gven: ε 1 Y 1 ε F Y 1.A. (10) Wrte down the usual model and assumptons that are mpled by ths dagram. Soluton:

More information

/ n ) are compared. The logic is: if the two

/ n ) are compared. The logic is: if the two STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence

More information

Statistics Chapter 4

Statistics Chapter 4 Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment

More information

Chapter 11: Simple Linear Regression and Correlation

Chapter 11: Simple Linear Regression and Correlation Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests

More information

a. (All your answers should be in the letter!

a. (All your answers should be in the letter! Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal

More information

where I = (n x n) diagonal identity matrix with diagonal elements = 1 and off-diagonal elements = 0; and σ 2 e = variance of (Y X).

where I = (n x n) diagonal identity matrix with diagonal elements = 1 and off-diagonal elements = 0; and σ 2 e = variance of (Y X). 11.4.1 Estmaton of Multple Regresson Coeffcents In multple lnear regresson, we essentally solve n equatons for the p unnown parameters. hus n must e equal to or greater than p and n practce n should e

More information

The written Master s Examination

The written Master s Examination he wrtten Master s Eamnaton Opton Statstcs and Probablty SPRING 9 Full ponts may be obtaned for correct answers to 8 questons. Each numbered queston (whch may have several parts) s worth the same number

More information

Regression Analysis. Regression Analysis

Regression Analysis. Regression Analysis Regresson Analyss Smple Regresson Multvarate Regresson Stepwse Regresson Replcaton and Predcton Error 1 Regresson Analyss In general, we "ft" a model by mnmzng a metrc that represents the error. n mn (y

More information

January Examinations 2015

January Examinations 2015 24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)

More information

DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes

DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes 25/6 Canddates Only January Examnatons 26 Student Number: Desk Number:...... DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Module Code Module Ttle Exam Duraton

More information

Lecture 6 More on Complete Randomized Block Design (RBD)

Lecture 6 More on Complete Randomized Block Design (RBD) Lecture 6 More on Complete Randomzed Block Desgn (RBD) Multple test Multple test The multple comparsons or multple testng problem occurs when one consders a set of statstcal nferences smultaneously. For

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com PhscsAndMathsTutor.com phscsandmathstutor.com June 005 5. The random varable X has probablt functon k, = 1,, 3, P( X = ) = k ( + 1), = 4, 5, where k s a constant. (a) Fnd the value of k. (b) Fnd the eact

More information

Topic- 11 The Analysis of Variance

Topic- 11 The Analysis of Variance Topc- 11 The Analyss of Varance Expermental Desgn The samplng plan or expermental desgn determnes the way that a sample s selected. In an observatonal study, the expermenter observes data that already

More information

Limited Dependent Variables

Limited Dependent Variables Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages

More information

Expected Value and Variance

Expected Value and Variance MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or

More information

Linear Regression Analysis: Terminology and Notation

Linear Regression Analysis: Terminology and Notation ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented

More information

Professor Chris Murray. Midterm Exam

Professor Chris Murray. Midterm Exam Econ 7 Econometrcs Sprng 4 Professor Chrs Murray McElhnney D cjmurray@uh.edu Mdterm Exam Wrte your answers on one sde of the blank whte paper that I have gven you.. Do not wrte your answers on ths exam.

More information

Joint Statistical Meetings - Biopharmaceutical Section

Joint Statistical Meetings - Biopharmaceutical Section Iteratve Ch-Square Test for Equvalence of Multple Treatment Groups Te-Hua Ng*, U.S. Food and Drug Admnstraton 1401 Rockvlle Pke, #200S, HFM-217, Rockvlle, MD 20852-1448 Key Words: Equvalence Testng; Actve

More information

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models

More information

NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION MTH352/MH3510 Regression Analysis

NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION MTH352/MH3510 Regression Analysis NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION 014-015 MTH35/MH3510 Regresson Analyss December 014 TIME ALLOWED: HOURS INSTRUCTIONS TO CANDIDATES 1. Ths examnaton paper contans FOUR (4) questons

More information

x i1 =1 for all i (the constant ).

x i1 =1 for all i (the constant ). Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by

More information

Biostatistics 360 F&t Tests and Intervals in Regression 1

Biostatistics 360 F&t Tests and Intervals in Regression 1 Bostatstcs 360 F&t Tests and Intervals n Regresson ORIGIN Model: Y = X + Corrected Sums of Squares: X X bar where: s the y ntercept of the regresson lne (translaton) s the slope of the regresson lne (scalng

More information

[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.

[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students. PPOL 59-3 Problem Set Exercses n Smple Regresson Due n class /8/7 In ths problem set, you are asked to compute varous statstcs by hand to gve you a better sense of the mechancs of the Pearson correlaton

More information

Statistics for Economics & Business

Statistics for Economics & Business Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable

More information

Module Contact: Dr Susan Long, ECO Copyright of the University of East Anglia Version 1

Module Contact: Dr Susan Long, ECO Copyright of the University of East Anglia Version 1 UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PG Examnaton 016-17 ECONOMETRIC METHODS ECO-7000A Tme allowed: hours Answer ALL FOUR Questons. Queston 1 carres a weght of 5%; Queston carres 0%;

More information

Lecture Notes on Linear Regression

Lecture Notes on Linear Regression Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume

More information

Lecture 4 Hypothesis Testing

Lecture 4 Hypothesis Testing Lecture 4 Hypothess Testng We may wsh to test pror hypotheses about the coeffcents we estmate. We can use the estmates to test whether the data rejects our hypothess. An example mght be that we wsh to

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

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

# c i. INFERENCE FOR CONTRASTS (Chapter 4) It's unbiased: Recall: A contrast is a linear combination of effects with coefficients summing to zero:

# c i. INFERENCE FOR CONTRASTS (Chapter 4) It's unbiased: Recall: A contrast is a linear combination of effects with coefficients summing to zero: 1 INFERENCE FOR CONTRASTS (Chapter 4 Recall: A contrast s a lnear combnaton of effects wth coeffcents summng to zero: " where " = 0. Specfc types of contrasts of nterest nclude: Dfferences n effects Dfferences

More information

Chapter 14 Simple Linear Regression

Chapter 14 Simple Linear Regression Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng

More information

Negative Binomial Regression

Negative Binomial Regression STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...

More information

Chapter 7 Generalized and Weighted Least Squares Estimation. In this method, the deviation between the observed and expected values of

Chapter 7 Generalized and Weighted Least Squares Estimation. In this method, the deviation between the observed and expected values of Chapter 7 Generalzed and Weghted Least Squares Estmaton The usual lnear regresson model assumes that all the random error components are dentcally and ndependently dstrbuted wth constant varance. When

More information

STAT 3340 Assignment 1 solutions. 1. Find the equation of the line which passes through the points (1,1) and (4,5).

STAT 3340 Assignment 1 solutions. 1. Find the equation of the line which passes through the points (1,1) and (4,5). (out of 15 ponts) STAT 3340 Assgnment 1 solutons (10) (10) 1. Fnd the equaton of the lne whch passes through the ponts (1,1) and (4,5). β 1 = (5 1)/(4 1) = 4/3 equaton for the lne s y y 0 = β 1 (x x 0

More information

Interval Estimation in the Classical Normal Linear Regression Model. 1. Introduction

Interval Estimation in the Classical Normal Linear Regression Model. 1. Introduction ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model

More information

10-701/ Machine Learning, Fall 2005 Homework 3

10-701/ Machine Learning, Fall 2005 Homework 3 10-701/15-781 Machne Learnng, Fall 2005 Homework 3 Out: 10/20/05 Due: begnnng of the class 11/01/05 Instructons Contact questons-10701@autonlaborg for queston Problem 1 Regresson and Cross-valdaton [40

More information

PubH 7405: REGRESSION ANALYSIS. SLR: INFERENCES, Part II

PubH 7405: REGRESSION ANALYSIS. SLR: INFERENCES, Part II PubH 7405: REGRESSION ANALSIS SLR: INFERENCES, Part II We cover te topc of nference n two sessons; te frst sesson focused on nferences concernng te slope and te ntercept; ts s a contnuaton on estmatng

More information

Two-factor model. Statistical Models. Least Squares estimation in LM two-factor model. Rats

Two-factor model. Statistical Models. Least Squares estimation in LM two-factor model. Rats tatstcal Models Lecture nalyss of Varance wo-factor model Overall mean Man effect of factor at level Man effect of factor at level Y µ + α + β + γ + ε Eε f (, ( l, Cov( ε, ε ) lmr f (, nteracton effect

More information

Probability and Random Variable Primer

Probability and Random Variable Primer B. Maddah ENMG 622 Smulaton 2/22/ Probablty and Random Varable Prmer Sample space and Events Suppose that an eperment wth an uncertan outcome s performed (e.g., rollng a de). Whle the outcome of the eperment

More information

Modeling and Simulation NETW 707

Modeling and Simulation NETW 707 Modelng and Smulaton NETW 707 Lecture 5 Tests for Random Numbers Course Instructor: Dr.-Ing. Magge Mashaly magge.ezzat@guc.edu.eg C3.220 1 Propertes of Random Numbers Random Number Generators (RNGs) must

More information

Introduction to Regression

Introduction to Regression Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes

More information

Resource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis

Resource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques

More information

See Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)

See Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition) Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes

More information

CIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M

CIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute

More information

STATISTICS QUESTIONS. Step by Step Solutions.

STATISTICS QUESTIONS. Step by Step Solutions. STATISTICS QUESTIONS Step by Step Solutons www.mathcracker.com 9//016 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to

More information

Statistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation

Statistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear

More information

University of California at Berkeley Fall Introductory Applied Econometrics Final examination

University of California at Berkeley Fall Introductory Applied Econometrics Final examination SID: EEP 118 / IAS 118 Elsabeth Sadoulet and Daley Kutzman Unversty of Calforna at Berkeley Fall 01 Introductory Appled Econometrcs Fnal examnaton Scores add up to 10 ponts Your name: SID: 1. (15 ponts)

More information

Answers Problem Set 2 Chem 314A Williamsen Spring 2000

Answers Problem Set 2 Chem 314A Williamsen Spring 2000 Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%

More information

CS-433: Simulation and Modeling Modeling and Probability Review

CS-433: Simulation and Modeling Modeling and Probability Review CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown

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

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

Complex Numbers Alpha, Round 1 Test #123

Complex Numbers Alpha, Round 1 Test #123 Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test

More information

THE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE

THE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE THE ROYAL STATISTICAL SOCIETY 6 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutons to assst canddates preparng for the eamnatons n future years and for

More information

ANOVA. The Observations y ij

ANOVA. The Observations y ij ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2

More information

Lecture 6: Introduction to Linear Regression

Lecture 6: Introduction to Linear Regression Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6

More information

Hydrological statistics. Hydrological statistics and extremes

Hydrological statistics. Hydrological statistics and extremes 5--0 Stochastc Hydrology Hydrologcal statstcs and extremes Marc F.P. Berkens Professor of Hydrology Faculty of Geoscences Hydrologcal statstcs Mostly concernes wth the statstcal analyss of hydrologcal

More information

Lecture 3 Stat102, Spring 2007

Lecture 3 Stat102, Spring 2007 Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture

More information

Comparison of Regression Lines

Comparison of Regression Lines STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence

More information

Chapter 11: I = 2 samples independent samples paired samples Chapter 12: I 3 samples of equal size J one-way layout two-way layout

Chapter 11: I = 2 samples independent samples paired samples Chapter 12: I 3 samples of equal size J one-way layout two-way layout Serk Sagtov, Chalmers and GU, February 0, 018 Chapter 1. Analyss of varance Chapter 11: I = samples ndependent samples pared samples Chapter 1: I 3 samples of equal sze one-way layout two-way layout 1

More information

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6 Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.

More information

Y = β 0 + β 1 X 1 + β 2 X β k X k + ε

Y = β 0 + β 1 X 1 + β 2 X β k X k + ε Chapter 3 Secton 3.1 Model Assumptons: Multple Regresson Model Predcton Equaton Std. Devaton of Error Correlaton Matrx Smple Lnear Regresson: 1.) Lnearty.) Constant Varance 3.) Independent Errors 4.) Normalty

More information

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4) I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes

More information

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for

More information

Estimation: Part 2. Chapter GREG estimation

Estimation: Part 2. Chapter GREG estimation Chapter 9 Estmaton: Part 2 9. GREG estmaton In Chapter 8, we have seen that the regresson estmator s an effcent estmator when there s a lnear relatonshp between y and x. In ths chapter, we generalzed the

More information

MATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2)

MATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2) 1/16 MATH 829: Introducton to Data Mnng and Analyss The EM algorthm (part 2) Domnque Gullot Departments of Mathematcal Scences Unversty of Delaware Aprl 20, 2016 Recall 2/16 We are gven ndependent observatons

More information

STAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression

STAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,

More information

since [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation

since [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation Econ 388 R. Butler 204 revsons Lecture 4 Dummy Dependent Varables I. Lnear Probablty Model: the Regresson model wth a dummy varables as the dependent varable assumpton, mplcaton regular multple regresson

More information

Homework 9 for BST 631: Statistical Theory I Problems, 11/02/2006

Homework 9 for BST 631: Statistical Theory I Problems, 11/02/2006 Due Tme: 5:00PM Thursda, on /09/006 Problem (8 ponts) Book problem 45 Let U = X + and V = X, then the jont pmf of ( UV, ) s θ λ θ e λ e f( u, ) = ( = 0, ; u =, +, )! ( u )! Then f( u, ) u θ λ f ( x x+

More information

A be a probability space. A random vector

A be a probability space. A random vector Statstcs 1: Probablty Theory II 8 1 JOINT AND MARGINAL DISTRIBUTIONS In Probablty Theory I we formulate the concept of a (real) random varable and descrbe the probablstc behavor of ths random varable by

More information

Properties of Least Squares

Properties of Least Squares Week 3 3.1 Smple Lnear Regresson Model 3. Propertes of Least Squares Estmators Y Y β 1 + β X + u weekly famly expendtures X weekly famly ncome For a gven level of x, the expected level of food expendtures

More information

F8: Heteroscedasticity

F8: Heteroscedasticity F8: Heteroscedastcty Feng L Department of Statstcs, Stockholm Unversty What s so-called heteroscedastcty In a lnear regresson model, we assume the error term has a normal dstrbuton wth mean zero and varance

More information

Learning Objectives for Chapter 11

Learning Objectives for Chapter 11 Chapter : Lnear Regresson and Correlaton Methods Hldebrand, Ott and Gray Basc Statstcal Ideas for Managers Second Edton Learnng Objectves for Chapter Usng the scatterplot n regresson analyss Usng the method

More information