STAT 3340 Assignment 1 solutions. 1. Find the equation of the line which passes through the points (1,1) and (4,5).
|
|
- Gwenda Virginia Palmer
- 6 years ago
- Views:
Transcription
1 (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 ), where (x 0, y 0 ) s a pont on the lne. Usng the pont (1,1), the equaton s y 1 = 4(x 1), or y = 4x 1. Could also use the pont (4,5), and would get the same equaton.. Suppose you are gven three data ponts (1,4), (,6) and (3,7) and the lne y = 1 + 4x. Gve the three resduals and ther sum of squares. (5 ponts for resduals, 5 ponts for resdual sum of squares) > x=c(1,,3) > y=c(4,6,7) > yhat=1+4*x > yhat # predcted values [1] > resds=y-yhat #resduals > resds [1] > sum(resds^) # resdual sum of squares [1] Some data gves the summares: n = 10, x y = 100, x = 0 and y = 10. Suppose that the response y s temperature n degrees Celcus. (3) (3) (4) (a) What s S xy? S xy = x y n xȳ = (0/10)(10/10) = 80. (b) If the response was converted to temperature n degrees Fahrenhet y, so that y = y, what s y? y = ( y ) = 3(10) y = (10) = 338 (c) If the response was converted to temperature n degrees Fahrenhet y, so that y = y, what s S xy? x y = x ( y ) = 3 x x y = 3(0) + 1.8(100) = 80 Then S xy = x y ( 10 x 10 y )/10 = 80 (0)(338)/10 =
2 4. In a smple lnear regresson, the sum of squares functon s S(β 0, β 1 ) = β 0 700β β 0 β β β 1. Fnd the least squares values for β 0 and β 1. Frst dfferentate wrt β 0 and β 1. (5 ponts for dervatves) S β 0 = β 1 + (50)β 0 S β 1 = β 0 + (70)β 1 Settng the partal dervatves eqch equal to 0 gves the followng two equatons, after a bt of smplfyng: β 1 + β 0 = 1 = β 0 = 1 β 1 1.4β 1 + β 0 = 7 Substtutng the frst equaton nto the second, 1.4β 1 +(1 β 1 ) = 7 =.4β 1 = 6 = β 1 = 15. Substtutng ths back nto the frst equaton gves β 0 = 1 β 1 = 14. The least squares soluton s ( 14, 15). (5 ponts for soluton)
3 5. A random sample of 11 elementary school students s selected, and each student s measured on a creatvty score (x) usng a well-defned testng nstrument and on a task score (y) usng a new nstrument. The task score s the mean tme taken to perform several hand-eye coordnaton tasks. The data are: STUDENT CREATIVITY(X) TASKS(Y) FR HT IO DP YR QD 40.1 DF ER RR TG EF Use R to do the followng questons. Show your commands. Make sure your output s ntegrated nto your responses. (Cut and paste as necessary.) (a) Plot Tasks versus Creatvty and comment on the form and strength of the assocaton. Be sure to label the axes. To make the plot, use: x=c(35,37,50,69,84,40,9,4,51,45,31) y=c(3.9,3.9,6.1,4.3,8.8,.1,5.7,3.0,7.1,7.3,3.3) plot(x,y,xlab="creatvty",ylab="tasks") Plot s below, together wth added least squares lne. (b) Calculate the summares S xx, S xy, S yy and X and Ȳ. (1 pont for each of the 5 summary statstcs) > x=c(35,37,50,69,84,40,9,4,51,45,31) > y=c(3.9,3.9,6.1,4.3,8.8,.1,5.7,3.0,7.1,7.3,3.3) > xbar=mean(x) > xbar [1] > ybar=mean(y) > ybar [1] > Sxx=sum((x-mean(x))^) > Sxx [1] > Syy=sum((y-mean(y))^) > Syy [1]
4 () (4) > Sxy=sum((x-mean(x))*(y-mean(y))) > Sxy [1] (c) Use these data summares to calculate the correlaton coeffcent. Does the value agree wth your vsual assessment n (a)? ( ponts for the correlaton coeffcent). > corrxy=sxy/sqrt(sxx*syy) > corrxy [1] from the plot followng, t looks lke there s a moderately strong ncreasng relatonshp between x and y, and ths s born out by the moderate sze of r. (d) Use these summares to calculate the least squares values for the ntercept and slope. ( ponts each for ˆbeta 1 and ˆbeta 0 ) > b1=sxy/sxx #estmated slope > b1 [1] > b0=ybar-b1*xbar #estmated ntercept > b0 [1] (e) Add the least squares lne to the plot n (a). (A convenent way to add the lne s the command ablne(ntercept,slope) (5 ponts for the plot wth added lne) > x=c(35,37,50,69,84,40,9,4,51,45,31) > y=c(3.9,3.9,6.1,4.3,8.8,.1,5.7,3.0,7.1,7.3,3.3) > plot(x,y,xlab="creatvty",ylab="tasks") > ablne(b0,b1) 4
5 Tasks Creatvty 5
6 (6) (f) Obtan the resduals, e = y ŷ. Calculate ther sample mean to verfy t s zero, and the correlaton wth X to verfy t s also zero. ( ponts for the resduals, ponts for showng the mean of resduals s 0, ponts for showng correlaton of resduals and x s 0.) (g) Plot the resduals versus X. Do the resduals look random? (5 ponts for the resdual plot) > yhat = b0+b1*x #predcted values > resds=y-yhat #resduals > resds [1] [7] > prnt(mean(resds)) #0 to round off error [1] e-17 > prnt(cor(x,resds)) #0 to round off error [1] e-16 6
7 > #<<fg=t,echo=true,keep.source=t>>= > plot(x,resds,man="resdual plot", xlab="x",ylab="resduals") resdual plot resduals x The resduals look random, wth no evdence that mean or varance change wth x. 7
8 (6) () (h) Obtan the resdual, regresson and total sums of squares, usng the data summares. ( ponts for each of SSE, SST, SSR) > SSE=sum(resds^) #resdual sum of squares > SSE [1] > SST=Syy #total sum of squares > SST [1] > SSR=SST-SSE #regresson sum of squares > SSR [1] > b1*sxy #another way to get the regresson sum of squares [1] () What s the value of the coeffcent of determnaton? > R=corrxy^ > R [1]
9 6. Use the data summares calculated for the prevous queston and the formulae from the book or notes to do the followng questons. (6) () () (a) Assess the null hypothess that there s no relatonshp between task score and creatvty. Use a test based on the normal assumpton. State the hypotheses, show calculaton of the test statstc, calculate the P value and draw a concluson. ( ponts for hypotheses, ponts for observed test statstc, ponts for p-value) H 0 : β 1 = 0, H 0 : β 1 0 > MSE=SSE/(11-) > tobs=b1/sqrt(mse/sxx) > tobs [1] > pvalue=*(1-pt(tobs,11-)) > pvalue [1] (b) Calculate the 95% confdence nterval for the mean task score when the creatvty s 50. ( ponts for CI) > c(b0+b1*50 - qt(.975,11-)*sqrt(mse)*sqrt(1/11+(50-xbar)^/sxx), + b0+b1*50 + qt(.975,11-)*sqrt(mse)*sqrt(1/11+(50-xbar)^/sxx)) [1] (c) Calculate the 95% predcton nterval for a new value for task score when the creatvty s 50. ( ponts for predcton nterval) > c(b0+b1*50 - qt(.975,11-)*sqrt(mse)*sqrt(1+1/11+(50-xbar)^/sxx), + b0+b1*50 + qt(.975,11-)*sqrt(mse)*sqrt(1+1/11+(50-xbar)^/sxx)) [1]
10 7. Suppose X and Y are random varables wth µ x = 10, σ x = 3, µ y = 4, σ y = 1, and Cov[X, Y ] = 1.5. Calculate: () (4) (4) (a) E[X Y ] E[X Y ] = E[X] E[Y ] = (10) 4 = 16 (b) V ar[x Y ] V [X Y ] = V [X] Cov[X, Y ] + V [Y ] = 4V [X] 4Cov[X, Y ] + V [Y ] = 4(3 ) 4(1.5) + 1 = 31 (c) Cor[X, Y ], where Cor stands for correlaton. Cor[X, Y ] = Cov[X,Y ] = Cov[X,Y ] = 1.5/(3(1)) =.5 V [X]V [Y ] 4V [X]V [Y ] 8. Suppose you have data (x, y ), = 1,..., n and want to ft the lne wth known ntercept, y = + β 1 x + ɛ wth the usual assumptons about ɛ. The least squares estmate for β 1 s ˆβ 1 = x y x. x (a) Fnd the expected value of ˆβ 1. Is ˆβ 1 unbased? (5 ponts for the expected value) E[ ˆβ 1 ] = x E[y ] x x = x ( + β 1 x ) x x = x (β 1 x ) x = β 1 x x = β 1 β 1 s unbased. (b) Fnd the varance of ˆβ 1. (5 ponts for the varance) V [ ˆβ 1 ] = x V [y ] ( = V [Y x ] ) x σ = ) ( x x 9. Suppose you have data (x, y ), = 1,..., n and want to ft the lne wth known slope equal to 1 y = β 0 + x + ɛ (10) Derve the least squares estmator of β 0. The error sum of squares s S(β 0 ) = n (y β 0 x ). Dfferentatng wth respect to β 0 gves d dβ 0 S(β 0 ) = n (y β 0 x ) = Settng the dervatve equal to zero and solvng gves β 0 = ȳ x n y + nβ 0 + n x 10
11 10. An experment s to be run to determne the lnear assocaton between x and y. Two possble arrangements of the x values are proposed (6) (4) (a) x = (1, 1, 1, 1, 1, 10, 10, 10, 10, 10) and (b) x = (1,, 3, 4, 5, 6, 7, 8, 9, 10).. Calculate S xx for each proposal. (3 ponts for each of the sums of squares.) > x1=c(1,1,1,1,1,10,10,10,10,10) > SSx1=sum((x1-mean(x1))^) > SSx1 [1] 0.5 > x=c(1,,3,4,5,6,7,8,9,10) > SSx=sum((x-mean(x))^) > SSx [1] 8.5. Whch arangement wll lead to the most precse (.e. smallest varance) estmator of the slope? Justfy your answer. (4 ponts for a reasonable justfcaton.) The varance of β σ 1 s gven by SS XX where SS XX s the sum of squares of the x s. The frst confguraton has a larger value of the X sum of squares, so a smaller value of V [ β 1 ], whch means a more precse estmator. 11
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 informationStatistics for Business and Economics
Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear
More informationStatistics 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 informationSTAT 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 informationDepartment 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 informationStatistics 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 information1. 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 informationChapter 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 information17 - LINEAR REGRESSION II
Topc 7 Lnear Regresson II 7- Topc 7 - LINEAR REGRESSION II Testng and Estmaton Inferences about β Recall that we estmate Yˆ ˆ β + ˆ βx. 0 μ Y X x β0 + βx usng To estmate σ σ squared error Y X x ε s ε we
More informationBiostatistics 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 informationa. (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 informationStatistics 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 informationIntroduction 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 informationLecture 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 informationChapter 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 informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More informationNANYANG 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[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 informationECONOMICS 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 informationLINEAR REGRESSION MODELS W4315
LINEAR REGRESSION MODELS W4315 HOMEWORK ANSWERS February 15, 010 Instructor: Frank Wood 1. (0 ponts) In the fle problem1.txt (accessble on professor s webste), there are 500 pars of data, where the frst
More informationβ0 + β1xi. You are interested in estimating the unknown parameters β
Ordnary Least Squares (OLS): Smple Lnear Regresson (SLR) Analytcs The SLR Setup Sample Statstcs Ordnary Least Squares (OLS): FOCs and SOCs Back to OLS and Sample Statstcs Predctons (and Resduals) wth OLS
More informationEconomics 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 informationY = 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 informationβ0 + β1xi. You are interested in estimating the unknown parameters β
Revsed: v3 Ordnar Least Squares (OLS): Smple Lnear Regresson (SLR) Analtcs The SLR Setup Sample Statstcs Ordnar Least Squares (OLS): FOCs and SOCs Back to OLS and Sample Statstcs Predctons (and Resduals)
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More information/ 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 informationLinear Feature Engineering 11
Lnear Feature Engneerng 11 2 Least-Squares 2.1 Smple least-squares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19
More informationCorrelation 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 informationChapter 15 - Multiple Regression
Chapter - Multple Regresson Chapter - Multple Regresson Multple Regresson Model The equaton that descrbes how the dependent varable y s related to the ndependent varables x, x,... x p and an error term
More informationRegression 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 informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More informationLearning 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 informationProperties 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 informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables
More informationChapter 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 informationLinear regression. Regression Models. Chapter 11 Student Lecture Notes Regression Analysis is the
Chapter 11 Student Lecture Notes 11-1 Lnear regresson Wenl lu Dept. Health statstcs School of publc health Tanjn medcal unversty 1 Regresson Models 1. Answer What Is the Relatonshp Between the Varables?.
More informationSTATISTICS 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 informationStatistics 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 informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationFirst Year Examination Department of Statistics, University of Florida
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
More information18. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III US Domestc Beers: Calores vs. % Alcohol Ftted Values and Resduals To each observed x, there corresponds a y-value on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.
More informationComparison 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 informationInterval 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 informationx yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.
The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator
More informationPolynomial 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 informationHere 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 informationEcon Statistical Properties of the OLS estimator. Sanjaya DeSilva
Econ 39 - Statstcal Propertes of the OLS estmator Sanjaya DeSlva September, 008 1 Overvew Recall that the true regresson model s Y = β 0 + β 1 X + u (1) Applyng the OLS method to a sample of data, we estmate
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Recall: man dea of lnear regresson Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 8 Lnear regresson can be used to study an
More information28. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III Ftted Values and Resduals US Domestc Beers: Calores vs. % Alcohol To each observed x, there corresponds a y-value on the ftted lne, y ˆ = βˆ + βˆ x. The are called ftted
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 008 Recall: man dea of lnear regresson Lnear regresson can be used to study
More informationLecture 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 informationRegression. The Simple Linear Regression Model
Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance Usng the Estmated Regresson Equaton for Estmaton and Predcton Resdual Analss: Valdatng
More informationx 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 informationF8: 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 informationSTAT 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 informationwhere 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 informationβ0 + β1xi and want to estimate the unknown
SLR Models Estmaton Those OLS Estmates Estmators (e ante) v. estmates (e post) The Smple Lnear Regresson (SLR) Condtons -4 An Asde: The Populaton Regresson Functon B and B are Lnear Estmators (condtonal
More informationBiostatistics. Chapter 11 Simple Linear Correlation and Regression. Jing Li
Bostatstcs Chapter 11 Smple Lnear Correlaton and Regresson Jng L jng.l@sjtu.edu.cn http://cbb.sjtu.edu.cn/~jngl/courses/2018fall/b372/ Dept of Bonformatcs & Bostatstcs, SJTU Recall eat chocolate Cell 175,
More informationTHE 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 informationResource 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 informationTopic 7: Analysis of Variance
Topc 7: Analyss of Varance Outlne Parttonng sums of squares Breakdown the degrees of freedom Expected mean squares (EMS) F test ANOVA table General lnear test Pearson Correlaton / R 2 Analyss of Varance
More informationx = , 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 informationLinear Correlation. Many research issues are pursued with nonexperimental studies that seek to establish relationships among 2 or more variables
Lnear Correlaton Many research ssues are pursued wth nonexpermental studes that seek to establsh relatonshps among or more varables E.g., correlates of ntellgence; relaton between SAT and GPA; relaton
More informationAnswers 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 informationProfessor 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 informationLecture 16 Statistical Analysis in Biomaterials Research (Part II)
3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan
More informationLecture 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 informationUNIVERSITY OF TORONTO. Faculty of Arts and Science JUNE EXAMINATIONS STA 302 H1F / STA 1001 H1F Duration - 3 hours Aids Allowed: Calculator
UNIVERSITY OF TORONTO Faculty of Arts and Scence JUNE EXAMINATIONS 008 STA 30 HF / STA 00 HF Duraton - 3 hours Ads Allowed: Calculator LAST NAME: FIRST NAME: STUDENT NUMBER: Enrolled n (Crcle one): STA30
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationDepartment 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 informationChapter 3. Two-Variable Regression Model: The Problem of Estimation
Chapter 3. Two-Varable Regresson Model: The Problem of Estmaton Ordnary Least Squares Method (OLS) Recall that, PRF: Y = β 1 + β X + u Thus, snce PRF s not drectly observable, t s estmated by SRF; that
More informationLinear 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 informationSIMPLE LINEAR REGRESSION
Smple Lnear Regresson and Correlaton Introducton Prevousl, our attenton has been focused on one varable whch we desgnated b x. Frequentl, t s desrable to learn somethng about the relatonshp between two
More informationJAB Chain. Long-tail claims development. ASTIN - September 2005 B.Verdier A. Klinger
JAB Chan Long-tal clams development ASTIN - September 2005 B.Verder A. Klnger Outlne Chan Ladder : comments A frst soluton: Munch Chan Ladder JAB Chan Chan Ladder: Comments Black lne: average pad to ncurred
More informationScatter Plot x
Construct a scatter plot usng excel for the gven data. Determne whether there s a postve lnear correlaton, negatve lnear correlaton, or no lnear correlaton. Complete the table and fnd the correlaton coeffcent
More informationDifferentiating Gaussian Processes
Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the
More information10-701/ 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 informationSome basic statistics and curve fitting techniques
Some basc statstcs and curve fttng technques Statstcs s the dscplne concerned wth the study of varablty, wth the study of uncertanty, and wth the study of decsonmakng n the face of uncertanty (Lndsay et
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More informationPubH 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 informationThis column is a continuation of our previous column
Comparson of Goodness of Ft Statstcs for Lnear Regresson, Part II The authors contnue ther dscusson of the correlaton coeffcent n developng a calbraton for quanttatve analyss. Jerome Workman Jr. and Howard
More informationEcon107 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 informationJanuary 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 informationSTAT 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 informationLECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
More informationGravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)
Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng
More informationCorrelation and Regression
Correlaton and Regresson otes prepared by Pamela Peterson Drake Index Basc terms and concepts... Smple regresson...5 Multple Regresson...3 Regresson termnology...0 Regresson formulas... Basc terms and
More informationLecture 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 informationActivity #13: Simple Linear Regression. actgpa.sav; beer.sav;
ctvty #3: Smple Lnear Regresson Resources: actgpa.sav; beer.sav; http://mathworld.wolfram.com/leastfttng.html In the last actvty, we learned how to quantfy the strength of the lnear relatonshp between
More informationSystematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal
9/3/009 Sstematc Error Illustraton of Bas Sources of Sstematc Errors Instrument Errors Method Errors Personal Prejudce Preconceved noton of true value umber bas Prefer 0/5 Small over large Even over odd
More informationUnit 10: Simple Linear Regression and Correlation
Unt 10: Smple Lnear Regresson and Correlaton Statstcs 571: Statstcal Methods Ramón V. León 6/28/2004 Unt 10 - Stat 571 - Ramón V. León 1 Introductory Remarks Regresson analyss s a method for studyng the
More informationThe SAS program I used to obtain the analyses for my answers is given below.
Homework 1 Answer sheet Page 1 The SAS program I used to obtan the analyses for my answers s gven below. dm'log;clear;output;clear'; *************************************************************; *** EXST7034
More informationSociology 301. Bivariate Regression. Clarification. Regression. Liying Luo Last exam (Exam #4) is on May 17, in class.
Socology 30 Bvarate Regresson Lyng Luo 04.28 Clarfcaton Last exam (Exam #4) s on May 7, n class. No exam n the fnal exam week (May 24). Regresson Regresson Analyss: the procedure for esjmajng and tesjng
More informationInterpreting Slope Coefficients in Multiple Linear Regression Models: An Example
CONOMICS 5* -- Introducton to NOT CON 5* -- Introducton to NOT : Multple Lnear Regresson Models Interpretng Slope Coeffcents n Multple Lnear Regresson Models: An xample Consder the followng smple lnear
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationMidterm Examination. Regression and Forecasting Models
IOMS Department Regresson and Forecastng Models Professor Wllam Greene Phone: 22.998.0876 Offce: KMC 7-90 Home page: people.stern.nyu.edu/wgreene Emal: wgreene@stern.nyu.edu Course web page: people.stern.nyu.edu/wgreene/regresson/outlne.htm
More informationANSWERS. 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 informationCathy Walker March 5, 2010
Cathy Walker March 5, 010 Part : Problem Set 1. What s the level of measurement for the followng varables? a) SAT scores b) Number of tests or quzzes n statstcal course c) Acres of land devoted to corn
More information