MATH 644: Regression Analysis Methods
|
|
- Hilda Lang
- 5 years ago
- Views:
Transcription
1 MATH 644: Regression Analysis Methods FINAL EXAM Fall, 2012 INSTRUCTIONS TO STUDENTS: 1. This test contains SIX questions. It comprises ELEVEN printed pages. 2. Answer ALL questions for a total of 100 marks. 3. This is an open-book and open-note test; you can use any materials you have. 4. Write your name on the front of your answer booklet and on any additional sheets you write on. 1
2 1. True/False. Please read each statement and put T(True)/F(False) in the beginning. Notice: the standard least squares estimator is applied whenever needed in the statements. Define the standard multiple linear regression model as follows: Y i = β 0 + β 1 X i1 + + β p X ip + ε i, where ε i i.i.d. N(0, σ 2 ). (a) related. A coefficient of determination zero indicates that X and Y are not (b) (c) (d) (e) A high coefficient of determination indicates that the estimated regression line is a good fit. If two multiple linear regression models have the same mean squared error (MSE), we prefer the model with less variables. For any F-test associated with the multiple linear regression model, we can find an equivalent t-test. In a standard multiple linear regression model, the variance of the prediction becomes larger as X j deviates from the sample mean X j. (f) (g) In a standard multiple linear regression model, define the residuals to be e i = Y i Ŷi, we have n i=1 e ix ij = 0 for all j = 1,..., p 1. In a standard multiple linear regression model, the prediction for a new observation with predictors X (new) = ( X 1, X 2,..., X p ) is Ȳ = n 1 n i=1 Y i, where X j = n 1 n i=1 X ji, j = 1,..., p is the sample mean. 2. Yes/No. Suppose you have four possible predictor variables X 1, X 2, X 3, and X 4 that could be used in a regression analysis. You run a forward selection procedure, and the variables are entered as follows: Step 1: X 2 Step 2: X 4 Step 3: X 1 Step 4: X 3 In other words, after Step 1, the model is E{Y } = β 0 + β 1 X 2. After Step 2, the model is E{Y } = β 0 + β 1 X 2 + β 2 X 4. 2
3 And so on. You also run an all subsets regression analysis using R 2 as the criterion for the best model for each possible number of predictors. Would the same models result from this analysis as from the forward selection procedure? In other words, would all subsets regression definitely identify the following as the best models for 1, 2, 3, and 4 variables? Choose Yes or No in each case. (a) β variable, the best model would be E{Y } = β 0 + β 1 X 2. (b) β variables, the best model would be E{Y } = β 0 + β 1 X 2 + β 2 X 4. (c) β variables, the best model would be E{Y } = β 0 + β 1 X 2 + β 2 X 4 + β 3 X 1. (d) β variables, the best model would be E{Y } = β 0 + β 1 X 2 + β 2 X 4 + β 3 X 1 + β 4 X Given data pairs (X i, Y i ), where i = 1,..., n. We fit the simple linear regression Y i = β 0 + β 1 X i + ε i. Suppose, in addition, ε i. are independent, normally distributed with mean 0 and variance σ 2. For each of the following three scenarios, how are b 0, b 1, σ 2, R 2 and the t-test of H 0 : β 1 = 0 v.s. H a : β 1 0 affected? Please answer accordingly and make necessary explanations. (a) X i is replaced by 2X i and Y i remains the same. (b) Y i is replaced by 2Y i and X i remains the same. (c) X i is replaced by 2X i and Y i is replaced by 2Y i. 3
4 (Extra Space for Answers) 4
5 4. Suppose we have the following two multiple linear regression models: Y i = β 0 + β 1 X 1i + β 2 X 2i + ε i (1) and Y i = β 0 + β 1 X 1i + β 2 X 2i + β 3 X 3i + ε i, (2) where ε i i.i.d. N(0, σ 2 ). We first perform the analysis for model (1) in R: > fit12 = lm(y ~ X1 + X2) > summary(fit12) Call: lm(formula = Y ~ X1 + X2) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) X <2e-16 X <2e-16 Residual standard error: on 97 degrees of freedom Multiple R-squared: 0.938, Adjusted R-squared:??? F-statistic: on 2 and 97 DF, p-value: < 2.2e-16 (a) Calculate the adjusted R-square value from the output. (b) Calculate the SSR (Regression Sum of Squares) from the output. (c) Perform the hypothesis test, H 0 : β 1 = β 2 = 0 v.s. H 1 : not both β 1 and β 2 equal zero. Write down the test method and calculate the test statistic. 5
6 Now, we perform the analysis for model (2) in R: > fit = lm(y ~ X1 + X2 + X3) > summary(fit) Call: lm(formula = Y ~ X1 + X2 + X3) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) X <2e-16 X <2e-16 X Residual standard error: on 96 degrees of freedom Multiple R-squared: 0.938, Adjusted R-squared: F-statistic: 484 on 3 and 96 DF, p-value: < 2.2e-16 (d) Calculate the Extra Sum of Squares SSR(X 3 X 1, X 2 ) and the coefficient of partial correlation RY (e) Compare model (1) and model (2), which one do you prefer and explain the reasons. 6
7 (Extra Space for Answers) 7
8 5. An analyst decided to fit the multiple regression model Y i = β 0 + β 1 X i1 + β 2 X i2 + β 3 X i3 + β 4 X i1 X i2 + β 5 X i1 X i3 + β 6 X i2 X i3 + ε i, where ε i N(0, σ 2 ), i = 1,..., 20. To reduce correlation between the covariates in this model, the centered variables x i1 = X i1 X 1 = X i , x i2 = X i2 X 2 = X i , and x i3 = X i3 X 3 = X i are used. The fitted regression equation is given by Ŷ = x x x x 1 x x 1 x x 2 x 3, MSE = 6.745, where the true model is Y i = β 0 + β 1 x i1 + β 2 x i2 + β 3 x i3 + β 4 x i1 x i2 + β 5 x i1 x i3 + β 6 x i2 x i3 + ε i. One would like to test whether the interaction terms between the three predictor variables should be included in the regression model. Use the above information and the following Table to conduct a F-test at 5% significance level. Clearly state the null and alternate hypotheses, test statistic, decision rule and the conclusion. Variable Extra Sum of Squares Value x 1 SSR(x 1 ) = x 2 SSR(x 2 x 1 ) = x 3 SSR(x 3 x 1, x 2 ) = x 1 x 2 SSR(x 1 x 2 x 1, x 2, x 3 ) = x 1 x 3 SSR(x 1 x 3 x 1, x 2, x 3, x 1 x 2 ) = x 2 x 3 SSR(x 2 x 3 x 1, x 2, x 3, x 1 x 2, x 1 x 3 ) = F(0.975, 3, 13) = , F(0.95, 3, 13) = , F(0.975, 7, 19) = , F(0.95, 7, 19) = , F(0.95, 4, 13) = , F(0.975, 4, 13) = , F(0.95, 4, 19) = , F(0.975, 4, 19) = , F(0.975, 3, 19) = , F(0.95, 3, 19) =
9 (Extra Space for Answers) 9
10 6. Suppose we have the following two multiple linear regression models: Y i = β 0 + β 1 X i1 + + β p 1 X i,p 1 + ε i (3) and Y i = β 0 + β 1 X i1 + + β p 1 X i,p 1 + β p X i,p + ε i, (4) where ε i i.i.d. N(0, σ 2 ). (a) Denote the R 2 (the coefficient of multiple determination) for the two models (3) and (4) as R 2 (3) and R 2 (4). Is it true that R 2 (3) R 2 (4) always holds? If yes, prove it. If not, give a counter example. (If you are providing a counter example, please write down the design matrix X and the response vector Y explicitly. The reasoning of R 2 (3) > R 2 (4) is required.) (b) Denote the Ra 2 (the adjusted coefficient of multiple determination) for the two models as Ra(3) 2 and Ra(4). 2 Is it true that Ra(3) 2 Ra(4) 2 always holds? If yes, prove it. If not, give a counter example. (If you are providing a counter example, please write down the design matrix X and the response vector Y explicitly. The reasoning of Ra(3) 2 > Ra(4) 2 is required.) 10
11 (Extra Space for Answers) 11
Stat 5102 Final Exam May 14, 2015
Stat 5102 Final Exam May 14, 2015 Name Student ID The exam is closed book and closed notes. You may use three 8 1 11 2 sheets of paper with formulas, etc. You may also use the handouts on brand name distributions
More informationApplied Regression Analysis
Applied Regression Analysis Chapter 3 Multiple Linear Regression Hongcheng Li April, 6, 2013 Recall simple linear regression 1 Recall simple linear regression 2 Parameter Estimation 3 Interpretations of
More informationSTATISTICS 110/201 PRACTICE FINAL EXAM
STATISTICS 110/201 PRACTICE FINAL EXAM Questions 1 to 5: There is a downloadable Stata package that produces sequential sums of squares for regression. In other words, the SS is built up as each variable
More informationInference for Regression
Inference for Regression Section 9.4 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 13b - 3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationST430 Exam 2 Solutions
ST430 Exam 2 Solutions Date: November 9, 2015 Name: Guideline: You may use one-page (front and back of a standard A4 paper) of notes. No laptop or textbook are permitted but you may use a calculator. Giving
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationST430 Exam 1 with Answers
ST430 Exam 1 with Answers Date: October 5, 2015 Name: Guideline: You may use one-page (front and back of a standard A4 paper) of notes. No laptop or textook are permitted but you may use a calculator.
More informationMath 3330: Solution to midterm Exam
Math 3330: Solution to midterm Exam Question 1: (14 marks) Suppose the regression model is y i = β 0 + β 1 x i + ε i, i = 1,, n, where ε i are iid Normal distribution N(0, σ 2 ). a. (2 marks) Compute the
More informationOutline. Remedial Measures) Extra Sums of Squares Standardized Version of the Multiple Regression Model
Outline 1 Multiple Linear Regression (Estimation, Inference, Diagnostics and Remedial Measures) 2 Special Topics for Multiple Regression Extra Sums of Squares Standardized Version of the Multiple Regression
More informationLecture 2. The Simple Linear Regression Model: Matrix Approach
Lecture 2 The Simple Linear Regression Model: Matrix Approach Matrix algebra Matrix representation of simple linear regression model 1 Vectors and Matrices Where it is necessary to consider a distribution
More informationCh 3: Multiple Linear Regression
Ch 3: Multiple Linear Regression 1. Multiple Linear Regression Model Multiple regression model has more than one regressor. For example, we have one response variable and two regressor variables: 1. delivery
More informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
More informationConcordia University (5+5)Q 1.
(5+5)Q 1. Concordia University Department of Mathematics and Statistics Course Number Section Statistics 360/1 40 Examination Date Time Pages Mid Term Test May 26, 2004 Two Hours 3 Instructor Course Examiner
More informationMODELS WITHOUT AN INTERCEPT
Consider the balanced two factor design MODELS WITHOUT AN INTERCEPT Factor A 3 levels, indexed j 0, 1, 2; Factor B 5 levels, indexed l 0, 1, 2, 3, 4; n jl 4 replicate observations for each factor level
More informationSTAT 350: Summer Semester Midterm 1: Solutions
Name: Student Number: STAT 350: Summer Semester 2008 Midterm 1: Solutions 9 June 2008 Instructor: Richard Lockhart Instructions: This is an open book test. You may use notes, text, other books and a calculator.
More informationLecture 6 Multiple Linear Regression, cont.
Lecture 6 Multiple Linear Regression, cont. BIOST 515 January 22, 2004 BIOST 515, Lecture 6 Testing general linear hypotheses Suppose we are interested in testing linear combinations of the regression
More informationNo other aids are allowed. For example you are not allowed to have any other textbook or past exams.
UNIVERSITY OF TORONTO SCARBOROUGH Department of Computer and Mathematical Sciences Sample Exam Note: This is one of our past exams, In fact the only past exam with R. Before that we were using SAS. In
More informationSTAT 512 MidTerm I (2/21/2013) Spring 2013 INSTRUCTIONS
STAT 512 MidTerm I (2/21/2013) Spring 2013 Name: Key INSTRUCTIONS 1. This exam is open book/open notes. All papers (but no electronic devices except for calculators) are allowed. 2. There are 5 pages in
More informationTests of Linear Restrictions
Tests of Linear Restrictions 1. Linear Restricted in Regression Models In this tutorial, we consider tests on general linear restrictions on regression coefficients. In other tutorials, we examine some
More informationMS&E 226: Small Data
MS&E 226: Small Data Lecture 15: Examples of hypothesis tests (v5) Ramesh Johari ramesh.johari@stanford.edu 1 / 32 The recipe 2 / 32 The hypothesis testing recipe In this lecture we repeatedly apply the
More informationFinal Review. Yang Feng. Yang Feng (Columbia University) Final Review 1 / 58
Final Review Yang Feng http://www.stat.columbia.edu/~yangfeng Yang Feng (Columbia University) Final Review 1 / 58 Outline 1 Multiple Linear Regression (Estimation, Inference) 2 Special Topics for Multiple
More informationMultiple Linear Regression
Multiple Linear Regression Simple linear regression tries to fit a simple line between two variables Y and X. If X is linearly related to Y this explains some of the variability in Y. In most cases, there
More informationStatistics for Engineers Lecture 9 Linear Regression
Statistics for Engineers Lecture 9 Linear Regression Chong Ma Department of Statistics University of South Carolina chongm@email.sc.edu April 17, 2017 Chong Ma (Statistics, USC) STAT 509 Spring 2017 April
More information22s:152 Applied Linear Regression. Take random samples from each of m populations.
22s:152 Applied Linear Regression Chapter 8: ANOVA NOTE: We will meet in the lab on Monday October 10. One-way ANOVA Focuses on testing for differences among group means. Take random samples from each
More informationUNIVERSITY OF MASSACHUSETTS. Department of Mathematics and Statistics. Basic Exam - Applied Statistics. Tuesday, January 17, 2017
UNIVERSITY OF MASSACHUSETTS Department of Mathematics and Statistics Basic Exam - Applied Statistics Tuesday, January 17, 2017 Work all problems 60 points are needed to pass at the Masters Level and 75
More informationCorrelation Analysis
Simple Regression Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the
More information22s:152 Applied Linear Regression. There are a couple commonly used models for a one-way ANOVA with m groups. Chapter 8: ANOVA
22s:152 Applied Linear Regression Chapter 8: ANOVA NOTE: We will meet in the lab on Monday October 10. One-way ANOVA Focuses on testing for differences among group means. Take random samples from each
More informationSimple Linear Regression
Simple Linear Regression In simple linear regression we are concerned about the relationship between two variables, X and Y. There are two components to such a relationship. 1. The strength of the relationship.
More informationSTAT420 Midterm Exam. University of Illinois Urbana-Champaign October 19 (Friday), :00 4:15p. SOLUTIONS (Yellow)
STAT40 Midterm Exam University of Illinois Urbana-Champaign October 19 (Friday), 018 3:00 4:15p SOLUTIONS (Yellow) Question 1 (15 points) (10 points) 3 (50 points) extra ( points) Total (77 points) Points
More informationSCHOOL OF MATHEMATICS AND STATISTICS
SHOOL OF MATHEMATIS AND STATISTIS Linear Models Autumn Semester 2015 16 2 hours Marks will be awarded for your best three answers. RESTRITED OPEN BOOK EXAMINATION andidates may bring to the examination
More informationUNIVERSITY OF TORONTO SCARBOROUGH Department of Computer and Mathematical Sciences Midterm Test, October 2013
UNIVERSITY OF TORONTO SCARBOROUGH Department of Computer and Mathematical Sciences Midterm Test, October 2013 STAC67H3 Regression Analysis Duration: One hour and fifty minutes Last Name: First Name: Student
More informationChapter 14 Simple Linear Regression (A)
Chapter 14 Simple Linear Regression (A) 1. Characteristics Managerial decisions often are based on the relationship between two or more variables. can be used to develop an equation showing how the variables
More informationReview: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.
1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately
More informationBiostatistics 380 Multiple Regression 1. Multiple Regression
Biostatistics 0 Multiple Regression ORIGIN 0 Multiple Regression Multiple Regression is an extension of the technique of linear regression to describe the relationship between a single dependent (response)
More information14 Multiple Linear Regression
B.Sc./Cert./M.Sc. Qualif. - Statistics: Theory and Practice 14 Multiple Linear Regression 14.1 The multiple linear regression model In simple linear regression, the response variable y is expressed in
More informationCoefficient of Determination
Coefficient of Determination ST 430/514 The coefficient of determination, R 2, is defined as before: R 2 = 1 SS E (yi ŷ i ) = 1 2 SS yy (yi ȳ) 2 The interpretation of R 2 is still the fraction of variance
More informationVariance Decomposition and Goodness of Fit
Variance Decomposition and Goodness of Fit 1. Example: Monthly Earnings and Years of Education In this tutorial, we will focus on an example that explores the relationship between total monthly earnings
More informationSTAT763: Applied Regression Analysis. Multiple linear regression. 4.4 Hypothesis testing
STAT763: Applied Regression Analysis Multiple linear regression 4.4 Hypothesis testing Chunsheng Ma E-mail: cma@math.wichita.edu 4.4.1 Significance of regression Null hypothesis (Test whether all β j =
More informationFigure 1: The fitted line using the shipment route-number of ampules data. STAT5044: Regression and ANOVA The Solution of Homework #2 Inyoung Kim
0.0 1.0 1.5 2.0 2.5 3.0 8 10 12 14 16 18 20 22 y x Figure 1: The fitted line using the shipment route-number of ampules data STAT5044: Regression and ANOVA The Solution of Homework #2 Inyoung Kim Problem#
More informationLinear Regression Model. Badr Missaoui
Linear Regression Model Badr Missaoui Introduction What is this course about? It is a course on applied statistics. It comprises 2 hours lectures each week and 1 hour lab sessions/tutorials. We will focus
More informationSCHOOL OF MATHEMATICS AND STATISTICS
RESTRICTED OPEN BOOK EXAMINATION (Not to be removed from the examination hall) Data provided: Statistics Tables by H.R. Neave MAS5052 SCHOOL OF MATHEMATICS AND STATISTICS Basic Statistics Spring Semester
More informationDensity Temp vs Ratio. temp
Temp Ratio Density 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Density 0.0 0.2 0.4 0.6 0.8 1.0 1. (a) 170 175 180 185 temp 1.0 1.5 2.0 2.5 3.0 ratio The histogram shows that the temperature measures have two peaks,
More informationComparing Nested Models
Comparing Nested Models ST 370 Two regression models are called nested if one contains all the predictors of the other, and some additional predictors. For example, the first-order model in two independent
More informationSTATISTICS 174: APPLIED STATISTICS FINAL EXAM DECEMBER 10, 2002
Time allowed: 3 HOURS. STATISTICS 174: APPLIED STATISTICS FINAL EXAM DECEMBER 10, 2002 This is an open book exam: all course notes and the text are allowed, and you are expected to use your own calculator.
More informationSCHOOL OF MATHEMATICS AND STATISTICS. Linear and Generalised Linear Models
SCHOOL OF MATHEMATICS AND STATISTICS Linear and Generalised Linear Models Autumn Semester 2017 18 2 hours Attempt all the questions. The allocation of marks is shown in brackets. RESTRICTED OPEN BOOK EXAMINATION
More informationMatrices and vectors A matrix is a rectangular array of numbers. Here s an example: A =
Matrices and vectors A matrix is a rectangular array of numbers Here s an example: 23 14 17 A = 225 0 2 This matrix has dimensions 2 3 The number of rows is first, then the number of columns We can write
More informationCAS MA575 Linear Models
CAS MA575 Linear Models Boston University, Fall 2013 Midterm Exam (Correction) Instructor: Cedric Ginestet Date: 22 Oct 2013. Maximal Score: 200pts. Please Note: You will only be graded on work and answers
More informationLecture 10 Multiple Linear Regression
Lecture 10 Multiple Linear Regression STAT 512 Spring 2011 Background Reading KNNL: 6.1-6.5 10-1 Topic Overview Multiple Linear Regression Model 10-2 Data for Multiple Regression Y i is the response variable
More informationExam Applied Statistical Regression. Good Luck!
Dr. M. Dettling Summer 2011 Exam Applied Statistical Regression Approved: Tables: Note: Any written material, calculator (without communication facility). Attached. All tests have to be done at the 5%-level.
More informationLeverage. the response is in line with the other values, or the high leverage has caused the fitted model to be pulled toward the observed response.
Leverage Some cases have high leverage, the potential to greatly affect the fit. These cases are outliers in the space of predictors. Often the residuals for these cases are not large because the response
More informationLecture 18: Simple Linear Regression
Lecture 18: Simple Linear Regression BIOS 553 Department of Biostatistics University of Michigan Fall 2004 The Correlation Coefficient: r The correlation coefficient (r) is a number that measures the strength
More informationMultiple Linear Regression
Multiple Linear Regression ST 430/514 Recall: a regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates).
More informationRegression and the 2-Sample t
Regression and the 2-Sample t James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) Regression and the 2-Sample t 1 / 44 Regression
More informationChapter 4: Regression Models
Sales volume of company 1 Textbook: pp. 129-164 Chapter 4: Regression Models Money spent on advertising 2 Learning Objectives After completing this chapter, students will be able to: Identify variables,
More informationRegression Review. Statistics 149. Spring Copyright c 2006 by Mark E. Irwin
Regression Review Statistics 149 Spring 2006 Copyright c 2006 by Mark E. Irwin Matrix Approach to Regression Linear Model: Y i = β 0 + β 1 X i1 +... + β p X ip + ɛ i ; ɛ i iid N(0, σ 2 ), i = 1,..., n
More information(ii) Scan your answer sheets INTO ONE FILE only, and submit it in the drop-box.
FINAL EXAM ** Two different ways to submit your answer sheet (i) Use MS-Word and place it in a drop-box. (ii) Scan your answer sheets INTO ONE FILE only, and submit it in the drop-box. Deadline: December
More informationVariance Decomposition in Regression James M. Murray, Ph.D. University of Wisconsin - La Crosse Updated: October 04, 2017
Variance Decomposition in Regression James M. Murray, Ph.D. University of Wisconsin - La Crosse Updated: October 04, 2017 PDF file location: http://www.murraylax.org/rtutorials/regression_anovatable.pdf
More informationCategorical Predictor Variables
Categorical Predictor Variables We often wish to use categorical (or qualitative) variables as covariates in a regression model. For binary variables (taking on only 2 values, e.g. sex), it is relatively
More informationFormal Statement of Simple Linear Regression Model
Formal Statement of Simple Linear Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters X i is a known constant, the value of the predictor
More informationT.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS
ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only
More informationSTA 4210 Practise set 2a
STA 410 Practise set a For all significance tests, use = 0.05 significance level. S.1. A multiple linear regression model is fit, relating household weekly food expenditures (Y, in $100s) to weekly income
More informationSTAT 525 Fall Final exam. Tuesday December 14, 2010
STAT 525 Fall 2010 Final exam Tuesday December 14, 2010 Time: 2 hours Name (please print): Show all your work and calculations. Partial credit will be given for work that is partially correct. Points will
More informationLecture 4 Multiple linear regression
Lecture 4 Multiple linear regression BIOST 515 January 15, 2004 Outline 1 Motivation for the multiple regression model Multiple regression in matrix notation Least squares estimation of model parameters
More informationHomework 2: Simple Linear Regression
STAT 4385 Applied Regression Analysis Homework : Simple Linear Regression (Simple Linear Regression) Thirty (n = 30) College graduates who have recently entered the job market. For each student, the CGPA
More informationPubH 7405: REGRESSION ANALYSIS. MLR: INFERENCES, Part I
PubH 7405: REGRESSION ANALYSIS MLR: INFERENCES, Part I TESTING HYPOTHESES Once we have fitted a multiple linear regression model and obtained estimates for the various parameters of interest, we want to
More informationSCHOOL OF MATHEMATICS AND STATISTICS Autumn Semester
RESTRICTED OPEN BOOK EXAMINATION (Not to be removed from the examination hall) Data provided: "Statistics Tables" by H.R. Neave PAS 371 SCHOOL OF MATHEMATICS AND STATISTICS Autumn Semester 2008 9 Linear
More informationLecture 15 Multiple regression I Chapter 6 Set 2 Least Square Estimation The quadratic form to be minimized is
Lecture 15 Multiple regression I Chapter 6 Set 2 Least Square Estimation The quadratic form to be minimized is Q = (Y i β 0 β 1 X i1 β 2 X i2 β p 1 X i.p 1 ) 2, which in matrix notation is Q = (Y Xβ) (Y
More informationSwarthmore Honors Exam 2012: Statistics
Swarthmore Honors Exam 2012: Statistics 1 Swarthmore Honors Exam 2012: Statistics John W. Emerson, Yale University NAME: Instructions: This is a closed-book three-hour exam having six questions. You may
More informationRegression, Part I. - In correlation, it would be irrelevant if we changed the axes on our graph.
Regression, Part I I. Difference from correlation. II. Basic idea: A) Correlation describes the relationship between two variables, where neither is independent or a predictor. - In correlation, it would
More informationStat 401B Exam 2 Fall 2015
Stat 401B Exam Fall 015 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed ATTENTION! Incorrect numerical answers unaccompanied by supporting reasoning
More informationy i s 2 X 1 n i 1 1. Show that the least squares estimators can be written as n xx i x i 1 ns 2 X i 1 n ` px xqx i x i 1 pδ ij 1 n px i xq x j x
Question 1 Suppose that we have data Let x 1 n x i px 1, y 1 q,..., px n, y n q. ȳ 1 n y i s 2 X 1 n px i xq 2 Throughout this question, we assume that the simple linear model is correct. We also assume
More informationOct Simple linear regression. Minimum mean square error prediction. Univariate. regression. Calculating intercept and slope
Oct 2017 1 / 28 Minimum MSE Y is the response variable, X the predictor variable, E(X) = E(Y) = 0. BLUP of Y minimizes average discrepancy var (Y ux) = C YY 2u C XY + u 2 C XX This is minimized when u
More informationChapter 12: Linear regression II
Chapter 12: Linear regression II Timothy Hanson Department of Statistics, University of South Carolina Stat 205: Elementary Statistics for the Biological and Life Sciences 1 / 14 12.4 The regression model
More informationBNAD 276 Lecture 10 Simple Linear Regression Model
1 / 27 BNAD 276 Lecture 10 Simple Linear Regression Model Phuong Ho May 30, 2017 2 / 27 Outline 1 Introduction 2 3 / 27 Outline 1 Introduction 2 4 / 27 Simple Linear Regression Model Managerial decisions
More informationSimple Linear Regression
Simple Linear Regression September 24, 2008 Reading HH 8, GIll 4 Simple Linear Regression p.1/20 Problem Data: Observe pairs (Y i,x i ),i = 1,...n Response or dependent variable Y Predictor or independent
More informationFinal Exam. Question 1 (20 points) 2 (25 points) 3 (30 points) 4 (25 points) 5 (10 points) 6 (40 points) Total (150 points) Bonus question (10)
Name Economics 170 Spring 2004 Honor pledge: I have neither given nor received aid on this exam including the preparation of my one page formula list and the preparation of the Stata assignment for the
More informationST505/S697R: Fall Homework 2 Solution.
ST505/S69R: Fall 2012. Homework 2 Solution. 1. 1a; problem 1.22 Below is the summary information (edited) from the regression (using R output); code at end of solution as is code and output for SAS. a)
More informationStat 411/511 ESTIMATING THE SLOPE AND INTERCEPT. Charlotte Wickham. stat511.cwick.co.nz. Nov
Stat 411/511 ESTIMATING THE SLOPE AND INTERCEPT Nov 20 2015 Charlotte Wickham stat511.cwick.co.nz Quiz #4 This weekend, don t forget. Usual format Assumptions Display 7.5 p. 180 The ideal normal, simple
More informationWISE International Masters
WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are
More informationChapter 2 Multiple Regression (Part 4)
Chapter 2 Multiple Regression (Part 4) 1 The effect of multi-collinearity Now, we know to find the estimator (X X) 1 must exist! Therefore, n must be great or at least equal to p + 1 (WHY?) However, even
More informationLinear Model Specification in R
Linear Model Specification in R How to deal with overparameterisation? Paul Janssen 1 Luc Duchateau 2 1 Center for Statistics Hasselt University, Belgium 2 Faculty of Veterinary Medicine Ghent University,
More informationStatistics & Data Sciences: First Year Prelim Exam May 2018
Statistics & Data Sciences: First Year Prelim Exam May 2018 Instructions: 1. Do not turn this page until instructed to do so. 2. Start each new question on a new sheet of paper. 3. This is a closed book
More informationLec 1: An Introduction to ANOVA
Ying Li Stockholm University October 31, 2011 Three end-aisle displays Which is the best? Design of the Experiment Identify the stores of the similar size and type. The displays are randomly assigned to
More informationBusiness 320, Fall 1999, Final
Business 320, Fall 1999, Final name You may use a calculator and two cheat sheets. You have 3 hours. I pledge my honor that I have not violated the Honor Code during this examination. Obvioiusly, you may
More information1 Multiple Regression
1 Multiple Regression In this section, we extend the linear model to the case of several quantitative explanatory variables. There are many issues involved in this problem and this section serves only
More information1 Use of indicator random variables. (Chapter 8)
1 Use of indicator random variables. (Chapter 8) let I(A) = 1 if the event A occurs, and I(A) = 0 otherwise. I(A) is referred to as the indicator of the event A. The notation I A is often used. 1 2 Fitting
More informationFirst Year Examination Department of Statistics, University of Florida
First Year Examination Department of Statistics, University of Florida August 19, 010, 8:00 am - 1:00 noon Instructions: 1. You have four hours to answer questions in this examination.. You must show your
More information6. Multiple Linear Regression
6. Multiple Linear Regression SLR: 1 predictor X, MLR: more than 1 predictor Example data set: Y i = #points scored by UF football team in game i X i1 = #games won by opponent in their last 10 games X
More informationChapter 12: Multiple Linear Regression
Chapter 12: Multiple Linear Regression Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 55 Introduction A regression model can be expressed as
More informationBasic Business Statistics 6 th Edition
Basic Business Statistics 6 th Edition Chapter 12 Simple Linear Regression Learning Objectives In this chapter, you learn: How to use regression analysis to predict the value of a dependent variable based
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science
UNIVERSITY OF TORONTO Faculty of Arts and Science December 2013 Final Examination STA442H1F/2101HF Methods of Applied Statistics Jerry Brunner Duration - 3 hours Aids: Calculator Model(s): Any calculator
More informationSchool of Mathematical Sciences. Question 1
School of Mathematical Sciences MTH5120 Statistical Modelling I Practical 8 and Assignment 7 Solutions Question 1 Figure 1: The residual plots do not contradict the model assumptions of normality, constant
More informationBusiness Statistics. Tommaso Proietti. Linear Regression. DEF - Università di Roma 'Tor Vergata'
Business Statistics Tommaso Proietti DEF - Università di Roma 'Tor Vergata' Linear Regression Specication Let Y be a univariate quantitative response variable. We model Y as follows: Y = f(x) + ε where
More informationIntroduction and Single Predictor Regression. Correlation
Introduction and Single Predictor Regression Dr. J. Kyle Roberts Southern Methodist University Simmons School of Education and Human Development Department of Teaching and Learning Correlation A correlation
More information36-707: Regression Analysis Homework Solutions. Homework 3
36-707: Regression Analysis Homework Solutions Homework 3 Fall 2012 Problem 1 Y i = βx i + ɛ i, i {1, 2,..., n}. (a) Find the LS estimator of β: RSS = Σ n i=1(y i βx i ) 2 RSS β = Σ n i=1( 2X i )(Y i βx
More informationMeasuring the fit of the model - SSR
Measuring the fit of the model - SSR Once we ve determined our estimated regression line, we d like to know how well the model fits. How far/close are the observations to the fitted line? One way to do
More informationLinear Modelling in Stata Session 6: Further Topics in Linear Modelling
Linear Modelling in Stata Session 6: Further Topics in Linear Modelling Mark Lunt Arthritis Research UK Epidemiology Unit University of Manchester 14/11/2017 This Week Categorical Variables Categorical
More informationProblems. Suppose both models are fitted to the same data. Show that SS Res, A SS Res, B
Simple Linear Regression 35 Problems 1 Consider a set of data (x i, y i ), i =1, 2,,n, and the following two regression models: y i = β 0 + β 1 x i + ε, (i =1, 2,,n), Model A y i = γ 0 + γ 1 x i + γ 2
More informationTwo Hours. Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER. 26 May :00 16:00
Two Hours MATH38052 Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER GENERALISED LINEAR MODELS 26 May 2016 14:00 16:00 Answer ALL TWO questions in Section
More informationLecture 5: Clustering, Linear Regression
Lecture 5: Clustering, Linear Regression Reading: Chapter 10, Sections 3.1-3.2 STATS 202: Data mining and analysis October 4, 2017 1 / 22 .0.0 5 5 1.0 7 5 X2 X2 7 1.5 1.0 0.5 3 1 2 Hierarchical clustering
More information