stat5044Exam1fall10v1sol

stat5044Exam1fall10v1sol - STAT5044: Regression and ANOVA,...

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Unformatted text preview: STAT5044: Regression and ANOVA, Fall 2010 Exam 1 on Nov 3 Your Name: Please make sure to specify all of your notations in each problem GOOD LUCK! 1 Problem# 1. (1.a) Write down a simple linear regression model. Specify each term as random or fixed. Specify each term as unknown term or known term. Y i =random, known, x i =fixed, known, i =random, unknown, , 1 , 2 =unknown, fixed. (1.b) In the simple regression model, suppose the value of the predictor X is replaced by cX , where c is some non zero constant. How are , 1 , 2 , and R 2 , and the t-test of H : 1 = 0 affected by this change? Let x i , new = cx i . Then , new = , 1 , new = 1 c 1 , 2 new = 2 , R 2 new = R 2 , t new = t (1.c) Suppose each value of the response of Y is replaced by dY , for some d 6 = 0. How are , 1 , 2 , and R 2 , and the t-test of H : 1 = 0 affected by this change? Let x i , new = dx i , , new = d , 1 , new = d 1 , 2 = d 2 2 , R 2 new = R 2 , t new = t 2 Problem# 2. Consider the following model Y i = + 1 X i + i , i = 1 ,..., n where E ( i ) = 0, Var ( i ) = 2 (unknown), Cov ( i , j ) = 0. (2.a) I would like to predict conditional mean of Y given a x new and a new observa- tion y * given a x new . First I estimated a point estimator + 1 x new . What estimator would you use , 1 , and 2 ? Why? I would like to use LSE (below) because they are unbiased and have minimum vari- ance among all the unbiased estimators. = ( X X ) T X Y , X = [ 1 , x ] 2 = Y ( I- H ) Y n- 2 (2.b) Under the assumption of i N ( , 2 ) , what is the distribution of + 1 x new ? Why? Let X new = ( 1 , x new ) and = ( , 1 ) T . Then Y new = X new = X new ( X X ) T X Y which is a linear form of Y. Since Y is following normal distribution and any linear form of Y is also following normal distribution, Y new is following normal distribution with E ( Y new ) = X new and Var ( Y new ) = 2 X new ( X X )- 1 X new...
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This note was uploaded on 01/02/2012 for the course STAT 5044` taught by Professor Staff during the Fall '11 term at Virginia Tech.

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stat5044Exam1fall10v1sol - STAT5044: Regression and ANOVA,...

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