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Unformatted text preview: x 1 x 2 y 3 5 13 3 7 18 2 5 10 2 6 10 2 7 14 1 5 8 1. Consider the model of the form: Y i = β + β 1 X i 1 + β 2 X i 2 + ε i , i = 1, 2, … , 7, where ε i ’s are independent N ( , σ 2 ) random variables. 1 7 11 X T X = & & & & ¡ ¢ £ £ £ £ ¤ ¥ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ 2 2 2 1 2 2 1 2 1 1 2 1 i i i i i i i i i i x x x x x x x x x x n = & & & ¡ ¢ £ £ £ ¤ ¥ 258 84 42 84 32 14 42 14 7 , ( X T X ) – 1 = & & & ¡ ¢ £ £ £ ¤ ¥ ⋅ 28 168 42 84 168 84 1200 168 1 , X T Y = & & & ¡ ¢ £ £ £ ¤ ¥ ¦ ¦ ¦ 2 1 i i i i i y x y x y = & & & ¡ ¢ £ £ £ ¤ ¥ 516 180 84 . a) Fill in the blanks. (You can give a range for the pvalues.) Call: lm(formula = Y ~ X1 + X2) Residuals: 1 2 3 4 5 6 7 0 1 0 2 0 1 0 Coefficients: Estimate Std. Error t value Pr(>t) (Intercept) X1 X2  Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: on degrees of freedom Multiple RSquared: , Adjusted Rsquared: Fstatistic: on and DF, pvalue: b) Construct a 95% prediction interval for the value of Y at x 1 = 3 and x 2 = 6. 2. Listed below are the price quotations of used cars along with their age and odometer mileage. A multiple linear regression analysis was performed using R, the output is given below. Age (years) X1 Mileage (thousand miles) X2 Price (thousand dollars) Y 1 1 8.1 9.45 2 2 17 8.4 3 2 12.6 8.6 4 3 18.4 6.8 5 3 19.5 6.5 6 4 29.2 5.6 7 6 40.4 4.75 8 7 51.6 3.89 9 8 62.6 2.7 10 10 80.1 1.47 > autos.fit < lm(Y ~ X1 + X2) > summary(autos.fit) Call: lm(formula = Y ~ X1 + X2) Residuals: Min 1Q Median 3Q Max 0.7390 0.2545 0.1114 0.3066 0.5674 Coefficients: Estimate Std. Error t value Pr(>t) (Intercept) 9.98543 0.34289 29.121 1.45e08 *** X1 1.38474 & ¡ X2 0.06481 ¢ £ Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: ¤ on ¥ degrees of freedom Multiple RSquared: ¦ , Adjusted Rsquared: § Fstatistic: ¨ on © 1 and © 2 DF, pvalue: > X < cbind(c(rep(1,10)), X1, X2) > X [,1] [,2] [,3] [1,] 1 1 8.1 [2,] 1 2 17.0 [3,] 1 2 12.6 [4,] 1 3 18.4 [5,] 1 3 19.5 [6,] 1 4 29.2 [7,] 1 6 40.4 [8,] 1 7 51.6 [9,] 1 8 62.6 [10,] 1 10 80.1 > solve(t(X) %*% X) [,1] [,2] [,3] [1,] 0.47166883 0.3716589 0.03940979 [2,] 0.37165893 0.9238623 0.11422997 [3,] 0.03940979 0.1142300 0.01431659 > sum(autos.fit$residuals^2) # RSS [1] 1.744894 > sum((Ymean(Y))^2) # SYY [1] 62.55944 a) Fill in & and ¡ . b) Fill in ¢ and £ . c) Fill in ¤ and ¥ . Is regression significant at a 1% level of significance?...
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This note was uploaded on 02/10/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 STEPANOV
 Statistics

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