In a multiple linear regression analysis, ( y i , x 1 i , x 2 i ), i 1, , n , are statistically independent and satisfy the model (M1) given by y B 0 B 1 x 1 B 2 x 2 , where the response variable y is continuous, the regressor vector X ( x 1 , x 2 ) T has mean vector ( x 1 , x 2 ) T and positive definite variance covariance matri x X , the random errors i conditional on X are statistically independent and normally distributed with mean zero and variance 2 which does not depend on X After trying many multiple linear regression analyses based on the model M1 in Problem 1 above, a data analyst obtained the following statistics The regression sum of squares SS ( B 0) by e x cluding the two regressors x 1 , x 2 The regression sums of squares SS ( B 0), SS ( B 0 , B 1 ), by e x cluding the regressor x 2 The regression sums of squares SS ( B 0), SS ( B 0 , B 2 ), by e x cluding the regressor x 1 The regression sums of squares SS ( B 0), SS ( B 0 , B 1 , B 2 ) that is, include all the regressors a) Discuss with mathematical proof whether the hypothesis that no regressor has a nonzero effect on the response variable can be tested b) Discuss with mathematical proof whether SS ( B 0 , B 1 , B 2 ) SS ( B 0 , B 1 ) SS ( B 0 , B 2 ) c) Discuss with mathematical proof whether S ( B 2 B 0 , B 1 ) SS ( B 2 B 0 ) In a multiple linear regression analysis, ypxli , x21), i 1, , n , are statistically independent and satisfy the model (M1) given by 3 30 31 32 5 where the response variable y is continuous, the regressor vector X (x1, x2 r has mean vector (,uxl, p)' and positive definite variance covariance matrix EX , the random errors a, conditional on X, are statistically independent and normally distributed with mean zero and variance 02 which does not depend on X After trying many multiple linear regression analyses based on the model M1 a data analyst obtained the following statistics The regression sum of squares 55( 30 by excluding the two regressors x1, x2 The regression sums of squares 55(30),SS(30, 31), by excluding the regressor x2 The regression sums of squares 55(30),SS(30, 52), by excluding the regressor x1 The regression sums of squares 55(30),SS(30,51,,82) that is, include all the regressors a Discuss with mathematical proof whether the hypothesis that no regressor has a nonzero effect on the response variable can be tested b) Discuss with mathematical proof whether 55(50, 51,32) 55(n,,81) 55(30, 5 ) (1 Discuss with mathematical proof whether 55(32 30,31) 55(32 ,8 )

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In a multiple linear regression analysis, ( y i , x 1 i , x 2 i ), i = 1, ... , n ,

In a multiple linear regression analysis, (y_i , x_1i , x_2i ), i = 1, ... , n , are statistically independent and satisfy the model (M1) given by:

y = B₀ + B₁x₁ + B₂x₂ + ? ,

where the response variable y is continuous, the regressor vector X = (x₁, x₂)^T has mean vector (?_x1, ?_x2)^T and positive definite variance-covariance matrix ?_X , the random errors ?_i conditional on X_? are statistically independent and normally distributed with mean zero and variance ?² which does not depend on X .

After trying many multiple linear regression analyses based on the model M1 in Problem 1 above, a data analyst obtained the following statistics.

The regression sum of squares SS(B0) by excluding the two regressors x₁, x₂ .

The regression sums of squares SS(B0), SS(B₀, B₁), by excluding the regressor x₂.

The regression sums of squares SS(B0), SS(B₀, B₂), by excluding the regressor x₁.

The regression sums of squares SS(B0), SS(B₀, B₁, B₂); that is, include all the regressors.

a) Discuss with mathematical proof whether the hypothesis that no regressor has a nonzero effect on the response variable can be tested.

b) Discuss with mathematical proof whether SS(B₀, B₁, B₂) = SS(B₀, B₁) + SS(B₀, B₂) .

c) Discuss with mathematical proof whether S?(B₂ | B₀, B₁ ) = SS(B₂ | B₀ ) .

In a multiple linear regression analysis, {ypxli , x21), i = 1, ..., n , are statistically independent and satisfy the model (M1) given by: 3" :30 +31% +32% +5; where the response variable y is continuous, the regressor vector X = (x1, x2]r has mean vector (,uxl, p)' and positive definite variance-covariance matrix EX , the random errors a, conditional on X,- are statistically independent and normally distributed with mean zero and variance 02 which does not depend on X . After trying many multiple linear regression analyses based on the model M1 a data analyst obtained the following statistics. The regression sum of squares 55([30] by excluding the two regressors x1, x2 . The regression sums of squares 55(30),SS(30, {31), by excluding the regressor x2 . The regression sums of squares 55(30),SS(30, 52), by excluding the regressor x1 . The regression sums of squares 55(30),SS(30,51,,82); that is, include all the regressors. a} Discuss with mathematical proof whether the hypothesis that no regressor has a nonzero effect on the response variable can be tested. b) Discuss with mathematical proof whether 55(50, 51,32) : 55(n,,81) + 55(30, 5?) . (1] Discuss with mathematical proof whether 55(32 | 30,31) = 55(32 | ,8\")