You have fit a multiple linear regression model and the (X\'X)^-1 matrix is (X\'

Question

You have fit a multiple linear regression model and the (X'X)^-1 matrix is (X'X)^-1 = [0.893758 -0.0282448 -0.0175641 -0.028245 0.0013329 0.0001547 -0.0175641 0.0001547 0.0009108] Please write the model of this multiple regression Assume the error sum of squares is 307, and there are 16 observations, what is the estimate of sigma^2 Please calculate the standard error of the regression coefficients beta_1^circ and beta_2^circ Assume the estimated regression coefficients beta_1^circ and beta_2^circ are 2.74427 and 0.012528, respectively. Please calculate the 95% confidence interval for each of them. Assume the intercept is 2.264, the two estimated regression coefficients are 2.74427 and 0.012528. Please construct a 95% prediction interval when X_1 = 4 and X_2 = 250.

Explanation / Answer

1) since dimension of (X'X) -1  is 3*3

hence there are three unknowns in multiple regression

hence the model is

y = b0 + b1 x1 + b2 x2

b) estimate of variance = (Error sum of square ) / )(n-k-1)

here k+1 = 3 , n = 16

hence 307/(16-3) = 307/13 =23.615

c)standard error of b1^ = sqrt ( a22) = sqrt(0.0013329) = 0.0365089

standard error of b2^ = sqrt ( a33) = sqrt(0.0009108) =0.030179

d) 95 % confidence interval is

(estimate - t * se,estimate+t*se)

t = 2.16 for 13 degree of freedom

hence for b1

(2.74427-2.16*0.0365089,2.74427+2.16*0.0365089,)

=(2.66541,2.82312922)

for b2

(0.012528-2.16*0.030179, 0.012528+2.16*0.030179,)

(-0.0526,0.07771)

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