Use hand calculations to fit the multiple linear regression model 1
y = β0 + β1x1 + β2x2
to the data set in DS 13.6.2.
(a) Write down the vector of observed values of the response variable Y and the design matrix X.
(b) Calculate X'X.
(c) Verify that

(d) Verify that β^0 = 4, β^1 = -3, and β^2 = 1.
(e) Calculate the vector of predicted values of the response variable Y and the vector of residuals e.
(f) What is the sum of squares for error?
(g) Show that the estimate of the error variance is σ^2 = 54/7.
(h) What is the standard error of β^1? Of β^2? Should either of the input variables be dropped from the model?
(i) What is the fitted value of the response variable when
x1 = 2 and x2 = -2?
What is the standard error of this fitted value?
(j) Construct a 95% prediction interval for a future value of the response variable obtained with x1 =2 and x2 = -2.

73/680-1/136 l/136 0 (XX)-1 =1-1/136