91Ó°ÊÓ

A linear model describes the correlation between the dependent and independent variables as a straight line.

\(y = {a_0} + {a_1}{x_1} + {a_2}{x_2} + \tilde A,\hat A1/4 + {a_n}{x_n}\)

Models using only one predictor are simple linear regression models. Multiple predictors are used in multiple linear regression models. For many response variables, multiple regression analysis models are used.

All the necessary quantities can be found in the covariance matrix which can be found as,

The design matrix is

\(Z = \left( {\begin{array}{*{20}{c}}1&{{x_{1,1}}}&{{x_{1,2}}}\\1&{{x_{2,1}}}&{{x_{2,2}}}\\ \vdots & \vdots & \vdots \\1&{{x_{10,1}}}&{{x_{10,2}}}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&{1.9}&{66}\\1&{0.8}&{62}\\1&{1.1}&{64}\\1&{0.1}&{61}\\1&{ - 0.1}&{63}\\1&{4.4}&{70}\\1&{4.6}&{68}\\1&{1.6}&{62}\\1&{5.5}&{68}\\1&{3.4}&{66}\end{array}} \right]\)

This is because

\({\mathop{\rm cov}} (\hat \beta ) = \left[ {\begin{array}{*{20}{c}}{Var\left( {{{\hat \beta }_0}} \right)}&{{\mathop{\rm cov}} \left( {{{\hat \beta }_0},{{\hat \beta }_1}} \right)}&{{\mathop{\rm cov}} \left( {{{\hat \beta }_0},{{\hat \beta }_2}} \right)}\\{{\mathop{\rm cov}} \left( {{{\hat \beta }_1},{{\hat \beta }_0}} \right)}&{Var\left( {{{\hat \beta }_1}} \right)}&{{\mathop{\rm cov}} \left( {{{\hat \beta }_1},{{\hat \beta }_2}} \right)}\\{{\mathop{\rm cov}} \left( {{{\hat \beta }_2},{{\hat \beta }_0}} \right)}&{{\mathop{\rm cov}} \left( {{{\hat \beta }_2},{{\hat \beta }_1}} \right)}&{Var\left( {{{\hat \beta }_2}} \right)}\end{array}} \right]\)

Hence the desired matrix is \(\left( {\begin{array}{*{20}{c}}1&{1.9}&{66}\\1&{0.8}&{62}\\1&{1.1}&{64}\\1&{0.1}&{61}\\1&{ - 0.1}&{63}\\1&{4.4}&{70}\\1&{4.6}&{68}\\1&{1.6}&{62}\\1&{5.5}&{68}\\1&{3.4}&{66}\end{array}} \right)\)

The desired values can be found in the upper triangle of the matrix found above.

We can find that,

Finally we have

Hence the values are \(\begin{array}{*{20}{c}}{Var\left( {{{\hat \beta }_0}} \right) = 222.725{\sigma ^2},Var\left( {{{\hat \beta }_1}} \right) = 0.135{\sigma ^2},Var\left( {{{\hat \beta }_2}} \right) = 0.058{\sigma ^2}}\\{{\mathop{\rm cov}} \left( {{{\hat \beta }_0},{{\hat \beta }_1}} \right) = 4.832{\sigma ^2},{\mathop{\rm cov}} \left( {{{\hat \beta }_0},{{\hat \beta }_2}} \right) = - 3.598{\sigma ^2},{\mathop{\rm cov}} \left( {{{\hat \beta }_1},{{\hat \beta }_2}} \right) = - 0.079{\sigma ^2}}\end{array}\)