Question

Question:For the conditions of Exercise \({\bf{12}}\), and the data in Table \({\bf{11}}{\bf{.2}}\), determine the value of \({{\bf{R}}^{\bf{2}}}\), as defined by Eq.\({\bf{(11}}{\bf{.5}}{\bf{.26)}}\).

Short answer

The value of \({R^2}\)is \(0.6627\).

Step-by-step solution

Define linear statistical models

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.

Find the value of \({R^2}\)

For the model of one of the previous exercises and the given data, we need to determine the value of multiple \({R^2}\).

It can be calculated as

For this we have fitted the regression function,

\(y = - 11.453 + 0.450{x_1} + 0.173{x_2}\)

Now we can find the mean.

Now we can calculate that

which means

\({R^2} = 1 - \frac{{8.875}}{{26.309}} = 0.6627{\rm{\;or \;}}66.27{\rm{\% }}\)

Hence the value of \({R^2}\)is \(0.6627\).