Question

Question: Consider a problem of simple linear regression as described in Sec.\({\bf{11}}{\bf{.2}}\) and let \({{\bf{R}}^{\bf{2}}}\), be defined by Eq.\({\bf{(11}}{\bf{.5}}{\bf{.26)}}\) of this section. Show that

\({{\bf{R}}^{\bf{2}}}{\bf{ = }}\frac{{{{\left( {\mathop {\sum {\left( {{{\bf{x}}_{\bf{i}}}{\bf{ - \bar x}}} \right)\left( {{{\bf{y}}_{\bf{i}}}{\bf{ - \bar y}}} \right)} }\limits_{{\bf{i = 1}}}^{\bf{n}} } \right)}^{\bf{2}}}}}{{\left( {\mathop {\sum {{{\left( {{{\bf{x}}_{\bf{i}}}{\bf{ - \bar x}}} \right)}^{\bf{2}}}} }\limits_{{\bf{i = 1}}}^{\bf{n}} } \right)\left( {\mathop {\sum {{{\left( {{{\bf{y}}_{\bf{i}}}{\bf{ - \bar y}}} \right)}^{\bf{2}}}} }\limits_{{\bf{i = 1}}}^{\bf{n}} } \right)}}\)

Short answer

Using the regression formulas we proved that

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 \({R^2}\)

We need to show that in case when the regression function is

\(y = {\beta _0} + {\beta _1}x\)

It can be calculated as

can be expressed as

According to the Theorem 11.1.1

Find the proof of the given statement.

Substitute the value obtained in the previous step in the initial expression for \({R^2}\)yields,

Hence the given statement is proved.