In exercise 3, the following estimated regression equation...

In exercise $3,$ the following estimated regression equation based on 30 observations was presented. $$\hat{y}=17.6+3.8 x_{1}-2.3 x_{2}+7.6 x_{3}+2.7 x_{4}$$ $\begin{array}{l}{\text { The values of SST and SSR are } 1805 \text { and } 1760 \text { , respectively. }} \\ {\text { a. Compute } R^{2} .} \\ {\text { b. Compute } R_{3}^{2} \text { . }} \\ {\text { c. Comment on the goodness of fit. }}\end{array}$

In exercise $3,$ the following estimated regression equation based on 30 observations was
presented.
$$\hat{y}=17.6+3.8 x_{1}-2.3 x_{2}+7.6 x_{3}+2.7 x_{4}$$
$\begin{array}{l}{\text { The values of SST and SSR are } 1805 \text { and } 1760 \text { , respectively. }} \\ {\text { a. Compute } R^{2} .} \\ {\text { b. Compute } R_{3}^{2} \text { . }} \\ {\text { c. Comment on the goodness of fit. }}\end{array}$

Key Concepts

Goodness of Fit

Goodness of fit refers to how well a statistical model fits a set of observations. In the context of regression, it is usually evaluated with R²—values closer to 1 indicate that the model explains a large proportion of the variability in the response variable. Assessing goodness of fit helps determine the model’s overall effectiveness in capturing the underlying relationship between the independent and dependent variables.

Coefficient of Determination (R²)

R² measures the proportion of the total variance in the dependent variable that is explained by the independent variables in the regression model. It is calculated as the regression sum of squares (SSR) divided by the total sum of squares (SST), or equivalently, one minus the ratio of the error sum of squares (SSE) to SST. This concept is fundamental in assessing how well the model captures the variation in the data.

Partial Coefficient of Determination

The partial R² quantifies the additional explanatory power provided by a particular independent variable when it is added to a model already containing other explanatory variables. It represents the unique contribution of that variable to the total variation in the dependent variable, after accounting for the effects of the other predictors. This concept is critical when evaluating the importance of individual predictors within a multiple regression context.

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