Table 12-8 presents statistics for the National Hockey...

Table $12-8$ presents statistics for the National Hockey League teams from the $2008-2009$ season (source: The Sports Network). Fit a multiple linear regression model that relates Wins to the variables GF through $F G$. Because teams play 82 games $W=82-L-T-O T L,$ but such a model docs not help build a better team. Estimate $\sigma^{2}$ and find the standard errors of the regression coefficients for your model.

Key Concepts

Standard Errors of Regression Coefficients

The standard errors of the regression coefficients measure the precision of the estimated parameters, reflecting how much the estimated coefficients would vary if the model were fitted to different samples. They are computed using the estimated error variance along with the elements of the inverse of the design matrix's cross-product, specifically the diagonal elements of (X?X)?¹.

Degrees of Freedom

Degrees of freedom in the context of regression are defined as the number of observations minus the number of parameters estimated (including the intercept). They are used to adjust the estimation of ?² and affect the reliability of the hypothesis tests and confidence intervals for the regression coefficients.

Multiple Linear Regression

Multiple linear regression models the relationship between one dependent variable and two or more independent variables. It estimates how each predictor influences the response, holding other factors constant, by fitting a linear equation to the observed data.

Estimation of Error Variance (?²)

In multiple regression analysis, the error variance ?² quantifies the dispersion of the observed values around the regression line. It is typically estimated by dividing the residual sum of squares (RSS) by the degrees of freedom, which accounts for the number of estimated parameters in the model.

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