4.3 The coefficient of Determination

Total Deviation:

Difference between observed value and mean values of the response variable:y-y(line over it)

Explained Deviation:

Difference between predicted value and mean values of the response variable: Y(arrow over it)-Y(line over it)

Unexplained Deviation:

Difference between the observed value and the predicted value of the response variable, the residual:y-y(Arrow). The unexplained deviation is due to random chance and possible lurking variables.

Coefficient of Determination Definition:

Measures the proportion of total variation in the response variable that is explained by the value of the explanatory variable x, and the least-squares regression line: Y(arrow over it)=b1x+b0

Compute and Interpret the coefficient of Determination:

R^2

For the least-squares regression model, Y(arrow over it)=b1x +bo, the coefficient of determination is the square of the linear correlation coefficient:

R^2=r^2

The smaller the residuals, the bigger the what?

The bigger the coefficient of determination.

since r is negative, we say it is a what?

We say it is a negative linear correlation.

If R^2=1 the least squares regression line explains 100% of what?

of the variation in the response variable. -> o</= R^2 </= 1