Q.4.116 - Elementary Statistics

Question

What can you say about SSE, SSR and the  utility of the regression equation for making predictions if 

a. $r^2=1?$   

b.$r^2=0?$

Step-by-Step Solution

Answer

 (a) $SSR=SST$

     $SSE=0$

The regression equation is very useful to determine predictions.

(b) $SSE=SST$

     $SSR=0$

   The regression equation is not useful in making predictions.

Part (a) Step 1: Given Information

The given equation is:

$r^2=1$

we have to determine SSE, SSR and the  utility of the regression equation .

Part (a) Step 2: Explanation

The coefficient of determination is given by $r^2=\frac{SSR}{SST}$

It is given that $r^2=1$, this means that SSR and SST will be equal to each other. The relationship between SST, SSR and SSE is given by

$SST=SSR+SSE$

Therefore, SSE$=0$. As it can be seen that the coefficient of determination is equal to $1$ , therefore, the regression equation is very useful in making the predictions.

Part (b) Step 1: Given Information

The given equation is:

$r^2=0$

we have to determine SSE, SSR and the  utility of the regression equation. 

Part (b) Step 2: Explanation

The coefficient of determination is given by $r^2=\frac{SSR}{SST}$

 It is given that $r^2=0$; this means that $SSR=0$ because SST cannot be equal to zero. The relationship between SST, SSR and SSE is given by

$SST=SSR+SSE$

Therefore, SST = SSE.. As it can be seen that the coefficient of determination is equal to $0$ , therefore, the regression equation is not useful in making the predictions.