Question

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

a. $r^2=1?$

b.$r^2=0?$

Short 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.

Step-by-step solution

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.