Q. 4.93 - Elementary Statistics

Question

a. compute the three sums of squares, $SST,SSR,SSE$, using the defining formulas

b. verify the regression identity, $SST=SSR+SSE$

c. compute the coefficient of determination.

d. determine the percentage of variation in the observed values of the response variable that is required by the regression

e. State how useful the regression equation appears to be for making predictions.

$\hat{y}=5-x$

Step-by-Step Solution

Answer

(a) $$\begin{gathered} SST=10 \\ SSR=4 \\ SSE=6 \end{gathered}$$

(b) $SST=10$

(c) $0.4$

(d) $40\%$

(e) Utilising the regression equation to generate predictions is useless, as the regression can only explain $40\%$ of the variation.

Part (a) Step 1: Given information

The given data is 

$\hat{y}=5-x$

Part (a) Step 2: Explanation

The regression equation is 

$\hat{y}=5-x$

The formulas to calculate the sum of squares is 

$$\begin{gathered} SST=\sum {\left(y_i-\overline{y}\right)}^2 \\ SST=\sum {\left({\hat{y}}_i-\overline{y}\right)}^2 \\ SST=\sum {\left(y_i-\hat{y}\right)}^2 \end{gathered}$$

As shown in the table below, the relevant sums can be determined. 

$$\begin{gathered} SST=10 \\ SSR=4 \\ SSE=6 \end{gathered}$$

Part (b) Step 1: Given information

The given data is 

$\hat{y}=5-x$

Part (b) Step 2: Explanation

$SST=SSR+SSE$

        $$\begin{gathered} =4+6 \\ =10 \end{gathered}$$

Part (c) Step 1: Given information

The given data is 

$\hat{y}=5-x$

Part (c) Step 2: Explanation

The coefficient of determination is 

$r^2=\frac{SSR}{SST}$

    $$\begin{gathered} =\frac{4}{10} \\ =0.4 \end{gathered}$$

Part (d) Step 1: Given information

The given data is 

$\hat{y}=5-x$

Part (d) Step 2: Explanation

The coefficient of determination restated as a percentage is the percentage of variation: 

$0.4=40\%$

Part (e) Step 1: Given information

The given data is 

$\hat{y}=5-x$

Part (e) Step 2: Explanation

The regression equation can be used to generate predictions if the estimated $r^2$ is near to $1$.

The estimated $r^2$ value is $0.4$, which is not equal to $1$.

As a result, utilising the regression equation to generate predictions is useless, as the regression can only explain $40\%$ of the variation.