Question

$\hat{y}=1+2x$

a. Compute the three sums of square SST,SSR, SSE using defining formulas

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

c. Compute the coefficient of determination.

d. Determine the percentage of variation in the observed values of the response variable that is explained by the regression.

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

Short answer

(a) $SST=46,SSE=6,SSR=40$

(b) $SST=46$

(c) The coefficient of determination $r^2=0.8696$

(d) The percentage of variation is$86.96\%$

(e) useful for The regression equation is useful for making predictions.

Step-by-step solution

Part(a) Step 1: Given Information  

The given data is

$\hat{y}=1+2x$we have to compute the three sums of square SST,SSR, SSE using defining formulas .

Part(a) Step 2: Explanation  

The table can be constructed as follows

$$\begin{gathered} SST=\sum {\left(y_i-\overline{y}\right)}^2 \\ =46 \end{gathered}$$

$$\begin{gathered} SSR=\sum {\left(y_i-\overline{y}\right)}^2 \\ =40 \end{gathered}$$

role="math" localid="1652086154935" $$\begin{gathered} SSE=SST-SSR \\ =46-40 \\ =6 \end{gathered}$$

Part(b) Step 1: Given Information  

The given data is

$\hat{Y}=1+2X$

we have to Verify the regression identity, $SST=SSE+SSR$

Part(b) Step 2: Explanation 

$$\begin{gathered} SST=SSE+SSR \\ =6+40 \\ =46 \end{gathered}$$

Part(c) Step 1: Given Information  

The given data is

$\hat{y}=1+2x$we have to compute the coefficient of determination.

Part(c) Step 2: Explanation 

$$\begin{gathered} r^2=\frac{SSR}{SST} \\ =\frac{40}{46} \\ =0.8696 \end{gathered}$$

Part(d) Step 1: Given Information  

The given data is

$\hat{y}=1+2x$

we have to determine the percentage of variation in the observed values of the response variable that is explained by the regression.

Part(d) Step 2: Explanation 

The coefficient of determination is the percentage of variation . it can be written as a percentage$86.96\%$.

Part(e) Step 1: Given Information   

The given data is

$\hat{y}=1+2x$

we have to State how useful the regression equation appears to be for making predictions

Part(e) Step 2: Explanation 

When the regression equation is useful to make the predictions if the calculated $r^2$is closed to $1$

The calculated $r^{2}is0.8696$, it is closed to 1 . so, it is useful to make the predictions using the regression equation.