Question

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

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

b. Verify the regression identity,

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)

$$\begin{gathered} SST=14 \\ SSR=8 \\ SSE=6 \end{gathered}$$

(b) $SST\left(14\right)=14$

(c) The coefficient of determination $r^2=0.571$

(d) The percentage of variation is $57.1\%$

(e) It is moderately useful for the regression equation appears to be for making predictions.

Step-by-step solution

Part(a) Step 1: Given Information 

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

$$\begin{gathered} SST=\sum _i{\left(y_i-\overline{y}\right)}^2 \\ =14 \end{gathered}$$

$$\begin{gathered} SSR=\sum _i{\left(y_i-\overline{y}\right)}^2 \\ =8 \end{gathered}$$

$$\begin{gathered} SSE=SST-SSR \\ =14-8 \\ =6 \end{gathered}$$

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  

$$\begin{gathered} SST=SSR+SSE \\ =8+6 \\ =14 \end{gathered}$$

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  

$$\begin{gathered} r^2=\frac{SSR}{SST} \\ =\frac{8}{14} \\ =0.571 \end{gathered}$$

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 

Because the coefficient of determination is $0.571$ the regression can explain$57.1\%$ of the variation in observed values.

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 

The regression equation is moderately useful for making predictions.