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.

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

Short answer

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

(b) The regression identity is verified

(c) $0.571$

(d) $57.1\%$

(e) Moderately effective.

Step-by-step solution

Part (a) Step 1: Given information

The given data is

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

Part (a) Step 2: Explanation

The given regression equation is

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

Formulas to calculate the sum of squares is

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

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

Part (b) Step 1: Given information

The given data is

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

Part (b) Step 2: Explanation

From the above answer

$SST=SSR+SSE$

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

Thus verified.

Part (c) Step 1: Given information

The given data is

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

Part (c) Step 2: Explanation

The formula for the coefficient of determination is

$$\begin{gathered} r^2=1-\frac{SSE}{SST} \\ {r}^2=1-\frac{6}{14} \\ {r}^2=1-0.429 \\ {r}^2=0.571 \end{gathered}$$

Part (d) Step 1: Given information

The given data is

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

Part (d) Step 2: Explanation

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

$0.571=57.1\%$

Part (e) Step 1: Given information

The given data is

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

Part (e) Step 2: Explanation

In this scenario, the regression equation is only somewhat effective in making predictions.