Q.4.90 - Elementary Statistics

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.

Step-by-Step Solution

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.

 

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

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

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

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

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 $r^2 is 0.8696$, it is closed to 1 . so, it is useful to make the predictions using the regression equation.