Q.4.89 - 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,

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)

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

(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.

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.