Q. 4.99 - Elementary Statistics

Question

Corvette Prices. Following are the age and price data for Corvettes from Exercise $4.59$:

a. Compute SST,SSR,SSE

b. Compute the coefficient of determination 

c. Determine the percentage of variation in the observed values of the response variable explained by the regression, and interpret your answer.

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

Step-by-Step Solution

Answer

(a) $$\begin{gathered} SST=25681.6 \\ SSR=24057.8913 \\ SSE=1623.7087 \end{gathered}$$

(b) $0.9368$

(c) $93.68\%$

(d) Utilising the regression equation to create predictions is quite effective, and the regression can explain roughly $94\%$ of the variation.

Part (a) Step 1: Given information

The given data is 

Part (a) Step 2: Explanation

The below table gives the prices of randomly selected $10$ Corvettes with their age between 1 and 6 years inclusively. The age is denoted by $x$, and the price is denoted by $y$ in hundreds of dollars.

The formulas to calculate the sum of squares is   

$SST=\sum y_i^2-{\left(\sum y_i\right)}^2/n$

$SSR=\frac{{\left[\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n\right]}^2}{\sum x_i^2-{\left(\sum x_i\right)}^2/n}$

$SSE=SST-SSR$

As shown in the table below, the relevant sums can be determined.   

$SST=1196690-\frac{{3422}^2}{10}$

         $=25681.6$

$$\begin{gathered} SSR=\frac{|13168-41\times 3422÷10{|}^2}{199-4{\rfloor }^2÷10} \\ SSR=24057.8913 \end{gathered}$$

$$\begin{gathered} SSE=25681.6-24057.8913 \\ SSE=1623.7087 \end{gathered}$$

Part (b) Step 1: Given information

The given data is 

Part (b) Step 2: Explanation

The coefficient of determination is   

$r^2=\frac{SSR}{SST}$

    $$\begin{gathered} =\frac{24057.8913}{25681.6} \\ =0.9368 \end{gathered}$$

Part (c) Step 1: Given information

The given data is 

Part (c) Step 2: Explanation

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

$0.9368=93.68\%$

Part (d) Step 1: Given information

The given data is 

Part (d) Step 2: Explanation

The regression equation can be used to generate predictions if the estimated $r^2$ is near to $1$.

The computed $r^2=0.9368$, which is extremely near to 1.

As a result, utilising the regression equation to create predictions is quite effective, and the regression can explain roughly 94%  of the variation.