Q. 4.99 - Elementary Statistics

Question

For Exercises 4.98 - 4.103,

(a) Compute SST, SSR, SSE using formula 4.2 on page 179.

(b) Compute the coefficient of determination,$r^2$.

(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.

Following are the age and price data for Corvettes from Exercise 4.59

Step-by-Step Solution

Answer

(a) The values are $SST=25681.6; SSR=24057.9; SSE=1623.7$

(b) The coefiicient of variation is $0.9368$.

(c) The percentage of variatioon is 93.7%. It means that 93.7% variation in price data is explained by age.

(d) The regression line appears to be extremely useful, becausse there is less portion of unexplained variation present in the given data. 

Part (a) Step 1. Given Information.

A table of values.

Part (a) Step 2. Construct the table.

Construct the table.

Part (a) Step 3. Compute SST, SSR, SSE.

Find SST, SSR, SSE. 

$$\begin{gathered} SST=\sum _{}{y}_i^2-\frac{{\left({\displaystyle \sum _{}}{y}_i\right)}^2}{2} \\ =1196690-\frac{{\left(3422\right)}^2}{10} \\ =25681.6 \end{gathered}$$

$$\begin{gathered} SSR=\frac{[{\displaystyle \sum _{}}{x}_i{y}_i-{\left({\displaystyle \frac{{\displaystyle \sum _{}{x}_i\sum _{}{y}_i}}{n}}\right)}^2}{{\displaystyle \sum _{}{x_i}^2-\frac{{\left({\displaystyle \sum _{}}{x}_i\right)}^2}{n}}} \\ =\frac{{[13168-{\displaystyle \frac{41\left(3422\right)}{10}}]}^2}{199-{\displaystyle \frac{{41}^2}{10}}} \\ =24057.9 \\ SSE=SST-SSR \\ =25681.6-24057.9 \\ =1623.7 \end{gathered}$$

Part (b) Step 1. FInd the coefficient of determination.

$$\begin{gathered} r^2=\frac{SSR}{SST} \\ =\frac{24057.9}{25681.9} \\ =0.9368 \end{gathered}$$

Part (c) Step 1. Find the percentage of variation.

The percentage of variatioon is $93.7\%$.

It means that 93.7% variation in price data is explained by age.

Part (d) Step 1. Useful for making predictions.

The regression line appears to be extremely useful, becausse there is less portion of unexplained variation present in the given data.