Q. 4.146 - Elementary Statistics

Question

Custom Homes. Following are the size and price data for

custom homes from Exercises 4.60 and 4.100.

Step-by-Step Solution

Answer

The value of r²   remains same in both cases .

Step 1.Given information

WE have been given a table 

Step 2. table for obtaining a linear correlation coefficient

The linear correlation coefficient of given set of data 

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

$\begin{array}{r}=\frac{169993-\frac{\left(270\right)\left(5552\right)}{9}}{\sqrt{\left.8316-\frac{(270{)}^2}{9}\right]\left[3504412-\frac{(5552{)}^2}{9}\right]}}\\ =0.829\end{array}$

Step 3. Theory for value of correlation

The value of correlation coefficient r suggests that there is a strong positive linear relationship

between size and price of custom homes .

Step 4.Graph for custom homes

The correlation coefficient ,r=0.829,suggestsastrong positive correlation between size and

price of custom homes

In particular, it indicates that as size of custom homes increases there is a strong tendency for

price of custom home to increase.

The graph from Exercise 14.90 is as follows

Step 5.Study of above graph

From the above graph also we can see that as size of custom homes increases there is a strong

tendency for price of custom home to increase.

Step 6. Finding mean from graph

Square of r is given by

From section 14.3(Exercise 14.52):

The mean of the observed tax efficiency is

$\begin{array}{r}\overline{y}=\frac{\sum y}{n}\\ =\frac{5552}{9}\\ =616.89\end{array}$

Step 7.Finding coefficient of variation

$\begin{array}{r}SSR=\sum {\left({\widehat{y}}_i-\overline{y}\right)}^2\\ =54565.76\\ SSR=\sum {\left(y_i-\overline{y}\right)}^2\\ =79444.89\end{array}$

Coefficient of determination,

$\begin{array}{r}{\mathrm{r}}^2=\frac{SSR}{SST}\\ =\frac{54565.76}{79444.89}\\ =0.687\end{array}$

The value of,²is same in both methods