Q.4.416 - Elementary Statistics

Question

Custom Homes Following are the size and price data for custom homes from exercise 4.60  and 4.100

a. Obtain the linear correlation coefficient. 

b. Interpret the value of $r$ in terms of the linear relationship between the two variables in question.

c. Discuss the graphical interpretation of the value of $r$and verify that it is consistent with the graph you obtained in the corresponding exercise in Section 4.2.

d. Square $r$ and compare the result with the value of the coefficient of determination you obtained in the corresponding exercise in Section 4.3.

Step-by-Step Solution

Answer

(a) The linear correlation coefficient is  $0.8287$

(b) Increase in home size leads to increase in price.

(c) 

(d) The coefficient of determination  is  $0.6868$

Part (a) Step 1: Given Information

The given table is

We have to obtain the linear correlation coefficient. 

Part (a) Step 2: Explanation

The formula of  linear correlation coefficient.  

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

Therefore,

$$\begin{gathered} r=\frac{169993-\left(270\right)\left(5552\right)/9}{\sqrt{\left[8316-(270{)}^2/9\right]\left[3504412-(5552{)}^2/9\right]}} \\ =0.8287 \end{gathered}$$

Part (b) Step 1: Given Information

The given table is

We have to interpret the value of $r$ in terms of the linear relationship between the two variables in question.  

Part(b) Step 2: Explanation

The variables are strongly correlated if the estimated $r$is near to $\pm 1$

Close to$1$ is the computed correlation coefficient. As a result, the variables are positively correlated. As a result, as the home size increases, the price increases.

Part (c) Step 1: Given Information

The given table is

We have to discuss the graphical interpretation of the value of $r$and verify that it is consistent with the graph you obtained in the corresponding exercise in Section 4.2. 

Part(c) Step 2: Explanation

If is near to , the data points are essentially scattered along a horizontal line. If is the father of , the data points are more widely dispersed around the regression line. If is close to, the data points cluster closely around the regression line.

The graph is shown below

jh

sssg