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.

Short 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 $0.6868$

Step-by-step solution

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

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