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.

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  $0.8272$

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

(c) 

(d) The coefficient of determination  is $\approx 0.6868$

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. is

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

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

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 $r$ is near to $0$, the data points are essentially scattered along a horizontal line. If $r$is the father of $\pm 1$, the data points are more widely dispersed around the regression line. If $r$is close to$\pm 1$, the data points cluster closely around the regression line.

 

The graph in shown below

$r$ is close to $1$ when calculated. As a result, the data points are closely clustered around the regression line. The are likewise clustered closely around the regression line in the graph created in Exercise 4.60. As a result, the calculated correlation coefficient matches the graph.

Part (d) Step 1: Given Information

Given table is

We have to square $r$ and compare the result with the value of the coefficient of determination you obtained in the corresponding exercise in Section 4.3.   

Part (d) Step 2: explanation

The square of  $r$ is

${\left(0.8272\right)}^2\approx 0.6868$