Q.145 - Elementary Statistics

Question

Following are the age and price data for Corvettes from exercises 4.59 and 4.99

a. obtain the linear correlation coefficient.

b. interpret the value of $r$in terms of the linear relationship between the fwo 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 $-0.9679$

(b) If the age increases the price will  decline.

(c)

(d) The coefficient of determination is $0.9368$

Part (a) Step 1: Given Information

Given table is

We have to obtain the linear correlation coefficient. 

Part(a) Step 2: Explanation

The formula of correlation coefficient is

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

The appropriate value can be calculated in the below table

Therefore,

$$\begin{gathered} r=\frac{13168-\left(41\right)\left(3422\right)/10}{\sqrt{\left[199-(41{)}^2/10\right]\left[1196690-(3422{)}^2/10\right]}} \\ =-0.9679 \end{gathered}$$

$$\begin{gathered} r=\frac{13168-\left(41\right)\left(3422\right)/10}{\sqrt{\left[199-(41{)}^2/10\right]\left[1196690-(3422{)}^2/10\right]}} \\ =-0.9679 \end{gathered}$$

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 negatively connected. As a result, as the age of the person increases, the price decreases.

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  $r$close to$-1$, the data points cluster closely around the regression line.

Graph is given below

 

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

Part (d) Step 1: Given Information

The 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

$(-0.9679{)}^2=0.9368$