Q 4.145 - Elementary Statistics

Question

In below exercise, we repeat data from exercises in Section 4.2. For given exercise here.

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.

Corvette Prices. Following are the age and price data for Corvettes from Exercises 4.59 and 4.99.

Step-by-Step Solution

Answer

from the given data the calculated coefficient of determination $r^2=0.9368$

From the two values we observe that both the values are equal.

Step 1. Given

Step 2. Table for obtaining the linear correlation coefficient.

x	y	xy
6	290	1740	36	84100
6	280	1680	36	78400
6	295	1770	36	87025
2	425	850	4	180625
2	384	768	4	147456
5	315	1575	25	99225
4	355	1420	16	126025
5	328	1640	25	107584
1	425	425	1	180625
4	325	1300	16	105625
41	3422	13168	199	1196690

$$\begin{gathered} r=\frac{\underset{}{\overset{}{\sum x_i{y}_{i }- {\displaystyle \frac{\left(\underset{}{\overset{}{\sum x_i}}\right) \left(\underset{}{\overset{}{\sum y_i}}\right)}{n}}}} }{\sqrt{\left[\sum {x^2}_i-{\displaystyle \frac{{\left(\sum x_i\right)}^2}{n}}\right]\left[\sum {y^2}_i-\frac{{\left(\sum y_i\right)}^2}{n}\right]}} \\ =\frac{\underset{}{\overset{}{13168- {\displaystyle \frac{\left(41\right) \left(3422\right)}{10}}}} }{\sqrt{\left[199-{\displaystyle \frac{{\left(41\right)}^2}{10}}\right]\left[1196690-{\displaystyle \frac{{\left(3422\right)}^2}{10}}\right]}} \\ = -0.968 \end{gathered}$$

Therefore the linear correlation coefficient value is -0.968.

Step 3. Solution b)

The linear correlation coefficient, r=-0.968, suggests a strong negative linear correlation between age and price. In particular, it indicates that as age increases, there is a strong tendency for price to decrease, which is not surprising. It also implies that the regression equation $\hat{y}= 456.6 - 27.9x$ is extremely useful for making predictions.

Step 4. Solution c)

Because the correlation coefficient, r = -0.968 is very close to -1, the data points should be clustered closely about the regression line.

From above graph it is very clear the point clustered closely about the regression line.

Step 5. Solution d)

We have $r = -0.968$ then $r^2=0.9368$

from the given data the calculated coefficient of determination $r^2= 0.9368$

From the two values we observe that both the values are equal.