Q.4.144 - Elementary Statistics

Question

Tax efficiency. Following are the data on percentage of investments in energy securities and tax efficiency from exercise 4.58 and 4.98

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  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 $-0.9748$

(b) If energy investments increase, tax efficiency will decline.

(c) 

(d) The coefficient of determination is $\approx 9503$

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 correlation coefficient is 

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

Therefore,

$$\begin{gathered} r=\frac{4376.95-(55.9)(832.5)/10}{\sqrt{\left[365.05-(55.9{)}^2/10\right]\left[70838.49-(832.5{)}^2/10\right]}} \\ =-0.9748 \end{gathered}$$

Part (b) Step 1: Given Information

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 associated 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, if there is an increase in energy investments, the tax efficiency will fall.

Part (c) Step 1: Given Information

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.

$r$ 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 also clustered closely around the regression line. 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

 $$\begin{gathered} r={\left(0.9784\right)}^2 \\ =0.95023 \\ \approx 9503 \end{gathered}$$