Q. 4.144 - 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.

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

Step-by-Step Solution

Answer

The value of $r^2$ is same in both methods

Step 1. Given Information

Step 2. Table for obtaining the linear correlation coefficient.

x	y	xy	$x^2$	$y^2$
3.1	98.1	304.11	9.61	9623.61
3.2	94.7	303.04	10.24	8968.09
3.7	92	340.4	13.69	8464
4.3	89.8	386.14	18.49	8064.04
4	87.5	350	16	7656.25
5.5	85	467.5	20.25	7225
6.7	82	549.4	44.89	6724
7.4	77.8	575.72	54.76	6052.84
7.4	72.1	533.54	54.76	5198.41
10.6	53.5	567.1	112.36	2862.25
55.9	832.5	4376.95	365.05	70838.49

Step 3. The linear correlation coefficient of given set of data is

$$\begin{gathered} r=\frac{\underset{}{\overset{}{\sum x_i{y}_{i }- {\displaystyle \frac{\left(\underset{}{\overset{}{\sum x_i}}\right) \left(\underset{}{\overset{}{\sum y_i}}\right)}{n}}}} }{\sqrt{\left[\sum {x^2}_i-{\displaystyle \frac{{\left(\sum x_i\right)}^2}{n}}\right]\left[\sum {y^2}_i-\frac{{\left(\sum y_i\right)}^2}{n}\right]}} \\ =\frac{\underset{}{\overset{}{4376.95- {\displaystyle \frac{\left(55.9\right) \left(832.5\right)}{10}}}} }{\sqrt{\left[365.05-{\displaystyle \frac{{\left(55.9\right)}^2}{10}}\right]\left[70838.49-{\displaystyle \frac{{(832.5)}^2}{10}}\right]}} \\ = -0.975 \end{gathered}$$

Step 4. Solution b)

The value of correlation coefficient r suggests an extremely strong negative linear relationship between percentage of investment in ENERGY security and Tax EFFICIENCY.

Step 5. Solution c)

The correlation coefficient, r = -0.975, suggests a strong negative correlation between percentage of investment in ENERGY security & Tax EFFICIENCY.

In particular, it indicates that as investment in ENERGY increases there is a strong tendency for tax efficiency to decrease.

Step 6.

From the above graph also we can see that as an investment in ENERGY increase there is a strong tendency for tax efficiency to decrease.

Step 7. Solution d)

Square of r is given by 

$$\begin{gathered} r^2={(-0.975)}^2 \\ = 0.9501 \end{gathered}$$

The mean of observed tax efficiency is 

$$\begin{gathered} \overline{y}=\frac{\sum _{}^{}y}{n} \\ = \frac{832.5}{10} \\ = 83.25 \end{gathered}$$

x	y	$\hat{y}$	$\hat{y}-\overline{y}$	${(\hat{y}-\overline{y})}^2$	${(y-\overline{y} )}^2$
3.1 3.2 3.7 4.3 4 5.5 6.7 7.4 7.4 10.6	98.1 94.7 92 89.8 87.5 85 82 77.8 72.1 53.5	96.36 95.84 93.20 90.04 91.62 83.73 77.41 73.73 73.73 56.88	13.11 12.59 9.95 6.79 8.37 0.48 -5.84 -9.52 -9.52 -26.37	171.91 158 39 99.07 46.17 70.12 0.23 34.09 90.70 90.70 695.29	220.52 131.10 76.56 42.90 18.06 3.06 1.56 29.70 124.32 885.06
				1456.67	1532.87

Step 8.

$$\begin{gathered} SSR = \underset{}{\overset{}{\sum {({\displaystyle \widehat{y_{i }}}-\overline{y})}^2}} \\ = 1456.67 \\ SST = \sum {(y_i -\overline{y})}^2 \\ = 1532.87 \end{gathered}$$

Coefficient of determination,

$$\begin{gathered} r^2=\frac{SSR}{SST} \\ = \frac{1456.67}{1532.87} \\ = 0.95 \end{gathered}$$

The value of $r^2$ is same in both methods