Chapter 2.5, Problem 8E

To determine

To find: The value of the correlation coefficient and describe it. And, check whether causation is possible here or not.

Answer to Problem 8E

The correlation coefficient is -0.858.

Explanation of Solution

Given:

Data set is:

Car Value	Miles per Gallon
1400	55
1600	48
18500	37
20000	35
22500	29
25000	33
29000	29
32000	19
35000	26
40000	33

Formula used:

Correlation coefficient formula is

  $r=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{\left(n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}\right)\left(n{\displaystyle \sum y^2-{\left({\displaystyle \sum y}\right)}^2}\right)}}$

Calculation:

Consider, car value = x and miles per gallon =y

The calculation for correlation is done as:

	x	y	$xy$	$x^2$	$y^2$
	1400	55	77000	1960000	3025
	1600	48	76800	2560000	2304
	18500	37	684500	342250000	1369
	20000	35	700000	400000000	1225
	22500	29	652500	506250000	841
	25000	33	825000	625000000	1089
	29000	29	841000	841000000	841
	32000	19	608000	1024000000	361
	35000	26	910000	1225000000	676
	40000	33	1320000	1600000000	1089
Sum	225000	344	6694800	6568020000	12820

Thus,

  ${\displaystyle \sum x=225000,}{\displaystyle \sum y=344},{\displaystyle \sum xy=6694800},{\displaystyle \sum x^2=\begin{array}{c}6568020000\end{array},}{\displaystyle \sum y^2=12820}$

Now, the correlation is computed as:

  $\begin{array}{c}r=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{\left(n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}\right)\left(n{\displaystyle \sum y^2-{\left({\displaystyle \sum y}\right)}^2}\right)}}\\ =\frac{10\left(6694800\right)-\left(225000\right)\left(344\right)}{\sqrt{\left(\left(10\left(6568020000\right)\right)-{\left(225000\right)}^2\right)\times \left(10\left(12820\right)\right)-{\left(12820\right)}^2}}\\ =-0.858\end{array}$

Thus, the value of the correlation coefficient ( r ) is -0.858.

Interpretation:

It shows that the strong negative relationship exists between the two variables. The causation is not implied by the correlation. Thus, causation is not possible here.