Chapter 2.5, Problem 11E

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 11E

The correlation coefficient is 1.

Explanation of Solution

Given:

Data set is:

Number of rings from pith to bark	Age of tree
5	5
10	10
15	15
20	20
30	30
40	40
50	50
60	60

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, number of rings = x and age of tree =y

The calculation for correlation is done as:

	x	y	$xy$	$x^2$	$y^2$
	5	5	25	25	25
	10	10	100	100	100
	15	15	225	225	225
	20	20	400	400	400
	30	30	900	900	900
	40	40	1600	1600	1600
	50	50	2500	2500	2500
	60	60	3600	3600	3600
sum	230	230	9350	9350	9350

Thus,

  ${\displaystyle \sum x=230,}{\displaystyle \sum y=230},{\displaystyle \sum xy=9350},{\displaystyle \sum x^2=9350,}{\displaystyle \sum y^2=9350}$

Now, the correlation is computed as:

  $\begin{array}{c}r=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{\left(n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}\right)\left(n{\displaystyle \sum y^2-{\left({\displaystyle \sum y}\right)}^2}\right)}}\\ =\frac{8\left(9350\right)-\left(230\right)\left(230\right)}{\sqrt{\left(\left(8\left(9350\right)\right)-{\left(230\right)}^2\right)\times \left(8\left(9350\right)\right)-{\left(230\right)}^2}}\\ =1\end{array}$

Thus, the value of the correlation coefficient ( r ) is 1.

Interpretation:

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