Chapter 2.5, Problem 9E

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 9E

The correlation coefficient is 0.890.

Explanation of Solution

Given:

Data set is:

Temp	Ice cream Sales
76	850
78	885
80	875
82	920
84	955
85	905
86	1005
87	1060
88	1105
89	1165

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, temp = x and miles per gallon = sales

The calculation for correlation is done as:

	x	y	$xy$	$x^2$	$y^2$
	76	850	64600	5776	722500
	78	885	69030	6084	783225
	80	875	70000	6400	765625
	82	920	75440	6724	846400
	84	955	80220	7056	912025
	85	905	76925	7225	819025
	86	1005	86430	7396	1010025
	87	1060	92220	7569	1123600
	88	1105	97240	7744	1221025
	89	1165	103685	7921	1357225
sum	835	9725	815790	69895	9560675

Thus,

  ${\displaystyle \sum x=835,}{\displaystyle \sum y=9725},{\displaystyle \sum xy=815790},{\displaystyle \sum x^2=\begin{array}{c}69895\end{array},}{\displaystyle \sum y^2=9560675}$

Now, the correlation is computed as:

  $\begin{array}{c}r=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{\left(n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}\right)\left(n{\displaystyle \sum y^2-{\left({\displaystyle \sum y}\right)}^2}\right)}}\\ =\frac{10\left(815790\right)-\left(835\right)\left(9725\right)}{\sqrt{\left(\left(10\left(69895\right)\right)-{\left(835\right)}^2\right)\times \left(10\left(9560675\right)\right)-{\left(9725\right)}^2}}\\ =0.890\end{array}$

Thus, the value of the correlation coefficient ( r ) is 0.890

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.