Question

Find the value of the linear correlation coefficient r. Round your answer to 3 decimal places if necessary. (1,2).(3,4),(5,6)

Answer 1

Solution-:

We prepare the following table:

	X	Y	X^2	Y^2	X*Y
	1	2	1	4	2
	3	4	9	16	12
	5	6	25	36	30
Total	9	12	35	56	44

η = 3,ΣΧ = 9,ΣΥ = 12,ΣΧ? = 35. ΣΥ? = 56, XY = 44

$\bar{X}=\frac{\sum X}{n}=\frac{9}{3}= 3$ and $\bar{Y}=\frac{\sum Y}{n}=\frac{12}{3}=4$

$Var(X)=\frac{\sum X^2}{n}-\bar{X}^2=\frac{35}{3}-3^2=2.67$

$Var(Y)=\frac{\sum Y^2}{n}-\bar{Y}^2=\frac{56}{3}-4^2=2.67$

$SD(X)=\sqrt{Var(X)}=\sqrt{2.67}=1.63$ and

$SD(Y)=\sqrt{Var(Y)}=\sqrt{2.67}=1.63$

$Cov(X,Y)=\frac{\sum XY}{n}-\bar{X}*\bar{Y}=\frac{44}{3}-3*4=2.67$

$Corr(X,Y)=r=\frac{Cov(X,Y)}{SD(X)*SD(Y)}=\frac{2.67}{1.63*1.63}=1$

Therefore, Correlation= 1.000

OR

By using R-Software:

> x=c(1,3,5);x
[1] 1 3 5
> y=c(2,4,6);y
[1] 2 4 6
> r=cor(x,y);r
[1] 1