Question

Consider the following sample data:

x	11	7	5	5	4
y	3	10	13	6	11

Click here for the Excel Data File

a.	Calculate the covariance between the variables. (Negative value should be indicated by a minus sign. Round your intermediate calculations to 4 decimal places and final answer to 2 decimal places.)

Covariance

b.	Calculate the correlation coefficient. (Round your intermediate calculations to 4 decimal places and final answer to 2 decimal places.)

Correlation coefficient

Answer 1

Solution-:

We prepare the following table:

	x	y	x^2	y^2	x*y
	11	3	121	9	33
	7	10	49	100	70
	5	13	25	169	65
	5	6	25	36	30
	4	11	16	121	44
Total	32	43	236	435	242

  From this table we get

$n=5,\sum X=32,\sum Y=43,\sum X^2=236,\sum Y^2=435,$$\sum XY=242$

ΣΧ 32 6.4000 5
and $\bar{Y}=\frac{\sum Y}{n}=\frac{43}{5}=8.6000$

$Var(X)=\frac{\sum X^2}{n}-\bar{X}^2=\frac{236}{5}-6.4^2=6.2400$

$Var(Y)=\frac{\sum Y^2}{n}-\bar{Y}^2=\frac{435}{5}-8.6^2=13.0400$

$SD(X)=\sqrt{Var(X)}=\sqrt{6.24}=2.4980$

$SD(Y)=\sqrt{Var(Y)}=\sqrt{13.04}=3.6111$

(a) $Cov(X,Y)=\frac{\sum XY}{n}-\bar{X}*\bar{Y}=\frac{242}{5}-6.4*8.6=-6.64$

(b)$Corr(X,Y)=r=\frac{-6.64}{(2.4980*3.6111)}=-0.74$

There is negative correlation between X and Y