Question

In the following data, one of the values of Y is missing. Arithmetic means of X and Y series are 6 and 8

X	6	2	10	4	8
Y	9	11	?	8	7

 (a) Estimate the missing observation.

 (b) Calculate correlation coefficient.

Answer 1

 

 

 

(a)First, we find the missing value of Y and let us denote it by a.

`barY=(SigmaY)/N=(9+11+a+8+7)/5=(35+a)/5`

`=>8=(35+a)/5=>40=35+a=>a=5`

Thus the completed series is 

X	6	2	10	4	8
Y	9	11	5	8	7

(b)Now we Find coefficient correlation

xi	yi	xiyi	`x_i^2`	`y_i^2`
6	9	54	36	81
2	11	22	4	121
10	5	50	100	25
4	8	32	16	64
8	7	56	64	49
Σxi = 30	Σyi = 40	Σxiyi = 214	Σ`x_i^2`=220	Σ`y_i^2` = 340

 Here, n= 5, ΣX = 30, Σx² = 40, ΣY=40, ΣY² = 20 and Σxy = -26

Now,

`barX = (SigmaX)/n =30/5=6 " and "barY =(SigmaY)/n=40/5=8`

`:. r = (1/n Σ x iyi - barx bary)/(sqrt[[(Σxi^2)/n - barx^2] [(Σyi^2)/n - bary^2]`

`r = (1/5 xx 214 - 8 xx 6)/(sqrt(220/5 - 6^2) xx sqrt(340/5 - 8^2)`

`= (42.8 - 48)/((sqrt (44 - 36) xx sqrt(68 - 64))`

= `(-5.2)/(sqrt8 xx sqrt4)`

= `(-5.2)/(4sqrt2)`

r = -0.92