Question

Find the value of the linear correlation coefficient. ху 62 158 53 176 64 151 52 164 52 164 54 174 58 162 0 - 0.775 -0.081 0.754

Answer 1

the following calculations are needed to compute the correlation coefficient:

	X	Y	X*Y	X2	Y2
	62	158	9796	3844	24964
	53	176	9328	2809	30976
	64	151	9664	4096	22801
	52	164	8528	2704	26896
	52	164	8528	2704	26896
	54	174	9396	2916	30276
	58	162	9396	3364	26244
Sum =	395	1149	64636	22437	189053

The correlation coefficient rr is computed using the following expression:

$r = \frac{ SS_{XY}}{\sqrt{ SS_{XX} SS_{YY}}}$r=SSXX​SSYY​​SSXY​​

where

$SS_{XY} = \sum_{i=1}^n {X_i Y_i}-\frac{1}{n}\left(\sum_{i=1}^n X_i\right) \left(\sum_{i=1}^n Y_i\right)$

$SS_{XX} = \sum_{i=1}^n {X_i^2}-\frac{1}{n}\left(\sum_{i=1}^n X_i\right)^2$

$SS_{YY} = \sum_{i=1}^n {Y_i^2}-\frac{1}{n}\left(\sum_{i=1}^n Y_i\right)^2$

HENCE:

SSXY 64636 – (395 x 1149) = -200.429

$SS_{XX} = 22437 - \frac{1}{ 7} (395)^2 = 147.714$

$SS_{YY} = 189053 - \frac{1}{ 7} (1149)^2 = 452.857$

Therefore, based on this information, the sample correlation coefficient is computed as follows

$r = \frac{ SS_{XY}}{\sqrt{ SS_{XX} SS_{YY}}} = \frac{ -200.429}{\sqrt{ 147.714 \times 452.857}} = -0.775$

HENCE option(B) -0.775 is correct.

please rate my answer and comment for doubts.