Question :
Calculate the correlation coefficient of the following heights(in inches) of fathers and their sons:
X: 65 66 67 67 68 69 70 72
Y: 67 68 65 68 72 72 69 71
Correlation Coefficient $\displaystyle \rho= \frac{cov(X,Y)}{\sigma_x \sigma_y}$
$\displaystyle \mu_x$ = $\displaystyle \sum_{i} \frac{x_i}{n_1}$ = $\displaystyle 68$
$\displaystyle \mu_y$ = $\displaystyle \sum_{i} \frac{y_i}{n_2}$ = $\displaystyle 69$
$\displaystyle \sigma_x ^2$ = $\displaystyle \sum_{i} \frac{(x_i - \mu_x)^2}{n_1} $ = $\displaystyle \frac{9}{2}$
$\displaystyle \sigma_y ^2$ = $\displaystyle \sum_{i} \frac{(y_i - \mu_y)^2}{n_1} $ = $\displaystyle \frac{11}{2}$
$\displaystyle cov(X,Y) \ = \ E(X,Y)$ ......I am stuck at this protion now ....
I dont know how to calculate the cov of x,y from the table provided ....please advice
Also please check if am i doing it correctly or no ....
This is what i have done
Sample Correlation coeff
$\displaystyle r_{xy} = \sum_{i=0}^{n} \frac{(x_i - \bar x)(y_i - \bar y)}{(n-1) S_x S_y}$
where
$\displaystyle \bar x , \bar y$ : Sample mean x,y
$\displaystyle S_x , S_y$ : Standard deviation for x , y
Is this the right formula that i am using???
then $\displaystyle r_{xy} = \frac{4}{33}$ Is this answer correct ???