If for a binomial bivariate data $\overline{x}$ = 10, $\overline{y}$= 12, $Var\left( X \right)=9$, ${{\sigma }_{Y}}=4$ and r = 0.6. Estimate y when x = 5.

Hint: The above question is about regression and correlation of bivariate data. To calculate the value of y at x =5 first we have to calculate the regression equation of y on x. We know that the regression equation is equal to $y={{b}_{yx}}x+a$ , and here ${{b}_{yx}}$ is the regression coefficient of y on x and ${{b}_{yx}}$ is equal to $r\dfrac{{{\sigma }_{y}}}{{{\sigma }_{x}}}$, and $a=\overline{y}-{{b}_{yx}}\overline{x}$ . After calculating the equation of regression of y on x we will put x = 5 in place of y to get x. And, also note that standard deviation (i.e.$\sigma $ ) is equal to square root variance, r is the correlation coefficient, $\overline{x}$ is the mean of x and or is the mean of y.

Complete step by step answer:
We can see from the above question is of regression and correlation of the bivariate data.
Since, from the question we can see that we have to find the value of y at x = 5 and for that we have to find the regression equation of y on x and the equation is given by \[\left( y-\overline{y} \right)={{b}_{yx}}\left( x-\overline{x} \right)\] where $\overline{y}$ is the mean of all value of y and $\overline{x}$ is the mean of all value of x and ${{b}_{yx}}$is the coefficient of regression coefficient of y on x.
So, before finding the equation we have to find the value coefficient of the regression coefficient of y on x. And, coefficient of regression is given by ${{b}_{yx}}=r\dfrac{{{\sigma }_{y}}}{{{\sigma }_{x}}}$. Here, $\sigma $ is the standard deviation and r is the correlation coefficient.
Since, we know that standard deviation is equal to square root of variance (i.e. $\sigma =\sqrt{Variance}$ ).
Since, we know that the value of variance of x (i.e. $Var\left( X \right)$ ) so standard deviation ${{\sigma }_{x}}=\sqrt{Var\left( X \right)}$ and $Var\left( X \right)$ = 9.
So, ${{\sigma }_{x}}=\sqrt{9}$ = 3 and it is given in question that ${{\sigma }_{Y}}=4$, and r = 0.6, hence ${{b}_{yx}}=0.6\times \dfrac{4}{3}$
$\therefore {{b}_{yx}}=0.8$
Now, also from the question we know that $\overline{x}$ = 10, $\overline{y}$= 12, and equation of regression of y on x is given by \[\left( y-\overline{y} \right)={{b}_{yx}}\left( x-\overline{x} \right)\].
So, we will get the equation after putting the value of $\overline{x}$, $\overline{y}$, and ${{b}_{yx}}$ in the above equation.
Hence, the equation is $\left( y-12 \right)=0.8\left( x-10 \right)$ .
Hence, y = 0.8x + 4.
Now, we will put x = 5 in the above equation to estimate the value of y.
So, when x = 5, $y=0.8\times 5+4$
Hence, y = 8.
This is our required solution.

Note: Students are required to note that ${{b}_{yx}}$ and ${{b}_{xy}}$ are slightly different. ${{b}_{yx}}$ is the coefficient of regression of y on x and ${{b}_{xy}}$ is the coefficient of regression of x on y. So, when we have to calculate the value of x when y is given we use ${{b}_{xy}}$ and when we calculate the value of y when x is given ${{b}_{yx}}$is used.