If the lines of regression of sample are x + 6y = 6 and 3x + 2y = 10.Find i) mean of x & y andii) correlation coefficient between x and yiii) estimate y when x = 12

The problem involves two regression lines:

1. x + 6y = 6, which can be rewritten as y = (1/6)x + 1. This is the regression of y on x.

2. 3x + 2y = 10, which can be rewritten as x = (1/1.5)y + 5. This is the regression of x on y.

i) The means of x and y can be found by equating the constants from the two equations.

For the mean of x (denoted as x̄), we have x̄ = 5.

For the mean of y (denoted as ȳ), we have ȳ = 1.

ii) The correlation coefficient (r) can be found using the slopes of the regression lines. If b1 is the slope of y on x and b2 is the slope of x on y, then r = sqrt(b1*b2).

For b1, we have b1 = 1/6.

For b2, we have b2 = 1/1.5 = 2/3.

So, r = sqrt((1/6)*(2/3)) = sqrt(1/9) = 1/3.

iii) To estimate y when x = 12, we can use the regression of y on x: y = (1/6)x + 1.

Substituting x = 12, we get y = (1/6)*12 + 1 = 2 + 1 = 3.