Question

In bivariate data following results were obtained.
Mean value of X = 53, Mean value of Y = 27, byx = -1.5, bxy = 0.2. Find the regression equation of X on Y and Y on X.

Hint:

The regression equations of X on Y and Y on X are used to estimate the relationship between two variables. The coefficients \(b_{xy}\) and \(b_{yx}\) determine the steepness of the regression lines.

Answer 1

Step 1: Define regression equations.
The regression equations of X on Y and Y on X can be written as: - Regression of X on Y: \[ X - \bar{X} = b_{xy}(Y - \bar{Y}) \] where \(\bar{X}\) and \(\bar{Y}\) are the mean values of X and Y, and \(b_{xy}\) is the regression coefficient of X on Y. - Regression of Y on X: \[ Y - \bar{Y} = b_{yx}(X - \bar{X}) \] where \(b_{yx}\) is the regression coefficient of Y on X.
Step 2: Calculate the regression equation of X on Y.
We are given the following: \[ \text{Mean of X} = 53, \quad \text{Mean of Y} = 27, \quad b_{xy} = 0.2 \] Substitute these values into the regression equation of X on Y: \[ X - 53 = 0.2(Y - 27) \] Simplify: \[ X = 53 + 0.2(Y - 27) \] \[ X = 53 + 0.2Y - 5.4 \] \[ X = 47.6 + 0.2Y \] Thus, the regression equation of X on Y is: \[ X = 47.6 + 0.2Y \]
Step 3: Calculate the regression equation of Y on X.
We are given \(b_{yx} = -1.5\). Substituting the known values into the regression equation of Y on X: \[ Y - 27 = -1.5(X - 53) \] Simplify: \[ Y = 27 - 1.5(X - 53) \] \[ Y = 27 - 1.5X + 79.5 \] \[ Y = 106.5 - 1.5X \] Thus, the regression equation of Y on X is: \[ Y = 106.5 - 1.5X \]