Regression Coefficient of X on Y and Ratio of Variance of X and Y

Regression analysis is a statistical tool used to estimate the relationship between two or more variables. In this question, we are given the regression coefficient of X on Y and the ratio of variance of X and Y. Using this information, we need to obtain the regression coefficient of Y on X.

Regression Coefficient of X on Y

The regression coefficient of X on Y is denoted by bxy. It measures the change in the value of X for a unit change in the value of Y. In this case, we are given that the regression coefficient of X on Y is 1.5. This means that for a unit change in Y, X changes by 1.5 units.

Ratio of Variance of X and Y

The variance of a variable measures the spread of its values around the mean. The ratio of variance of X and Y is given as 9:81. This means that the variance of X is one-ninth of the variance of Y. Mathematically, we can express this as:

Var(X) = 1/9 * Var(Y)

Regression Coefficient of Y on X

To obtain the regression coefficient of Y on X, we need to use the formula:

bxy * byx = Var(Y) / Var(X)

where byx is the regression coefficient of Y on X.

Substituting the given values, we get:

1.5 * byx = 81 / 9

Simplifying this equation, we get:

byx = 81 / (9 * 1.5) = 6

Therefore, the regression coefficient of Y on X is 6. This means that for a unit change in X, Y changes by 6 units.

Conclusion

In this question, we used the given information about the regression coefficient of X on Y and the ratio of variance of X and Y to obtain the regression coefficient of Y on X. By using the formula, we found that the regression coefficient of Y on X is 6.