The correct answer is: B. (A) is correct, but (R) is not correct
Assertion (A) is correct. The regression coefficient of X on Y is the slope of the line that best fits the data when X is the independent variable and Y is the dependent variable. If the regression coefficient of X on Y is greater than one, then the line slopes upwards from left to right, which means that as X increases, Y also increases. This implies that there is a positive correlation between X and Y.
Reason (R) is not correct. The geometric mean between two regression coefficients is not the coefficient of correlation. The coefficient of correlation is a measure of the strength of the linear relationship between two variables. It is calculated by taking the square root of the product of the two regression coefficients.
Here is an example to illustrate why assertion (A) is correct but reason (R) is not correct. Consider the following data:
X | Y
——- | ——–
1 | 2
2 | 4
3 | 6
4 | 8
The regression coefficient of X on Y is 2. This means that the line that best fits the data has a slope of 2. The regression coefficient of Y on X is 1. This means that the line that best fits the data has a slope of 1. The geometric mean of the two regression coefficients is $\sqrt{2 \times 1} = \sqrt{2}$. The coefficient of correlation is $\frac{2 \times 1}{\sqrt{2 \times 2}} = \frac{2}{\sqrt{4}} = \frac{1}{\sqrt{2}}$.
As you can see, the geometric mean of the two regression coefficients is not equal to the coefficient of correlation.