Is correlation coefficient greater than geometric mean of regression coefficients?Justify

To determine if the correlation coefficient is greater than the geometric mean of regression coefficients, we need to follow these steps:

Step 1: Calculate the correlation coefficient. The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where a value of +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.

Step 2: Calculate the regression coefficients. In regression analysis, the regression coefficients represent the slope of the regression line, which shows the relationship between the independent variable(s) and the dependent variable. The geometric mean of regression coefficients is calculated by taking the product of all the regression coefficients and then taking the nth root, where n is the number of coefficients.

Step 3: Compare the correlation coefficient with the geometric mean of regression coefficients. If the correlation coefficient is greater than the geometric mean of regression coefficients, it suggests that the strength of the linear relationship between the variables is greater than the average strength of the individual relationships between the independent variables and the dependent variable.

Justification: The correlation coefficient provides a measure of the overall strength of the linear relationship between two variables, while the geometric mean of regression coefficients represents the average strength of the individual relationships between the independent variables and the dependent variable. Therefore, if the correlation coefficient is greater than the geometric mean of regression coefficients, it indicates that the overall relationship between the variables is stronger than the average relationship between each independent variable and the dependent variable.