Which one of the following statements is false?

Answer: Option B

Option A: Both correlation and regression coefficients have the same sign is generally true. Both correlation and regression coefficients can be positive or negative, indicating the direction of the relationship between variables.

Option B: Arithmetic mean of the regression coefficients is always more than the correlation coefficient is false. The arithmetic mean of regression coefficients is not necessarily greater than the correlation coefficient.

Option C: Regression coefficients are independent of both the origin and scale is generally true. Regression coefficients are not affected by changes in the origin or scale of the variables.

Option D: Correlation coefficient is the square root of two regression coefficients is generally true. The correlation coefficient is the square root of the product of the two regression coefficients.

Therefore, the correct answer is Option B: Arithmetic mean of the regression coefficients is always more than the correlation coefficient.