The correct answer is $\boxed{\text{B}}$.
The arithmetic mean of the regression coefficients is always less than or equal to the correlation coefficient. This is because the regression coefficients are estimates of the slope of the line of best fit, and the correlation coefficient is a measure of the strength of the linear relationship between the two variables. The closer the correlation coefficient is to 1, the stronger the linear relationship is, and the closer the regression coefficients will be to the slope of the line of best fit. However, the regression coefficients are also affected by the variance of the data, and the variance of the data can cause the regression coefficients to be less than the slope of the line of best fit.
Here is a brief explanation of each option:
- Option A: Both correlation and regression coefficients have same sign. This is true. The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. The regression coefficients are estimates of the slope of the line of best fit, and the slope of the line of best fit is also a measure of the strength and direction of the linear relationship between two variables. Therefore, both the correlation coefficient and the regression coefficients will have the same sign.
- Option B: Arithmetic mean of the regression coefficients is always more than the correlation coefficient. This is false. The arithmetic mean of the regression coefficients is always less than or equal to the correlation coefficient. This is because the regression coefficients are estimates of the slope of the line of best fit, and the correlation coefficient is a measure of the strength of the linear relationship between the two variables. The closer the correlation coefficient is to 1, the stronger the linear relationship is, and the closer the regression coefficients will be to the slope of the line of best fit. However, the regression coefficients are also affected by the variance of the data, and the variance of the data can cause the regression coefficients to be less than the slope of the line of best fit.
- Option C: Regression coefficients are independent of both the origin and scale. This is true. The regression coefficients are estimates of the slope of the line of best fit, and the slope of the line of best fit is independent of both the origin and scale. This means that the regression coefficients will not change if the data is shifted up or down, or if the data is stretched or compressed.
- Option D: Correlation coefficient is the square root of two regression coefficients. This is false. The correlation coefficient is not the square root of two regression coefficients. The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables, and the regression coefficients are estimates of the slope of the line of best fit. The slope of the line of best fit is not the square root of two regression coefficients.