Final Answer:
The mean of x and the mean of y both lie on the simple regression line.
Step-by-step explanation:
The simple regression line is defined by the equation \( \hat{y} = b_0 + b_1x \), where \( \hat{y} \) is the predicted value of y, \( b_0 \) is the y-intercept, \( b_1 \) is the slope, and \( x \) is the independent variable. When calculating the mean of x and y, we find the central tendency of the data in both dimensions. Plugging the mean of x into the regression equation (\( \hat{y} = b_0 + b_1 \bar{x} \)), it simplifies to the mean of y. This occurs because the line passes through the point (\( \bar{x}, \bar{y} \)), where \( \bar{x} \) is the mean of x and \( \bar{y} \) is the mean of y. Therefore, both means lie on the regression line.
In essence, the mean of x represents the center of the x-axis, and when substituted into the regression equation, it aligns with the y-axis at the mean of y. This alignment is a fundamental characteristic of the regression line, indicating that the line accurately captures the relationship between x and y. This relationship is further emphasized by the fact that the line intersects at the point defined by the mean values of x and y.