Regression Line of y on x:

The regression line of y on x is a straight line that represents the relationship between two variables, y and x. It is used to predict the value of the dependent variable (y) based on the value of the independent variable (x). The regression line is derived through a process called linear regression.

Linear Regression:

Linear regression is a statistical technique used to model the relationship between two variables. In the case of simple linear regression, there is only one independent variable (x) and one dependent variable (y). The goal of linear regression is to find the best-fitting line that minimizes the differences between the observed data points and the predicted values.

Minimization of Vertical Distances:

The regression line of y on x is derived by minimizing the vertical distances between the observed data points and the predicted values. The vertical distance represents the difference between the actual value of y and the predicted value of y based on the regression line. By minimizing these vertical distances, the regression line is able to capture the overall trend and relationship between the variables.

Explanation:

When we minimize the vertical distances, we are essentially minimizing the errors or residuals between the observed data points and the predicted values. The residuals represent the unexplained variation in the dependent variable (y) that cannot be accounted for by the independent variable (x). The regression line is derived in such a way that the sum of the squared residuals is minimized.

By minimizing the vertical distances, the regression line is able to pass through the data points in a way that best represents the overall trend and relationship between the variables. It aims to capture the central tendency of the data and provide the best estimate of the dependent variable based on the independent variable.

Minimization of Horizontal Distances:

In contrast, minimizing the horizontal distances in the scatter diagram would not be appropriate for deriving the regression line. The horizontal distances represent the difference between the observed value of x and the predicted value of x based on the regression line. However, the regression line is not concerned with predicting the independent variable (x), but rather predicting the dependent variable (y) based on the independent variable (x).

Therefore, the correct answer is option 'A' - the regression line of y on x is derived by the minimization of vertical distances in the scatter diagram.