Regression must be defined. What is the significance of the two Regression Lines? Under what circumstances may just one Regression Line exist?

Linear Regression

Q1. Regression must be defined. What is the significance of the two Regression Lines? Under what circumstances may just one Regression Line exist?

Regression literally means "return" or "go back." In the nineteenth century, Francis Galton employed regression for the first time in his work "Regression towards Mediocrity in Hereditary Stature" to examine hereditary traits.

In current times, regression is not only used to research genetic traits, but it is also frequently utilised to study the anticipated dependency of one variable on another.

As a result, regression refers to the process of calculating the best likely values of unknown data for a variable based on the known values of another variable.

Forecasting, decision making, and examining two or more variables in the economic sector are all aided by regression.

It also displays the correlation's direction, quality, and degree.

Regression Line:

A regression line is defined as the line that provides the best estimate of the dependent variable for every given value of the independent variable. If we consider two variables, X and Y, we will have two regression lines: The regression of X on Y and the regression of Y on X.

Regression Line X and Y:

In this configuration, Y is an independent variable and X is a dependent variable, and the best predicted value of X is computed for the given value of Y. 

Regression Line Y on X:

In this case, Y is the dependent variable, and X is the independent variable; the best anticipated value of Y is judged to be comparable to the provided value of X.

One significant reason for having two regression lines is that they are based on the least square assumption, which states that the sum of squares of the deviations from different locations to that line must be the smallest.

The deviations from the line of best fit can be assessed in two ways: vertically, parallel to the Y axis, and horizontally, parallel to the X axis.

It is necessary to have two regression lines to minimise the total of the squares independently.

Single line of Regression:

When the two variables have a perfect positive or perfect negative correlation (r = 1), the regression lines will meet or overlap and create a single regression line.

Q2. What exactly is the distinction between Regression and Correlation?

Q3. How do Regression Equations get their names? Explain.

Computation of Regression Equations:

Regression equations are algebraic expressions of regression lines.

There are two equations that are like lines:

Computation of Regression Coefficients:

Q4. Provide an explanation of the Regression Coefficients. How is Correlation derived from Regression Coefficients?

Interpretation of Regression Coefficients:

           i. If both coefficients are positive, the correlation coefficient will be positive; if both coefficients are negative, the correlation coefficient will be negative.

                      ii. The sign of both regression coefficients will be the same.

                     iii. The sum of both regression coefficients cannot be more than one.

Calculation of Correlation Coefficient from Regression Coefficient: