Question

The least squares regression line minimizes the sum of the
A. Sum of Differences between actual and predicted Y values
B. Sum of Squared differences between actual and predicted X values
C. Sum of Absolute deviations between actual and predicted X values
D. Sum of Absolute deviations between actual and predicted Y values
E. Sum of Squared differences between actual and predicted Y values

Answer 1

Concepts and reason

Regression:

Regression is a technique that is used to determine relationship between two or more variables. That is, the change in the predictor variable influences the change in the dependent variable is determined. Moreover, in regression analysis which involves more than one independent variable, the change in the dependent is analyzed when the one independent variable is varied by keeping all other independent variables as constant.

If the data set is bivariate, then linear regression best suits the data. The straight line known as least squares regression line is obtained which best represents the data with two variables.

Slope:

The slope of a least squares regression line is interpreted as the predicted change in the average response variable for a one-unit change in the explanatory variable.

Intercept:

The y-intercept of a regression line is interpreted as the predicted value of the response variable when the explanatory variable has a value of zero (though be wary of extrapolation in interpreting the intercept or other values outside the original data range).

Least squares:

Least squares method is one of the approaches for regression analysis. Moreover, least squares minimize the sum of the squares of residuals. Also, it is used to find the values of constants in regression equation.

Fundamentals

If the data set is bivariate, then linear regression best suits the data. The straight line known as least squares regression line is obtained which best represents the data with two variables. The equation of the line is given by,

$\begin{array}{l}\\\hat y = a + bx\\\\{\rm{where,}}\\\\a{\rm{ - Intercept}}\\\\b{\rm{ - Slope}}\\\\\hat y{\rm{ - Predicted value of the dependent variable}}\\\\x{\rm{ - Independent or predictor variable}}\\\end{array}$

Theincorrect options are explained below:

In the regression analysis X is the independent variable and Y is the dependent variable. The value of Y is predicted in the regression analysis. The least square regression method minimizes the sums of the squared residuals but it would not reduce the sum of absolute deviations of the actual and predicted values.

This indicates that the options ‘Sum of Differences between actual and predicted Y values, Sum of Squared differences between actual and predicted X values, Sum of Absolute deviations between actual and predicted X values, Sum of Absolute deviations between actual and predicted Y values’ are incorrect.

Theincorrect option is explained below:

In the regression analysis X is the independent variable and Y is the dependent variable. The value of Y is predicted in the regression analysis. The least square regression method minimizes the sums of the squared residuals, that is the sum of squared differences of the original values and predicted values in regression.

This indicates that the option ‘Sum of Squared differences between actual and predicted Y values’ is correct.

Ans:

The least squares regression line minimizes the sum of the squared differences between actual and predicted Y values.