Question

In a linear regression, a line of best fit can be obtained by

• A. minimizing the sum of squares of the residuals
• B. minimizing the sum of the residuals
• C. minimizing the standard deviation from the mean
• D. minimizing the standard error of the mean

Hint:

In linear regression, always minimize the sum of the squares of the residuals to obtain the best fit line. This is the core principle of the least squares method.

Answer 1

The Correct Option is A

Step 1: Understanding the concept of residuals.
In linear regression, a residual is the difference between an observed value and the predicted value from the regression model. The residual for each data point is given by: \[ e_i = y_i - \hat{y}_i \] where \( y_i \) is the observed value and \( \hat{y}_i \) is the predicted value from the regression line.
Step 2: Least squares method.
The method used to find the line of best fit in linear regression is called the "least squares method." This method minimizes the sum of the squared residuals to obtain the best-fitting line. The sum of squared residuals is: \[ S = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 \] where \( n \) is the number of data points. 
Step 3: Why minimizing the sum of squares of residuals works.
Minimizing the sum of the squared residuals ensures that the line of best fit has the smallest possible overall error. Squaring the residuals amplifies larger errors, which helps to prevent the line from being influenced too much by smaller errors and ensures a better overall fit.