Question

The method of least squares dictates that we choose a regression line where the sum of the square of deviations of the points from the line is :

• A. Maximum
• B. Minimum
• C. Zero
• D. Negative

Hint:

The term "least squares" itself tells you the objective: "least" means minimum, and "squares" refers to the squared deviations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The Method of Least Squares is a standard mathematical optimization technique used to find the best-fitting curve or line for a set of data points.
Detailed Explanation:
Let us examine the objective of the least squares method:
1. For any data point $(x_i, y_i)$, the vertical distance from the point to the regression line $\hat{y}_i = a + bx_i$ is the residual (deviation): \[ e_i = y_i - \hat{y}_i \] 2. Since residuals can be positive or negative, summing them directly can cancel out errors. To prevent this, we square the deviations: \[ e_i^2 = (y_i - \hat{y}_i)^2 \] 3. The method of least squares minimizes the sum of these squared residuals: \[ S = \sum_{i=1}^{n} e_i^2 = \sum_{i=1}^{n} (y_i - a - bx_i)^2 \] 4. We choose parameters $a$ and $b$ that minimize this sum ($S \rightarrow \text{Minimum}$).

Step 2: Final Answer:
The method seeks to minimize this sum, which corresponds to Option (B).