Which of the following statements are correct?

(A) The method of least squares determines the position of the trend line of the given time series.

(B) The trend line is called the line of best fit.

(C) The line of best fit is a line in which the sum of deviations of the actual values of the variable from their corresponding trend value is always positive.

(D) The normal equations of the trend line $y = a + bx$ are $\sum y = na + b \sum x$ and $\sum xy = a \sum x + b \sum x^2$, where $n$ is the numbers of observations.

Choose the correct answer from the options given below:

Let's evaluate each statement one by one.

Statement (A): The method of least squares determines the position of the trend line of the given time series.

This is correct. The method of least squares is a mathematical technique that finds the exact position (i.e., the values of $a$ and $b$) of the trend line by minimizing the sum of squared deviations between actual and trend values.

Statement (B): The trend line is called the line of best fit.

This is correct. The trend line obtained using the method of least squares is called the line of best fit because no other line fits the data better in terms of minimizing squared errors.

Statement (C): The line of best fit is a line in which the sum of deviations of the actual values from their corresponding trend values is always positive.

This is incorrect.

The key property of the line of best fit is:

$\sum (y - \hat{y}) = 0$

The sum of deviations is always zero, not positive. Some actual values lie above the trend line (positive deviations) and some lie below it (negative deviations). The least squares method ensures these cancel out perfectly, giving a sum of zero.

Statement (D): The normal equations of the trend line $y = a + bx$ are $\sum y = na + b\sum x$ and $\sum xy = a\sum x + b\sum x^2$, where $n$ is the number of observations.

This is correct. These are the standard normal equations derived from the method of least squares for fitting a straight line $y = a + bx$.

The first equation is obtained by summing $y = a + bx$ over all $n$ observations.

The second equation is obtained by multiplying $y = a + bx$ by $x$, then summing over all $n$ observations.

Statements (A), (B), and (D) are correct.

Statement (C) is incorrect because the sum of deviations from the line of best fit is zero, not positive.