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 number of observations.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
The method of least squares chooses a and b in \(y_t = a + bx\) so that the sum of the squared deviations \(\sum (y - y_t)^2\) is the least.
Checking the statements
(A) The method of least squares determines the position of the trend line of the given time series. — Correct. Finding a and b fixes the position of the trend line.
(B) The trend line is called the line of best fit. — Correct. No other straight line gives a smaller sum of squared deviations, so it is 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. — Incorrect. For the least squares line the sum of deviations is zero: the first normal equation gives
\( \displaystyle \sum (y - y_t) = \sum y - (na + b\sum x) = 0\)
(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. — Correct. These are the two normal equations obtained by setting the partial derivatives of \(\sum (y - a - bx)^2\) with respect to a and b equal to zero.
CBSE Applied Mathematics (Class XII):
Computation of Straight-line Trend by Using Method of Least Squares
