Question

calculate the trend values by the method of least squares from the following data year: 2010 2011 2012 2013 2014 PRODUCTION IN TONS 830 920 710 900 1690

Answer 1

Answer:

Explanation:

Method of Least Squares

The line of best fit is a line from which the sum of the deviations of various points is zero. This is the best method for obtaining the trend values. It gives a convenient basis for calculating the line of best fit for the time series. It is a mathematical method for measuring trend. Further the sum of the squares of these deviations would be least when compared with other fitting methods. So, this method is known as the Method of Least Squares and satisfies the following conditions:

(i) The sum of the deviations of the actual values of Y and Ŷ (estimated value of Y) is Zero. that is Σ(Y–Ŷ) = 0.

(ii) The sum of squares of the deviations of the actual values of Y and Ŷ (estimated value of Y) is least. that is Σ(Y–Ŷ)2 is least ;

Procedure:

(i)  The straight line trend is represented by the equation Y = a + bX        …(1)

where  Y is the actual value, X is time, a, b are constants      

(ii)  The constants ‘a’ and ‘b’ are estimated by solving the following two normal

Equations   ΣY = n a + b ΣX   ...(2)

ΣXY = a ΣX + b ΣX2     ...(3)

Where ‘n’ = number of years given in the data.  

(iii) By taking the mid-point of the time as the origin, we get ΣX = 0

(iv) When ΣX = 0 , the two normal equations reduces to