8.12 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICSTime Series Analysis
(ii) For 2012 X = 7, so Y 2012 = 112 + 12 (7)
Y = 196.
(iii) The annual increase in the expected, sales of the firm is 12 (‘000 quintals)
Example9 :
The sales of a commodity (in ‘000 of `) are given below :
Year 2001 2002 2003 2004 2005 2006 2007
Sales (in’000 of `) 82 86 81 86 92 90 99
(i) Using the method of least squares, fit a straight line equation to the data
(ii) What is the average annual change in the sales?
(iii) Obtain the trend values for the years 2001-2007 and show that the sum of difference between the
actual and the trend values is equal to zero.
(iv) What are the expected sales for the year 2012?
Solution:
Table : Calculation of trend values
Year Sales (in ’000 Deviations from X^2 XY Trend values Y- Yc
of `) (Y) 2002 (X) Yc = 88+2.5X
2001 82 -3 9 -246 88 + 2.5(-3) = 80.5 1.5
2002 86 -2 4 -172 88 + 2.5(-2) = 83.0 3.0
2003 81 -1 1 -81 88 + 2.5(-l) = 85.5 -4.5
2004 86 0 0 0 88 + 2.5(0) = 88.0 -2.0
2005 92 1 1 92 88 + 2.5(1) = 90.5 1.5
2006 90 2 4 180 88 + 2.5(2) = 93.0 -3.0
2007 99 3 9 297 88 + 2.5(3) = 95.5 3.5
ΣY=616 ΣX=0 ΣX^2 =28 ΣXY=70 ΣY-YC=0
(i) The equation of the straight line trend is
Yc = a + bX
Since ΣX = 0; a = ΣΣXXY 70 2 = 28 =2.5
The trend line is Yc = 88 + 2.5X
(i) The average annual change in sales is 2.5 x 1000 = ` 2,500
(iii) Sum of difference between the actual- (Y) and trend line (Yc) is equal to
Σ(Y - Yc) = 0 as shown in last column of the table.
(iv) Expected sales for the year 2012—
For 2012 X = 8, so Y 2012 = 88 + 2.5(8) = 88 + 20
Y = ` 108
The expected sales for the year 2012 is ` 1,08,000.