Paper 4: Fundamentals of Business Mathematics & Statistic

8.10 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Time Series Analysis

Table : Finding the trend line by taking middle year (2006) as origin

Year Exports Deviation X^2 XY

(in tons) from 2006 (X)

2003 47 -3 9 -141

2004 50 -2 4 -100

2005 53 -1 1 -53

2006 65 0 0 0

2007 62 1 1 62

2008 64 2 4 128

2009 72 3 9 216

413 ΣX = 0 ΣX^2 = 28 ΣXY=112

The equation of the straight line trend is

Yc = a + bX

Since ΣX = 0; a = ΣN 7Y=^413 =^59

b = 2

XY 112 4

X 28

Σ = =

Σ

The trend line is Yc = 59 + 4X (origin 2006)

For 2012 X = 6, so Y 2012 = 59 + 4(6) = 59 + 24

Y = 83

The expected exports for the year 2012 are 83 tons.

Note : We see that though trend line equations are different depending upon the year of origin, but the

trend values would come out to be the same.

Example 7 :

Fit the Straight Line Trend by method of least squares and estimate the sales for 2012.

Year 2006 2007 2008 2009 2010 2011

Sales 12 13 14 15 22 26

Solution:

Table : Calculation of Trend & Trend Values

Year Sales Deviations from X = 2X’ X^2 XY Yc = a+bx

2008.5 (X’)

2006 12 -2.5 -5 25 -60 17 + 1.4(-5) = 10

2007 13 -1.5 -3 9 -39 17 + 1.4(-3)= 12.8

2008 14 -0.5 -1 1 -14 17 + 1.4(-1)=15.6

2009 15 0.5 1 1 15 17 + 1.4(1) =18.4

2010 22 1.5 3 9 66 17 + 1.4(3) = 21.2

2011 26 2.5 5 25 130 17 + 1.4(5) = 24.0

102 ΣX’ = 0 ΣX = 0 ΣX^2 = 70 ΣXY = 98