Step-by-Step Solution

Topic: Method of Least Squares (Time Series)

(I) EQUATION OF STRAIGHT LINE TREND

Since the number of years (n) is 6 (Even), we shift the origin to the center (between 1998 and 1999).
Let Origin = 1998.5.
Let X = 2(Year – 1998.5) to avoid decimals.

Year Sales (Y) X X2 XY
1996 6.5 -5 25 -32.5
1997 5.3 -3 9 -15.9
1998 4.3 -1 1 -4.3
1999 6.1 1 1 6.1
2000 5.6 3 9 16.8
2001 7.8 5 25 39.0
Sum (Σ) 35.6 0 70 9.2

Calculations:

a = ΣY / n = 35.6 / 6 = 5.933

b = ΣXY / ΣX2 = 9.2 / 70 = 0.131

Trend Equation (Yc) = 5.933 + 0.131X

(II)(A) TREND VALUES & 2002 FORECAST

Substituting X values into Yc = 5.933 + 0.131X:

  • 1996 (X=-5): 5.933 – 0.655 = 5.28
  • 1997 (X=-3): 5.933 – 0.393 = 5.54
  • 1998 (X=-1): 5.933 – 0.131 = 5.80
  • 1999 (X=1): 5.933 + 0.131 = 6.06
  • 2000 (X=3): 5.933 + 0.393 = 6.33
  • 2001 (X=5): 5.933 + 0.655 = 6.59

Forecast for 2002:

For Year 2002, X = 2(2002 – 1998.5) = 7.

Y2002 = 5.933 + 0.131(7) = 5.933 + 0.917

Expected Sales (2002) = 6.85 Lakhs (approx)

(II)(B) TREND FOR PROFIT (ODD YEARS)

Number of years n = 7 (Odd). Origin = Middle Year (2007).
Let X = Year – 2007.

Year Y X X2 XY
2004 114 -3 9 -342
2005 130 -2 4 -260
2006 126 -1 1 -126
2007 144 0 0 0
2008 138 1 1 138
2009 156 2 4 312
2010 164 3 9 492
Σ 972 0 28 214

Calculations:

a = ΣY / n = 972 / 7 = 138.86

b = ΣXY / ΣX2 = 214 / 28 = 7.64

Straight Line Trend: Yc = 138.86 + 7.64X
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