Page 413 - DMTH404_STATISTICS

Unit 30: Method of Least Square

Notes
Example 12: Convert the following annual trend equation, for the production of cloth in
a factory, into monthly average equation and predict the monthly averages for 1988 and 1989.
Y = 96 + 7.2X, with origin = 1986, unit of X = 1 year and Y denotes annual cloth production in '000
metres.

Solution.
The average monthly equation is given by
96 7.2
Y = + X = 8 + 0.6 X, where origin = 1986, unit of X = 1 year and Y denotes monthly average
12 12
production in the year.
The predicted values of Y are 8 + 0.6  2 = 9.2 thousand metres for 1988 and 9.2 + 0.6 = 9.8
thousand metres for 1989.

30.1.2 Fitting of Parabolic Trend

2
The mathematical form of a parabolic trend is given by Y = a + bt + ct or Y = a + bt + ct 2
t
(dropping the subscript for convenience). Here a, b and c are constants to be determined from
the given data.
Using the method of least squares, the normal equations for the simultaneous solution of a, b,
and c are :

Y = na + bt + ct 2
2
tY = at + bt + ct 3
2
2
3
t Y = at + bt + ct 4
By selecting a suitable year of origin, i.e., define X = t - origin such that X = 0, the computation
3
work can be considerably simplified. Also note that if X = 0, then X will also be equal to zero.
Thus, the above equations can be rewritten as:
SY = na + cSX 2 .... (i)
SXY = bSX 2 .... (ii)
2
2
SX Y = aSX + cSX 4 .... (iii)
å XY
From equation (ii), we get b = .... (iv)
å X 2

å Y cå X 2

Further, from equation (i), we get a = .... (v)
n
nå X Y  (å X 2 )(å Y )
2
And from equation (iii), we get c = 2 .... (vi)
4
nå X  (å X 2 )
Thus, equations (iv), (v) and (vi) can be used to determine the values of the constants a, b and c.

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