Page 410 - DMTH404_STATISTICS

Statistics

Notes Solution.

We note that n is even in the given example.
Calculation Table

Year ( ) Sales ( ) d =  1989.5 X = 2d XY X 2 Trend Values
t
t
Y
1987 15  2.5  5  75 25 15.45
1988 17  1.5  3  51 9 17.27
1989 20  0.5  1  20 1 19.09
1990 21 0.5 1 21 1 20.91
1991 23 1.5 3 69 3 22.73
1992 24 2.5 5 120 5 24.55
Total 120 0 64 70

From the above table, we can write
120 64
a = = 20 and b = = 0.91
6 70
 The fitted trend line is Y = 20 + 0.91 X
Year of origin : Middle of 1989 and 1990 or 1st Jan. 1990
1
Unit of X : year (Since X changes by 2 units in one year)
2
Nature of Y values : Annual figures of sales.

Calculation of trend values
Trend for 1989 = 20 - 0.91 = 19.09
Trend for 1988 = 19.09 - 2  0.91 = 17.27
Trend for 1990 = 20 + 0.91 = 20.91

Trend for 1991 = 20.91 + 2  0.91 = 22.73, etc.
To predict the sales for 1993, we note that X = 7
Thus, the predicted sales = 20 + 7  0.91 = Rs 26.37 (thousand).
Shifting of Origin of a Trend Equation

Let Y = a + bX be the equation of linear trend, with 1985 as the year of origin and unit of X equal
to 1 year.
To shift origin of the above equation, say to 1990, we proceed as follows : The associated value
of X for 1990 is 5. Thus, the trend for 1990 = a + 5b. We know that a linear trend equation is given
by Y = trend value in the year of origin + bX. Thus, we can write the trend equation, with origin
at 1990, as Y = a + 5b + bX = a + b (X + 5). This implies that the required equation can be obtained
by replacing X by X + 5 in the original trend equation.

Similarly, the trend equation with 1984 as origin can be written as Y = a + b (X - 1) = (a - b) + bX.
Further, if the unit of X is given to be half year, the trend equation with 1990 as the year of origin
can be written as Y = a + b (X + 10) = (a + 10b) + bX.

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