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Statistics Tutor Help Illustrations and Solutions

Illustration 1.

The parabolic trend equation for the sales (000 $) of a company is given below :

Yc = 20.4 1.5X + 0.62

Shift the trend origin from 1999 to 2004 given that the time unit is 1 Year.

Solution:

Here, the trend origin is to be shifted forward by 5 years i.e. from 1999 to 2004. Hence, k =

5

By the formula of shifting a trend we have,

Yc = a b (X + K) + c(X + K)2

( it is a parabolic equation of the given order.)

Substituting the respective values in the above we have,

Yc = 20.4 1.5 X +5) + 0.6 (X + 5)2

= 20.4 1.5X 7.5 + 0.6 (2 + 10X + 25)

= 20.4 1.5X 7.5 + 0.62 + 6X + 15

= 27.9 + 4.5X + 0.62

Hence, the shifted trend equation is Yc = 27.9 + 4.5X + 0.62 given that the trend origin is

shifted to 2004.

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Illustration 2.

Shift the trend origin from 2003 to 1st Jan 04 in the linear equation,

Yc = 14.5 + 0.4X, given that unit of time = 1 Year.

Solution:

Here, the origin 2003 means 1st July 03 (the middle part of the year).

Hence, the origin is to be shifted forward by only 1

2 year i.e. from 1st July 03 to 1st Jan 04.

Thus, k =

By the formula of shifting a trend we have,

Yc = a + b (X + k)

Substituting the given values we have,

Yc = 14.5 + 0.4 (X + )

= 14.5 + 0.4X + 0.2

= 14.7 + 0.4X

Hence Yc = 14.7 + 0.4X

Where the trend origin is shifted to 1st Jan 04.

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Conversion of a Trend Equation.

By conversion of a trend equation, we mean the change of a trend equation formulated on

one type of time basis into a trend equation on another type of time basis. Thus, if a trend

equation of yearly period, is transformed into a trend equation of monthly or quarterly

period, or vice versa, it will be called conversion of a trend equation.

Method of Conversion.

The following methods are to be adopted for converting a given trend equation into a

required trend Equation:

(i) To convert an annual trend equation into a monthly trend equation: Divide a

by 12, bX, by 12 (or 144), c2, by 12 12 12 (or 1728) and so on.

(ii) To convert a monthly trend equation into an annual trend equation. Multiply a

by 12, bX by 12 12 (or 144), c2 by 12 12 12 (or 1728) and so on.

(iii) To convert an annual trend equation into a Quarterly trend equation : Divide a

by 4, bX by 4 12 (or 48), c2,by 4 12 12 12 (or 576) and so on.

(iv) To convert a quarterly trend equation into an annual trend equation : Multiply

a by 4, bX by 4 12 and c2 by 4 12 12 and so on.

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Illustration 3.

Convert the following annual trend equation into a monthly one:

Yc = 24 + 1.8X

Given X unit = 1 Year, and Y unit = annual production.

Solution:

By the formula of conversion of an annual trend to a monthly one we have :

Yc = a/12 + bX/(12 12)

Substituting the respective values in the above we get,

Yc = 24/12 + 1.8X/144

Or Yc = 2 + 0.0125X

Where, X unit = 1 month and Y unit = monthly prodn.

Illustration 4.

Convert the following monthly trend equation into the annual one :

Yc = 2 + 0.0125X

Where, X unit = 1 month, and Y unit = monthly sales

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Solution

By the formula of conversion of a monthly trend into an annual one we have,

Yc = a 12 + bX (12 12)


Yc = 2 12 0.0125 144 or

Yc = 24 + 1.8X

Where, X unit = 1 Year, and Y unit = annual sales.

Illustration 5.

Convert the following annual trend equation into a monthly order:

Yc = 60 + 3.6X + 2.42

Solution

By the formula of conversion of an annual equation of second degree parabola into a

monthly one we have:

Yc =

12 +

12 12 +

2

12 12 12


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Yc = 60

12 +

3.6

144 +

2.42

1728 ,

= 5 + 0.025X + 0.0014 2

Where, X unit = 1 month, and Y unit = monthly value.

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