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Statistics Online Tutor Help Illustrations and Solutions

Illustration 1.

Convert the following annual trend equation into a quarterly one:

Yc = 32 + 4.8X

Solution:

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

Yc =

4 +

4 12

Substituting the respective values in the above we get,

Yc = 32

4 +

4.8

48

= 8 + 0.1 X

Where, X unit = 1 quarter, and Y unit = quarterly value of a variable.

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Illustration 2.

Given the trend equation, Yc = 240 1.8X, where X unit = 6 months, and Y unit = annual

production. Convert the equation into a monthly trend.

Solution:

Here, Y unit = annual production, and X unit is 6 months, So to convert the given equation

into a monthly trend, we have to divide a by 12, but bX by 12 6. Thus the required

equation will be :

Yc =

12 -

12 6

= 20 0.025X

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

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Illustration 3.

Convert the following trend equation into an annual basis:

Yc = 50 + 0.5X

Where, the point of origin is Jan 2004, X unit = one month and Y unit = monthly sales (in

000$)

Solution

Here, the origin of the equation is 15th 04 (i.e. the midpoint of January) which needs to be

shifted first to the 1st July 04 (i.e. the midpoint of the yr. 04) by 5.5 months forward.

Hence, K = 5.5

By the formula of shifting a trend we have,

Yc = a + b (X + K)

= 50 + 0.5 (X + 5.5)

= 50 + 0.5X + 2.75

= 52.75 + 0.5X

Now, by the formula of conversion of a monthly equation into an annual trend we have,

Yc = a 12 bX (12 12)

Substituting the respective values in the above we get,

Yc = 52.75 12 + 0.5 (144)

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= 633 + 72X

Where, the point of origin = 2004, X unit = 1 year and Y unit = annual sales (in 000 Rs).

Note. The above problems required both shifting of the trend origin, and conversion of the

trend equation. Since, the conversion was to be made from a lower time basis (i.e. monthly)

to a higher time basis (i.e. annual) shifting, was made first and the conversion at the last.

Illustration 4.

The equation for yearly sales (in 000$) for a commodit6y with the year, 2003 as origin is

Yc =91.6 + 28.8X. Determine the trend equation to give the monthly trend values with Jan

04. Since, the conversion is required to be made from a higher periodical base (annual) to

a lower periodical base (monthly), it is necessary to attempt on the conversion first, and

then on shifting of the trend origin as follows :

(ii) Conversion of the trend

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

Yc =

12 +

12 12

Substituting the respective values, in the above we get,

Yc = 91 .6

12 +

28.8

12 12 = 7.63 + 0.2X

Thus, Yc = 7.63 + 0.2 Y

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

(ii) Shifting of the Trend Origin

Here, it is required to shift forward the trend origin by 6.5 months i.e. from 1st July, 03 (the

midpoint of the year, 03) to the 15th Jan, 04 (the midpoint of the month).

Thus, K = 5.5

By the formula of shifting a trend we have,

Yc = a + b (X + K)

Substitution the respective values in the above we get,

Yc = 7.63 + 0.2 (X + 6.5)

= 7.63 + 0.2X + 1.3

= 8.93 + 0.2 Y

Yc = 8.93 + 0.2 X

Where, the point of origin is Jan 03, X unit = 1 month, and Y unit monthly sales.

(iii) Calculation of the trend value for March 04

The time deviation of March 04 from the trend origin Jan 04, or X = 2

Thus, when X = 2, Yc = 8.93 + 0.2 (2)

= 8.93 + 0.4 = 9.33

Hence, the trend value for March 04 will be 9.33.

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Illustration 5.

Convert the annual trend equation Yc = 48 + 1.2 X into monthly trend equation, and shift

the origin from 2003 to Jan 1, 2004.

Solution

Here, conversion is to be made first as it is required to be made from a higher period

(annual) to a lower period (monthly) basis.

(i) Conversion of the trend equation

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

Yc =

12 +

1212

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