Page 412 - DMTH404_STATISTICS

Statistics

Notes a b
A quarterly trend equation can also be obtained in a similar way. We can write Y = + X , as
4 4
a b
the quarterly average equation and Y = + X , as the quarterly trend equation.
4 16

Example 10: The equation for yearly sales (in '000 Rs) of a commodity with 1st July, 1971,
as origin is Y = 91.6 + 28.8X.
(i) Determine the trend equation to give monthly trend values with 15th January, 1972, as
origin.

(ii) Calculate the trend values for March, 1972 to August, 1972.
Solution.

91.6 28.8
(i) The monthly trend equation with 1st July, 1971, as origin is given by Y = + X =
12 144
7.63 + 0.2X, where unit of X is one month.
To shift the origin to 15th January, 1972, we replace X by X + 6.5 in the above equation.
Note that the associated value of X for 15th January, 1972, is 6.5. Thus, the required equation
is Y = 7.63 + 0.2(X + 6.5) = 8.93 + 0.2X
(ii) Calculation of trend values

Trend value for March, 1972 = 8.93 + 0.2  2 = Rs 9.33
Trend value for April, 1972 = 9.33 + 0.2 = Rs 9.53
Trend value for May, 1972 = 9.53 + 0.2 = Rs 9.73
Trend value for June, 1972 = 9.73 + 0.2 = Rs 9.93
Trend value for July, 1972 = 9.93 + 0.2 = Rs 10.13

Trend value for August, 1972 = 10.13 + 0.2 = Rs 10.33.

Example 11:
Convert the following into annual trend equation :
Y = 350 + 3X with origin = I - II Quarter, 1986, unit of X = one quarter and Y denotes quarterly
production.

Solution.
Important Note : To convert a quarterly (or monthly) equation into an annual equation, it is
necessary to first shift the origin to the middle of the year.

In the given example, since the middle of the year lies a quarter ahead, we shall replace X by X
+ 1 in the above equation. Thus, the quarterly equation with middle of the year as origin is Y =
350 + 3(X +1) = 353 + 3X.
Then, the annual trend equation can be written as
Y = 353  4 + 3  16X = 1412 + 48X

404 LOVELY PROFESSIONAL UNIVERSITY