Online Statistics Homework Help

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Online Statistics Homework Help Service Online Statistics Assignment Help

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Online Statistics Homework Help | Online Statistics Assignment Help

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Online Statistics Homework Illustrations and Solutions

Illustration 1.

Shift the trend origin form 2001 to 2004 in the straight line trend equation, Yc = 25 + 2X,

given that the time unit = 1 Year.

Solution

Here, the trend origin is to be shifted forward by 3 years i.e. from 2001 to 2004. Thus K =3

By the formula of trend shifting we have,

Yc = a + b (X + K)

Substituting the respective values in the above we get, Yc = 25 + 2 (X +3)

= 25 + 2X + 6

= 31 + 2X

Thus, the shifted equation is given by

Yc = 31 + 2X

Where, origin of X = 2004, and

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X unit = 1 Year.

Note. From the above shifted equation, it must be noted that the value of a only has been

changed from 25 to 31, whereas the value of b remains the same 2.

Illustration 2.

The trend equation for a time series is fitted as Yc = 60 + 1.5X

Shift the trend origin from 2004 to 1998, given that the time unit is 1 year.

Solution

Here, the trend origin is to be shifted back by 5 years i.e. from 2004 to 1999.

Thus K = -5

By the formula of trend shifting we have, Yc = a + b (X K)

= 60 + 1.5 (X 5)

= 60 + 1.5X 7.5

= 52.5 + 1.5X, where the origin of X = 1999.

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Illustration 3.

Shift the trend origin from 2002 to 2004 in the following trend equation:

Yc = 20.3 + 0.4X + 0.52

Given that the time unit = 1 Year.

Solution

Here, the trend origin is to be shifted forward by 2 years i.e. from 2002 to 2004. Hence, k =

2. By the formula of trend shifting we have, Yc = a +b (X + K) + c (X + K )2

( it is a parabolic equation of second degree)

Substituting the respective values in the above, we get Yc = 20.3 + 0.4 (X + 2) + 0.5 (X +

2)2

= 20.3 + 0.4X + 0.8 + 0.5 (2 + 4X + 4)

= 20.3 + 0.4X + 0.8 + 0.52 + 2X + 2

= Yc = 23.1 + 2.4X + 0.52

In the above equation, it must be seen that the values of both a and b have been changed,

whereas the value of c has remained unchanged This is because, it was the case of shifting

a parabolic equation of second degree.

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Illustration 4.

From the following data, calculate the 3 yearly weighted moving averages, the weights

being 1, 2 and 3 respectively.

Year : Value :

1998 2

1999 6

200 3

2001 5

2002 7

2003 4

2004 2

Solution:

Computation of the 3 yearly weighted moving averages (Weights being 1, 2 and 3

respectively)

Year Value X

3 Yearly weighted moving totals ( )

3 yearly weighted moving averages

i.e. ( / ) 1998 1999 2000 2001 2002

2003 2004

2 6 3 5 7

4 2

- (2 1 + 6 2 + 3 3) = 23 (6 1 + 3 2 + 5 3) = 27

(3 1 + 5 2 + 7 3) = 34 (5 1 + 7 2 + 4 3) = 31 (7 1 + 4 2 + 2 3) = 21 -

- 23/6 = 3.83 27/6 = 4.50 34/6 = 5.67 31/6 = 5.17

21/6 = 3.50 -

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Determination of Trend Values through Moving Average Method

The moving average method discussed above can be used as a simple device of reducing

the fluctuations, and of obtaining the trend values in a time series with a fair degree of

accuracy. Under this method, the moving averages calculated represent the trend values for

the middle point of the period of moving averages. The moving averages when plotted on a

graph paper gives us a trend line in the form of a smoothed curve by reducing the

fluctuations in her time series.

Period of Moving Average

As stated above, moving averages can be calculated for any period viz: 3-yearly, 5-yearly,

7-yearly, 9-yearly (monthly, weekly) etc. But in the context of a time-series, the purpose of

moving averages is to find the trend values by eliminating or reducing to the minimum

extent all the types of fluctuations, viz : seasonal, cyclical, and irregular. Hence, the period

of moving averages should be such as would fulfil the purpose. As such, in deciding the

period of moving averages, the following guide-lines should be taken into consideration, or

else the whole purpose of finding the moving average will be vitiated.

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