Below are given the figures of production (in thousand tons) of a sugar factory: Year 1999 2000 2001 2002 2003 2004 2005 Production 77 88 94 85 91 98 90 a) Fit a straight line trend by the method of ‘least squares’ and show the trend values. b) What is the monthly increase in production?
Answers
Answer:
This method gives the line which is the line of best fit. This method is applicable to give results either to fit a straight line trend or a parabolic trend.
The method of least squares as studied in time series analysis is used to find the trend line of best fit to a time series data.
Secular Trend Line
The secular trend line (Y) is defined by the following equation:
Y = a + b X
Where, Y = predicted value of the dependent variable
a = Y-axis intercept i.e. the height of the line above origin (when X = 0, Y = a)
b = slope of the line (the rate of change in Y for a given change in X)
When b is positive the slope is upwards, when b is negative, the slope is downwards
X = independent variable (in this case it is time)
To estimate the constants a and b, the following two equations have to be solved simultaneously:
ΣY = na + b ΣX
ΣXY = aΣX + bΣX2
To simplify the calculations, if the midpoint of the time series is taken as origin, then the negative values in the first half of the series balance out the positive values in the second half so that ΣX = 0. In this case, the above two normal equations will be as follows:
ΣY = na
ΣXY = bΣX2
The Given values are arranged in the table.
The straight-line equation is given as:
Y = a+ bX, ................(1)
Where a is the intercept at Y-axis i.e. when X=0, Y=a
and b is the slope of the line and X is the Independent Variable.
as there are 7 number of data points hence n = 7.
According to least square fitting method, to estimate the value of a and b the following equations are used:
∑Y = na + b∑X .......... (2)
∑XY = a ∑X + b ∑X² ...........(3)
The summation values are already calculated on the table:
∑Y= 623
∑X= 0
∑XY= 56
∑X²= 28
Putting the values in equations 2 and 3:
623 = 7a
56 = 28b
Solving the equations,
a = 623/7
a=89
and,
b=56/28
b=2
putting the values of a and b in the straight line equation:
Straight-line trend: Y = 89 + 2X.............(4)
To calculate the trend values, put the different values of X in equation 4
We get
Y1= 89 + (2*-3) = 83
Y2= 89 + (2*-2) = 85
Y3= 89 + (2*-1) = 87
Y4= 89 + (2*1) = 91
Y5= 89 + (2*2) = 93
Y1= 89 + (2*3) = 95
Hence, the trend values are 83, 85, 87, 91, 93, 95
On comparing equation 4 with equation 1 we get that 2(in thousand tons) is the slope of the equation and it shows the increase in production yearly.
For monthly increase in production:
2000/12 tons
= 166.6
Hence 166.6 Tons of increase in production every month
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