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Statistics assignment Sample Questions and Answers:
Question 1. Fit a parabolic equation of second degree to the following data, and obtain the
trend values including that of 2005. Also, represent the original and trend values graphically
Year :
Year in
Value :
1998
95
1999
160
200
255
2001
380
2002
535
2003
720
2004
935
(a) Solution. Determination of the Parabolic equation of second degree
Year
T
Value
Y
Time
dvn.
X
XY X2
X2
Y X3
X4
1998
1999
2000
2001
2002
2003
2004
95
160
255
380
535
720
935
-3
-2
-1
0
1
2
3
-285
-320
-255
0
535
1440
2805
9
4
1
0
1
4
9
855
640
255
0
535
2880
8415
-27
-8
-1
0
1
8
27
81
16
1
0
1
16
81
Total 𝑌 =
3080
𝑋 =0 𝑋𝑌 =
3920
𝑋2
= 28 𝑋2
Y =
13580
𝑋3
= 0 𝑋4
=
196
Working
The parabolic equation of second degree is given by Yc = a + bX + cX2
Since from the above table, the values of 𝑋 and 𝑋3
appear to be zero, the values of a, b,
and c will be determined by the reduced normal equations as follows:
a =
𝒀−𝒄 𝑿 𝟐
𝑵
=
𝟑𝟎𝟖𝟎−𝟐𝟖𝒄
𝟕
= 440 – 4c
= a + 4c = 440
b =
𝑿𝒀
𝑿 𝟐 =
𝟑𝟗𝟐𝟎
𝟐𝟖
= 140,

and
c =
𝑿 𝟐 𝒀−𝒂 𝑿 𝟐
𝑿 𝟒 =
𝟏𝟑𝟓𝟖𝟎−𝟐𝟖𝒂
𝟏𝟗𝟔
=
𝟒𝟖𝟓−𝒂
𝟕
= a + 7c = 485
Subtracting the eqn (i) from the eqn (iii) we get,
a + 7c = 485
(-) a + 4c = 440
3c = 45
∴ c = 15
Putting the value of c in the equation (i) we get,
a + 4(15) = 440
= a = 440 – 60 = 380
Thus, a = 380, b – 140, and c = 15
Putting the above values of the constants a, b, and c in the equation, we get the parabolic
equation of the second degree fitted as under:
Yc = 380 + 140X + 15X2
Where, X unit = time deviation, Y unit = annual value, and the trend origin is 2001.
Using the above parabola we obtain the trend values as under:

(a)Computation of the Trend values including that of 2004:
For 1998 when X = -3, Yc = 380 + 140 (-3) + 15 (-3)2 = 95
For 1999 when X = -2, Yc = 380 + 140 (-3) + 15 (-2)2 = 160
For 2000 when X = -1, Yc = 380 + 140 (-1) + 15 (-1)2 = 255
For 2001 when X = 0, Yc = 380 + 140 (0) + 15 (0)2 = 380
For 2002 when X = 1, Yc = 380 + 140 (1) + 15 (1)2 =535
For 2003 when X = 2, Yc = 380 + 140 (2) + 15 (2)2 = 720
For 2004 when X = 3, Yc = 380 + 140 (4) + 15 (4)2 =935
For 2005 when X = 4, Yc = 380 + 140 (5) + 15 (5)2 = 1180
Note. From the above trend values it must be marked that they are very much the same as
those of the observed values. This implies that there is no other fluctuation except the trend
in the given series.
(e) Graphic representation of the Parabolic Curve of the second degree.

Vii. Geometric, or Logarithmic Method of the Least square
Under this method, the trend equation is obtained Yc = aXb
Using the logarithms, the above equation is modified as under :
Log Yc = Log a + b log X
The above geometric curve equation should not, however, be used unless there is a clear
geometric progression in the value variable of a time series. Further, while using the
logarithm of X, the X origin cannot be taken at the middle of the period. This limitation is
overcome by the exponential trend fitting discussed below. The above trend equation can
also be put in the following modified form :
Yc = aXb
+ k
Using the logarithms, the modified form of the above can be presented as under :
Yc = {Anti log of (log a + b log X)} + k
Where, k is a constant
However, this method is rarely used in practice.
Vii. Exponential Method of the Least Square.
This method of trend fitting is resorted to only when the value variable Y shows a geometric
progression viz : 1,2,4,8,16,32, and so on, and the time variable (t) shows an arithmetic
progression viz : 1,2,3,4,5,6 and the like In such cases, the trend line is to be drawn on a
semi-logarithmic chart in the form of a straight line, or a non-linear curve to show the
increase, or decrease of the value of variable Y at a constant rate rather than a constant
amount. When the trend takes the form of a non-linear curve on a semi-logarithmic chart,
an upward curve indicates the increase at varying rates depending upon the shapes of the
slopes. The steeper the slope, the higher is the rate of increase.
However, under this method, the trend line is fitted by the following model:
Yc = abX
Using logarithmic operation, the above equation is modified as under :
Yc = A.L. (log a + X log b)
In the above equation, a and b are the two constants the values of which are determined by
solving the following two normal equations and finding the antilogarithms thereof:
log 𝑦 = N log a + log b 𝑋
𝑋 log 𝑦 = log a 𝑋 + log b 𝑋2
If by taking the time deviations X from the mid-point of the time variable t, 𝑋 could be
made zero, the logarithm of the two constants a and b can be determined directly as under:
Log a =
log 𝑌
𝑋2

Log b =
Xlog 𝑌
𝑋2
After obtaining the values of a and b in the above manner, and substituting their values in
the equation Yc = abX, we can fit the trend line equation under this method, and with such
an equation we can very well estimate the trend values of the time series, and predict the
value for any past and future year as well.
Question 2. Using the exponential method of the least square, fit a trend line equation for
the following data, and obtain the trend values for the series. Also, plot the trend lien on a
graph paper and estimate the value for the year 2004.
Year (t) :
Value (Y)
2000
4
2001
13
2002
40
2003
125
2004
380
(a)Solution. Determination of the Trend line by the Exponential Method of the Least
square.
Year t
(1)
Value Y
(2)
Time dvn
T – 2001
X
(3)
Log y
(4)
X log Y
(5)
𝑿 𝟐
(6)
Trend
Value
(7)
1999
2000
2001
2002
2003
4
13
40
125
380
-2
-1
0
1
2
0.6021
1.1139
1.6021
2.0969
2.5798
-1.2042
-1.1139
0.0000
2.0969
5.1596
4
1
0
1
4
4
13
40
124
386

Working
The exponential trend line is given by
Yc = abX
= AL (log a + X log b)
Since 𝑋 = 0, the logarithm of a and b the two constants in the above equation are directly
computed as under:
Log a =
𝑙𝑜𝑔𝑌
𝑁
=
7.9948
5
= 1.5990 app.
And log b =
𝑋𝑙𝑜𝑔𝑌
𝑋2 =
4.9384
10
= 0.4938 app.
Putting the above logs of a and b in the equation, we get the exponential trend lien titled as
under. :
Yc = AL ( 1.5990 + 0.4938X)
Where, X = time deviation, Y = annual value and 2002 is the trend origin i.e. the origin of
X. With the above equation, the trend values are computed as under:
(b)Computation of the Trend Values (as shown in the column 7 of the Table)
For 2000 when X = -2, Yc = AL [1.5990 + 0.4938 (-2)] = 4.087 = 4
For2001 when X = -1, Yc = AL [1.5990 + 0.4938 (-1)] = 12.750 = 13
For 2002 when X = 0, AL [1.5990 + 0.4938 (0)] = 39.72 = 40
For 2003 when X = 1, AL [1.5990 + 0.4938 (1)] = 123.8 = 124
For 2004 when X = 2, Yc = AL [1.5990 + 0.4938 (2)] 386.0 = 386
From the above results, it must be seen that the trend values almost approach the observed
values. The slight difference that appear between some of them are due to the error of
approximation involved in the logarithms.

(b)Graphic representation of the Exponential Trend line

(c)Forecasting the value for 2005
For 2005, X = 3
Thus, when X = 3, Yc = AL [1.5990 + 0.4938(3)] = 1203
Ix. Growth Curve Method of the Least Square.
The growth curves are some special type of curves which are plotted on graph papers for
analysis and estimation the trend values in the business and economic phenomena. Where
initially, the growth rate is very slow but gradually it picks up at a faster rate till it reaches a
point of stagnation, or saturation. Such situations are quite common in business fields
where new products are introduced for marketing. In such a situation, when a new product
say the book, “Statistical Methods” is introduced into the market, the growth rate of its sale
quite slow, and if the product proves to be worthwhile, the growth rate of its sale gradually
goes up fast and reaches relatively a higher level, and then it begins to decline. As such, a
curve representing this type of phenomena continues to grow more and more slowly
approaching the upper limit, but never reaching the same. It takes the shape of an
elongated f which indicates the pattern of growth in terms of the actual amount as small in
the early years increasingly greater in the middle years, and large but stabilized in the later
years. But when such curves are plotted on a semi logarithmic chart, they show a growth at
rapidly increasing rates in the earlier periods, and at a declining ratio in the later periods of
the series.
There are different types of growth curves used in different fields of business. And
economics, but the most popular among them are the following twos which have come into
being since 1920 :
1. Geometric Growth Curve.
2. Logistic or Pearl-Read Growth Curve.
A brief introduction to these curve is given as belows:
1. Gompertz Growth Curve. The fundamental trend equation of this curve is given by Yc
= kabx
When put to the logarithmic form, the above equation is modified as under:
Ye = AL of [log K + bX lag a]
Where, k represents the constant of the highest point, a the intercept of y i.e. the trend
value of the origin of X, and b the slope of the line i.e. the rate of growth.
2. Logistic, or Pearl-Read Growth Curve.
This curve is given by the following model
𝟏
𝒀 𝒄 = K + abX
The above equation may also be used in the alternative form as under :
Yc =
𝑲
𝟏+ 𝟏𝟎 𝒂+𝒃𝑿 ,or
𝑲
𝟏+ 𝒆 𝒂+𝒃𝑿

It may be noted that the Logistic curve is nothing but a modified exponential curve in terms
of the reciprocal of the Y variable. This curve, however, is very popular in the demographic
studies and in many business and economic analysis.
Merits and Demerits of the Method of the least square.
Like any other statistical methods, the method of the least square explained thus, has a
good number of merits and demerits. These are enumerated as under:
(i) This method is completely free from personal bias of the analyst as it is very
objective in nature. Anybody using this method is bound to fit the same type of
straight line, and find the same trend values for the series.
(ii) Unlike the moving average method, under this method, we are able to find the
trend values for the entire time series without any exception for the extreme
periods of the series even.
(iii) Unlike the moving average method, under this method, it is quite possible to
forecast any past or future values perfectly, since the method provides us with a
functional relationship between two variables in the form of a trend line equation,
viz. Yc = a + bX + c𝑋2
+ … or Yc = abx
etc.
(iv) This method provides us with a rate of growth per period i.e. b, as shown in the
equations cited above. With this rate of growth, we can very well determine the
value for any past, or previous year by the process of successive addition, or
deduction from the trend value of the origin of X.
(v) This method providers us with the line of the best fit from which the sum of the
positive and negative deviation is zero, and the sum of the squares (i) (𝑌 − 𝑌𝑐)2
= The least value. Function to a given set of observations.
(vi) This method is the most popular, and widely used for fitting mathematical
function to a given set of observations.
(vii) This method is very flexible in the sense that it allows for shifting the trend origin
from one point of time to another, and for the conversion of the annual trend
equation into monthly, or quarterly trend equation, and vice versa.
Demerits
(i) This method is very much rigid in the sense that if any item is added to, or
subtracted from the series, it will need a through revision of the trend equation to
fit a trend line, and find the trend values thereby.
(ii) In comparison to the other methods of trend determination, the method is bit
complicated in as much as it involves many mathematical tabulations,
computations, and solutions like those of simulation equations.
(iii) Under this method, we forecast the past and future values basing upon the trend
values only, and we do not take note of the seasonal, cyclical and irregular
components of the series for the purpose.
(iv) This method is not suitable for business, and economic data which conform to the
growth curves like Geometric cur5ve, Logistic Pearl-Read curve etc.
(v) It needs great care for the determination of the type of the trend curve to be
fitted in viz: linear, parabolic, exponential, or any other more complicated curve.
An erratic selection of the type of curve may lead to fallacious conclusions.
(vi) This method is quite inappropriate for both very short and very long series. It is
also unsuitable for a series in which the differences between the successive
observations are not found to be constant, or nearly so.

Shifting of a Trend Origin and Conversion of the Trend Equation
Shifting of a Trend Origin.
By shifting of a trend origin we mean changing the origin of X (the time from which time
deviations are taken) from one point of time to another point of time, whether earlier or
later to the present origin. This becomes necessary at times to facilitate comparison in
the trend values. To give effect to such shifting, the trend equation that is already fitted
is slightly modified by adding to, or subtracting from X, the time difference between the
existing origin and the proposed origin. In case, the shifting is made to a later period,
the difference in time is added to X, and in case, it is made to an earlier period, the
difference in time is subtracted from X. Thus the fitted trend equation is modified as
under:
Yc = a + b (X ± K).
Where, K represents the time difference in shifting.
By such modifications in a linear equation, the value of a (i.e. the value of the trend
origin) only, is affected and the value of b (i.e. the slope of the change) remains as it is.
But in a parabolic equation of second degree, such modification will affect the value of
both a and b, but not the value of c (i.e. the rate of change in slope).

Question 3. 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
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.

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