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Business Statistics Homework Help
Business Statistics Assignment Help
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Business Statistics Homework Help | Business Statistics Assignment Help
About Business Statistics: Business statistics is a science that
deals with the process of taking good decisions under the
condition of uncertainty. It is used in many business fields such as
banking, finance, stock market, econometrics, production process,
quality control, and marketing research. It is a well-known fact
that business environment is always more complex as it deals with
money and direct/indirect communication with people. This makes
the process of decision-making more difficult in any type of
Business Statistics Assignment Help. Due to this reason, the
businessman or the decision-maker do not have confidence on his decision that was taken
simply based on his observation and own experience in his business.
Business Statistics Homework Illustrations and Solutions
Illustration 1.
Production figures of a Textile industry are as follows:
Year
Production
(in units)
:
1998
12
1999
10
2000
14
2001
11
2002
13
2003
15
2004
16
For the above data.
(i) Determine the straight line equation by change of the origin under the least square
method.
(ii) Find the trend values, and show the trend line on a graph paper, and
(iii)Estimate the production for 2005 and 2007.
Solution
(i)Determination of the straight line equation by change of the origin under the least square
method.

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Year
T
Prodn.
Y
Successive
values of
time
variable X
XY 𝑿 𝟐
Trend
values
T
1998
1999
2000
2001
2002
2003
2004
12
10
14
11
13
15
16
1
2
3
4
5
6
7
12
20
42
44
65
90
112
1
4
9
16
25
36
49
10.745
11.50
12.25
13.00
13.75
14.50
15.25
Total 𝑌 = 91 𝑋 =28 𝑋𝑌 = 385 𝑋2
= 140 N = 7
Note. The successive values of time variable X, have been taken as a matter of change of
the origin to reduce their magnitude for the sake of convenience in calculations.
Working
The straight line equation is given by Yc = a + bX
Here, since 𝑥 ≠ 0, we are to work out the values of the two constants, a and b by
simultaneous solution of the following two normal equations:
𝑌 = Na + b 𝑋
𝑋𝑌 = a 𝑋 + b 𝑋2
Substituting the respective values obtained from the above table in the above equation we
get,
91 = 7a + 28b
385 = 28a + 140b
Multiplying the eqn. (i) by 4 under the equation (iii) and subtracting the same from the eqn.
(ii) we get,
28a + 140b = 385
=
− 28𝑎+112𝑏=364
28𝑏=21
∴ b =
𝟐𝟏
𝟐𝟖
= .75
Putting the above value of b in the eqn. (i) we get,
7a + 28 (.75) = 91
= 7a = 91 – 21 = 70
∴ a = 70/7 = 10

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Putting the above values of a and b in the relevant equation we get the straight line
equation naturalized as under:
Yc = 10 + 0.75 X
Where, X, represents successive values of the time variable, Y, the annual production and
the year of origin is 1997 the previous most year.
(ii)Calculation of the Trend values & Their Graphic Representation
1998 When X = 1, Yc = 10 + 0.75 (1) = 10.75
1999 When X =2, Yc = 10 + 0.75 (2) = 11.50
2000 When X = 3, Yc = 10 + 0.75 (3) = 12.25
2001 When X =4, Yc = 10 + 0.75 (4) = 13.00
2002 When X = 5, Yc = 10 + 0.75 (5) = 13.75
2003 When X = 6, Yc = 10 + 0.75 (6) = 14.50
2004 When X = 7, Yc = 10 + 0.75 (7)= 15.25

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Illustration 2.
State by using the method of First Differences, if the straight line model is suitable for
finding the trend values of the following time series:
Year :
Sales :
1997
30
1998
50
1999
72
2000
90
2001
107
2002
129
2003
147
2004
170
Solution
Determination of Suitability of the Straight line model by the method of First
Differences
Year
T
Sales
Y
First Differences
1997
1998
1999
2000
2001
2002
2003
2004
30
50
72
90
107
129
147
170
50-30 = 20
72 – 50 = 22
90 – 72 = 18
107 – 90 = 17
129 – 107 = 22
147 – 129 = 18
170 – 147 = 23
From the above table, it must be seen that the first differences in the successive
observations are almost constant by 20 or nearly so. Hence, the straight line model is quite
suitable for representing the trend components of the given series.
Vi. Parabolic Method of the least square
This method of least square is used only when the trend of a series is not linear, but
curvilinear. Under this method, a curve of parabolic type is fitted to the data to obtain their
trend values and to obtain such a curve, an equation of power series is determined in the
following model.
Yc = a + bX + cX2
+ dX3
+ ….+ mXn
It may be noted that the above equation can be carried to any power of X according to the
nature of the series. If the equation is carried only up to the second power of X, (i.e. X2
) it
is called the parabola of second degree, and if it is carried up to the 3rd power of X, (i.e.
X3) it is called the Parabola of 3rd degree. However, in actual practice the parabolic curve of
second degree is obtained in most of the cases to study the non-linear of a time series. For
this, the following equation is used.
Yc = a + bX +cX2
Where, Yc represents the computed trend value of the Y variable ‘a’ the intercept of Y, b’
the slope of the curve at the origin of X and c’, the rate of change in the slope.

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In the above equation, a, b, and c are the three constants, the value of which are
determined by solving simultaneously the following three normal equations:
𝑌 = 𝑁𝑎 + 𝑏 𝑋 + 𝑐 𝑋3
It may be noted here that the first normal equation has been derived by multiplying each
set of the observed relationship by the respective coefficients of a, and getting them all
totaled ; the second normal equation has been derived by multiplying each set of the
observed relationship by the respective coefficients of b, and getting them all totaled ; and
the third equation has been derived by multiplying each set of the observed relationship by
the respective coefficients of c and getting them all totaled.
Further, it may be noted that by taking the time deviations from the midpoint of the time
variable, if 𝑋 and 𝑋3
could be made zero, the above three normal equations can be
reduced to the simplified forms to find the values of the relevant constants as follow :
From the above, it must be noticed that the value of b can be directly obtained as b =
𝑋𝑌
𝑋2,
and the values of the other two constants a and c can be obtained by solving simultaneously
the rst of the following two normal equations:

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a =
𝑌− 𝑐 𝑋2
𝑋4
c =
𝑋2 𝑌−𝑎 𝑋2
𝑋4
Once, the values of the three constants a, b and c are determined in the above manner, the
trend line equation can be fitted to obtain the trend values of the given time series by
simply substituting the respective values of X therein.
Graphic Representation of the Trend Line & the Original Data
(ii) Estimation the Production Figure for 2004 and 2006.
Since the last successive value of X for 2004 is 7, the successive values of X for 2005 and
2007 are 8 and 10 respectively.
Thus, for 2005, when X = 8, Yc = 10 + 0.75 (8) = 16.00
And for 2007, when X = 10, Yc = 10 + 0.75 (10) = 17.50
Hence, the estimate figures of production for 2005 and 2007 are 16000 units and 17500
units respectively.
Test of Suitability of Straight line Method
If the differences between the successive observations of a series are found to be constant,
or nearly so, the straight line model is considered to be a suitable measure for
representation of trend components, otherwise not. This fact can be determined by the
method of First Differences illustrated as under:

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