Business Statistics Homework Help

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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 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 = 91 =28 = 385 2 = 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 0, we are to work out the values of the two constants, a and b by

simultaneous solution of the following two normal equations:

= Na + b

= a + b 2

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

= 28+112=364

28=21

b =

= .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:

= + + 3

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 and 3 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 =

2,

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 = 2

4

c = 2 2

4

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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