**Answer of Question no. 1 (a).**

**Dispersion and its characterictics**

**Dispersion:**Dispersion is the measure of variation of items. It measures the extent to which the items vary from central value. Dispersion is also known as average of the second order.

In the words
of Brooks and Dick,” Dispersion is the degree of the scatter or variation of
the variable about a central value.”

In the words
Simpson and Kafka,” The measurement of the Scatterness of the mass of figures
in a series about an average is called measure of variation or dispersion.”

Dispersion
includes range, mean deviation, quartile deviation, and standard deviation.

*The following are the important properties which a good measure of dispersion should satisfy:***a)**

**It should be simple to understand and easy to compute.**

**b)**

**It should be simple to compute.**

**c)**

**It should be based on all the items.**

**d)**

**It should not be affected by extreme values.**

**f)**

**It should be capable of further algebraic treatment.**

**g)**

**It should have sampling stability.**

**Answer of Question no. 1 (d).**

**Measure of Central tendency and Its Properties**

**In the words of Croxton and Cowden, “An average value is a single value within the range of the data that is used to represent all the values in the series.”**

**The value of average lies between the maximum and minimum values of the series. That is why it is also called measure of central tendency.**

*The following are the important properties which a good average should satisfy***a)**

**It should be easy to understand.**

**b)**

**It should be simple to compute.**

**c)**

**It should be based on all the items.**

**d)**

**It should not be affected by extreme values.**

**e)**

**It should be rigidly defined.**

**f)**

**It should be capable of further algebraic treatment.**

**Sample survey and its advantages**

**Sample:**If a part is selected out of the universe then the selected part or portion is known as sample. Sample is only a part of the universe.

**Sample survey:**It is a survey under which only a part taken out of the universe is investigated. It is not essential to investigate every individual item of the Universe.

*Advantages of Sample surevey*
a)
It is cheaper as compared to census survey

b) While using secondary data, time and labour are saved.

c) It may also be collected from unpublished form.

d)
If secondary Data are available, they are much
quicker to obtain than primary data.

**Answer of Question no.2(a).**

**Regression line**

A line
of regression by the method of “least square” shows an average relationship
between variables under study. This regression line can be drawn graphically or
derived algebraically. A line fitted by method of least square is known as the
line of best fit. There are two regression lines:-

Regression
line of x on y: - Regression line of x on y is used to predict x for a given
value of y. The regression equation of x on y is x=a+by.

Regression
line of y on x: - Regression line of y on x is used to predict y for a given
value of x. The regression equation of y on x is y=a+bx

*Two regression lines*

*:*
There are two lines
of regression: - x on y and y on x. For these lines, the sum of the square of the
deviations between the given values and their corresponding estimated values
obtained from the line is least as compared to other line. One regression line
cannot minimise the sum of squares for both the variables that is why we are
getting two regression lines. (We
get one regression line when r = +1 and Two regression lines will be at right
angles when r = 0.)

**Answer of Question no. 2(c)**

**Coefficient of Correlation**

To measure the degree of
association or relationship between two variables quantitatively, an index of
relationship is used and is termed as co-efficient of correlation.

Co-efficient
of correlation is a single number that tells us to what extent the two
variables are related and to what extent the variations in one variable changes
with the variations in the
other.

The co-efficient of correlation
is always symbolized either by r or p (Rho). The notion 'r' is known as product
moment correlation co-efficient or Karl Pearson's Coefficient of Correlation.
The symbdl 'P' (Rho) is known as Rank Difference Correlation
Coefficient or Spearman's Rank Correlation Coefficient.

*Range of Coefficient of Correlation*
The measurement of correlation
between two variables results in a maximum value that ranges from -1 to +I, through
zero.

**Answer of Question no. 3(a)**

**Index number**is an indicator of changes in prices and quantities. It is a specialized average designed to measure the change in a group of related variables over a period of time. It is also an indicator of inflationary or deflationary tendencies.

*Uses of Index number in Trade and commerce*
a)
Measurement
of change in the price level or the value of money: - Index number can be used
to know the impact of the change in the value of money on different sections of
the society.

b)
Knowledge of
the change in standard of living: - Index number helps to ascertain the living
standards of people. Money income may increase but if index number show a
decrease in the value if money. Living standard may even decline.

c)
Adjustment
in salaries and allowances: - Cost of living index number is a useful guide to
the government and private enterprises to make necessary adjustment in salaries
and allowances of the workers.

d)
Useful to
business community: - Price index numbers serve as a useful guide to the
business community in planning.

e)
Information
regarding foreign trade: - Index of exports and imports provides useful
information regarding foreign trade.

**Answer of Question no. 3(c)**

**Fisher’s index is regarded as ideal index because**:-

a)
It considers both base year and
current year’s price and quantity.

b)
It satisfies both time reversal and
factor reversal test.

c)
It is based on Geometric mean which is
theoretically considered to be the best average of constructing index number.

d)
It is free from bias as it considers
both current year and base year price and qty.

**Answer of Question no. 3(d)**

**Difference between Price index and Quantity index**

A measure
reflecting the average of the proportionate changes in the prices of a
specified set of goods and services between two periods of time. Usually a
price index is assigned a value of 100 in some selected base period and the values
of the index for other periods are intended to indicate the average percentage
change in prices compared with the base period. A quantity index is built up
from information on prices of various commodities.

A measure
reflecting the average of the proportionate changes in the quantities of a
specified set of goods and services between two periods of time. Usually a
quantity index is assigned a value of 100 in some selected base period and the
values of the index for other periods are intended to indicate the average
percentage change in quantities compared with the base period. A quantity index
is built up from information on quantities such as the number or total weight
of goods or the number of services.

**Limitation of index number**

a)
Not completely true: - Index number
not fully true. The index number simply indicate arithmetical tendency of the
temporal changes in the variable.

b)
International comparison is not
possible: - Different countries have different bass of index numbers; these do
not help international comparisons.

c)
Difference of time: - With the passage
of time, it is difficult to make comparison of index number. With the changing
time man’s habits.

d)
Limited use: - Index numbers are
prepared with certain specific objective. If they are used for another purpose
they may lead to wrong conclusion.

e)
Lack of retail price index number: -
Most of the index numbers are prepared on the basis of wholesaler prices. But
in real life, retail prices are most relevant, but it is difficult to collect
retail prices.

**Answer of Question no. 4(b)**

**Components of time series**

1.
Secular trend: - Secular trend is a long term trend which has the basic
tendency to grow or decline over a period of time. It may be due to population
change technological progress, large scale shifts in consumer tastes, discovery
of new things, etc.

2.
Seasonal variation: - Seasonal variations are those periodic movements in
business activity, which occur regularly every year and have their origin in
the nature of the year itself. It may be due to climate weather conditions,
customs, traditions and habits, festivals, etc.

3.
Cyclical variation: - The term cycle refers to the recurrent variations in time
series that usually last longer than a year and are regular neither in
amplitude nor in length. Cyclical fluctuations are long-term movements that
represent consistently recurring rises and declines in activity. It has four
important characteristics:

i)
Prosperity

ii)
Decline

iii)
Depression

iv)
Improvement

4. Irregular variation or irratic movement: - It is the variation in business activities, which do not repeat in a definite pattern. Floods, earthquakes, strikes and wars cause it.

*M*

*odels of time series*
In Traditional time series
analysis, it is ordinarily assumed that there is a multiplicative relationship
between the components of time series.

Symbolically, Y=T X S X C X I

Where T= Trend

S= Seasonal component

C= Cyclical component

I= Irregular
component

Y= Result of four
components.

Another approach is to
treat each observation of a time series as the sum of these four components Symbolically,
Y=T + S+ C + I

**Answer of Question no. 5(a).**

**Addictive Law of Probability**

Determining the
probability of two events occurring, use the additive law of probability. For
example if asked "What is the probability of drawing a 4 or a 7 from a
shuffled deck of cards?", Eight different cards would satisfy this
condition, 4 (4s) or 4 (7s). So the probability of this event is 8/52. The
equation below properly states the additive law of probability. p(A or
B)= p(A) + p(B) - p(AB)

**Multiplicative Law of Probability**

The multiplicative
law of probability is used to calculate the probability that two events will
occur in sequence. Remember that the additive law was used to find the
probability that one of two different events would occur on a single trial.
These two probabilities have key words to look for. In the additive law, you
will always see something like the following: Find the probability that event
(A) or event (B) will occur. The word or is key. With the multiplicative law,
the word and is key. Here is a typical problem requiring the multiplicative
law: What is the probability of drawing a 4 and then a 7 from a shuffled deck
of cards.

The
multiplicative law states that the probability of the occurrence of event A and
then event B, and then event C ... and so on is the product of the separate
simple probabilities of those events. The following equation expresses the multiplicative
law of probability's formula.

P(A and B and C)
= P(A). P(B). P(C)

**Answer of Question no. 5(b).**

**Methods of Business Forecasting**: The following are the main methods of business forecasting.

*1.*

*Business barometers*

*2.*

*Time series analysis*

*3.*

*Extrapolation*

*4.*

*Regression analysis*

*5.*

*Modern econometric methods*

*6.*

*Exponential smoothing method*

*Business barometers:***Business indices are constructed to study and analyse the business activities on the basis of which future conditions are predetermined. As business indices are the indicators of future conditions, so they are also known as “business barometers” or ‘economic barometers’. With the help of these business barometers the trend of fluctuations in business conditions are made known and by forecasting a decision can be taken relating to the problem.**

The
construction of business barometer consists of gross national product,
wholesale prices, consumer prices, industrial production, stock prices, and
bank deposits. These quantities may be converted into relatives on a certain
base. The relatives so obtained may be weighted and their average is computed.
The index thus arrived at in the business barometer.

*Time series analysis:***Time series analysis is also used for the purpose of making business forecasting. The forecasting through time series analysis is possible only when the business data of various years are available which reflects a definite trend and seasonal variation. By time series analysis the long term trend, secular trend, seasonal and cyclical variations are ascertained, analysed and separated from the data of various years.**

*Extrapolation:***Extrapolation is the simplest method of business forecasting. By extrapolation, a businessman finds out the possible trend of demand of his goods and also about the future price trends. The accuracy of extrapolation depends on two factors:**

i)
Knowledge about the fluctuations of the figures

ii)
Knowledge about the course of events relating to the problem under
consideration

Thus,
there are two assumptions on which extrapolations are based:

i)
There is no sudden jump in figures from one period to another

ii)
There is regularity in fluctuations and the rise and fall is uniform

In
extrapolation, we assume that the variable will follow the established pattern
of growth. For the purpose of business forecasting, it is to determine
accurately the appropriate trend curve and the values of its parameters.

**Answer of Question no. 5(c)**

*Objectives of forecasting in business:*

**(a) Future Prediction:**Forecasting is a part of human conduct. Businessmen also need to look to the future. Success in business depends on correct predictions. In fact when a man enters business, he automatically takes with it the responsibility for attempting to forecast the future.

To a
very large extent, his success or failure would depend upon the ability to
successfully forecast the future course of events. Without some element of
continuity between past, present and future, there would be little possibility
of successful prediction. But history is not likely to repeat itself and we
would hardly expect economic conditions next year or over the next ten years to
follow a clear cut prediction. Yet, frequently past patterns prevail
sufficiently to justify using the past as a basis for predicting the future.

(b)

**Reducing Uncertainty:**A businessman cannot afford to base his decisions on guesses. Forecasting helps a businessman in reducing the areas of uncertainty that surround management decision making with respect to costs, sales, production, profits, capital investment, pricing, expansion of production, extension of credit, development of markets, increase of inventories and curtailment of loans. These decisions cannot be made off-hand. They are to be based on present indications of future conditions.
(c)

**Measurement of Risk:**However, we should know that it is impossible to forecast the future precisely. There is a possibility of occurrence of some range of error in the forecast. Statistical forecasts are the methods in which we can use the mathematical theory of probability to measure the risks of errors in predictions.**Answer of Question no. 5(d)**

**Sales Forecasting**

In general terms, forecasting means “A statement made
about the future”. So, Sales forecasting is the estimation of sales made for
the future. Sales forecast is an estimate of sales in rupees or in units for
future period. A sales forecast is the prediction of sales volume that a
company can estimate to achieve in specified period of time in future.

According to American marketing Association, “Sales
forecast is an estimate of sales in dollars (Rupees in India) or physical units
for a specified future period”.

Sales forecasting is the basis of all business
activities. All the business activities may it a sales related matter,
production related matter, finance, advertising, etc depend on sales
forecasting. Any business firm starts its plan with sales forecasting .Sales
forecasting is a self-assessment tool for a company. The future direction of the
company depends on sales forecasting.
The importance of Sales forecasting can be stated as follows:

a)
Overstocking and under stocking of materials can be
maintained by a good inventory control.

b)
With the help of sales forecasting, sales opportunities
can be found out on the basis of the forecast.

c)
All the activities in an organization are controlled on
the basis of forecasting.

d)
Advertising and sales promotion expenses are based on
sales forecasting.

e)
Sales forecasting is also important in the field of
personnel department. The number of sales persons, executives etc can be
increased or decreased on the basis of sales forecasting.

**Demand Forecasting**

**Forecasting**is the process to predict how the future needs, which include the requirement in the size of the quantity, quality, time and location that is required in order to meet the demand for goods or services.

**Demand Forecasting**is the demand for products that are expected to be realized for a certain period in the future. Inforecasting we must pay attention to forecasting procedures should be implemented, namely :

a) Determining the purpose of
forecasting.

b)
Choose the
item “independent demand” to be predicted. Plot data intoscatter diagram.

c)
Selecting
“forecasting method” in accordance with the pattern of data for its intended
purpose.

d)
Counting
errors are to be performance of each method used, can be known.

e)
The
selection of the best method, which has the smallest error rate.

f)
Make
predictions of future demand, then perform the test verifies that the
forecasting results carried out a representative to the past data