Business Statistics Solved Question Paper 2021
[Dibrugarh University BCOM 3rd SEM CBCS Pattern]
2021 (Nov/Dec)
COMMERCE (Generic Elective)
Paper: GE – 303 (Business Statistics)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for
the questions
1. Answer any eight questions of the following: 2 x 8=16
a) Define
crosssectional data. Give an example.
Ans: Crosssectional data refer
to observations of many different individuals (subjects, objects) at a
given time, each observation belonging to a different individual. A simple
example of crosssectional data is the gross annual income for each of 500
randomly chosen households in Dibrugarh City for the year 2023.
b) If the
geometric mean of x, 4, 8 is 6; then find the value of x.
Ans:
c) What
are the limitations of the classical approach to probability?
Ans: Limitations of classical approach:
a) It is applicable only when the
total numbers of events are finite.
b) It is applicable only when all the
events are equally likely.
d) Define
equally likely events with an example.
Ans: Equally
Likely Events: Events are said to be equally likely when no one events can
be expected to occur in preference to the other events. For example, in tossing
a coin head and tail are equally likely events. All the cards of a well
shuffled pack of cards are equally likely when one card is drawn. Here, we can
also say that probability of event is same in case of equally likely items.
e) What do
you mean by regression analysis?
Ans: Regression
is the measure of the average relationship between two or more variable in
terms of the original units of the data. It is a statistical tool with the help
of which the unknown values of one variable can be estimated from known values
of another variable.
f) Why are
index numbers known as economic barometer?
Ans: Index
numbers are highly valuable in business and economics. They provide a good
basis for comparison as they are expressed in abstract units of measurement.
Some of the Use of Index number is listed below:
1. 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.
2. 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.
g) Define
price index number and quantity index number.
Ans: Price
Index: Price index
is 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.
Quantity Index: Quantity index is 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.
h) What
are the components of a time series?
Ans: The
four components of time series are: (FACTORS RESPONSIBLE FOR TREND IN TIMES
SERIES)
1.
Secular trend
2.
Seasonal variation
3.
Cyclical variation
4.
Irregular variation
i) Calculate
the range and its coefficient from the following data: 12, 8, 9, 10, 4, 14, 15.
Ans: Range = H – L = 15 – 4 = 11, Coefficient of range = (HL)/(H+L)
= (154)/(15+4) = 11/19
j) Give
the definitions of parameters and statistics.
Ans: Parameters are numbers that
describe the properties of entire populations. Statistics are numbers that
describe the properties of samples.
Statistics is the study and
manipulation of data, including ways to gather, review, analyze, and draw
conclusions from data.
k) Mention
the methods of nonrandom sampling.
Ans: Nonrandom sampling methods include purposive sampling,
convenience sampling, and stratified sampling.
2. (a) (1) What are the requisites of a good
average? 3
Ans: The
following are the important properties which a good average should satisfy
1.
It should be easy
to understand.
2.
It should be
simple to compute.
3.
It should be
based on all the items.
4.
It should not be
affected by extreme values.
5.
It should be
rigidly defined.
6.
It should be
capable of further algebraic treatment.
(2) In a factory employing 3,000 persons, 5
percent earn less than Rs. 150 per day, 580 earn from Rs. 151 to Rs. 200 per
day, 30 percent earn from Rs. 201 to Rs. 250 per day, 500 earn from Rs. 251 to
Rs. 300 per day, 20 percent earn from
Rs. 301 to Rs. 350 per day and the rest earn Rs. 351 or more per day.
Find the median wage of the employees in that factory. 4
Ans:
(3) Define skewness. 2
Ans: The
term ‘SKEWNESS’ refers to lack of symmetry, i.e., when a distribution is not
symmetrical (or is asymmetrical) it is called a skewed distribution. Any
measure of skewness indicates the difference between the manners in which items
are distributed in a particular distribution compared with a symmetrical (or
normal) distribution. If, for example, skewness is positive, the frequencies in
the distribution are spread out over a greater range of values on the
highvalue end of the curve (the righthand side) than they are on the low
value end. If the curve is normal spread will be the same on both sides of the
centre point and the mean, median and mode will all have the same value.
Or
(b) (1) Find the geometric mean of two numbers
if their arithmetic mean is 15 and the harmonic mean is 9.6. 1
Ans:
(2) Find
the standard deviation from the following frequency distribution: 5
Weight: 
44 – 46 
46 – 48 
48 – 50 
50 – 52 
52 – 54 
Frequency:

3 
24 
27 
21 
5 
Ans:
(3) Which is the best measure of dispersion?
Explain why. 1+2=3
Ans: There
are various advantages of Standard deviation due to which SD is regarded as the
best measure of dispersion. Some of the advantages of standard deviation are:
a)
It is based on
each and every item of the data and it is rigidly defined.
b) It is capable of
further algebraic treatment. Combined SD of two or more groups can be
calculated.
c) It is less
affected by fluctuations of sampling than most other measures of dispersion.
d) For comparing
variability of two or more series, coefficient of variation is considered as
most appropriate and this is based on SD and Mean.
3. (a) (1) Define event. 1
Ans: Events/Outcomes: Results of
a random experiment are called outcomes. Possible outcomes of a random
experiment are also called events.
(2) Can two events be mutually exclusive and
independent simultaneously? Support your answer with an example. 1+2
Ans:
(3) Find the probability that a leap year
selected at random will contain 53 Sundays. 3
Ans:
(4) Discuss the importance of probability
theory in business decision making. 4
Ans: Why do we use probability? – Applications
and importance of Probability
Initially the applications of probability
theories were restricted to games of chances. But the passage of time they are
used in taking various important business decisions. Also probability theory is
being applied in the solution of social, economic, political and business
problems. The insurance industry required precise knowledge about the risk of
loss in order to calculate premium. In fact, probability has become an
important part of our everyday lives. In personal and management decisions, we
face uncertainty and use probability theory. It is not possible to forecast the
future with 100 percent certainty in any decision problem. The probability
theory provides a tool to cope up with uncertainty.
(5) What are the assumptions or conditions for
binomial distribution? 2
Ans:
Assumptions of binomial distribution are:
a) An
experiment is performed under the same conditions for a fixed number or trials,
say, n.
b) In each
trial, there are only two possible outcomes of the experiment.
c) The
probability of a success denoted by p remains constant from trial to trial. The
probability of a failure denoted by q is equal to (1 – p). If the probability
of success is not the same in each trial, we will not have binomial
distribution.
d) The
trials are statistically independent, i.e., the outcomes of any trial or
sequence of trials do not affect the outcomes of subsequent trials.
Or
(b) (1) A bag contains 6 red and 8 green
balls. If two balls are drawn at random, then what is the probability that one
is red and the other is green? 3
Ans:
(2) State the Bayes’ theorem. 2
Ans: Bayes’ theorem also known as Bayes’ rule
was discovered in the year 1763 by Sir Thomas Bayes. This theorem is used to
determine the conditional probability of events. The concept of conditional
probability takes into account information about the occurrence of one event to
predict the probability of another event. Bayes’ theorem is simply an extension
of conditional probability. Bayes’ theorem is a revised probabilities based on
new information and to determine the probability that a particular effect was
due to a specific cause.
(3) Ten coins are tossed simultaneously. Find
the probability of getting at least seven heads. 6
Ans:
(4) Under what conditions normal distribution
is regarded as the limiting form of binomial distribution? 2
Ans: The
normal distribution is an approximation to binomial distribution. Whether or
not p is equal to q, the binomial distribution tends to the form of the
continuous curve and when n becomes large at least for the material part of the
range. As a matter of fact, the correspondence between the binomial and the
curve is surprisingly close even for comparatively low values of n, provided
that p and q are fairly near equality. The limiting frequency curve obtained as
n becomes large is called the normal frequency curve or simply the normal
curve.
4. (a) (1) State the properties of Karl
Pearson’s coefficient of correlation. Give the interpretations when the
correlation coefficient takes the values 0, 1 and – 1. 3+2=5
Ans:
(2) Given the two regression equations:
Find the coefficient of correlation between X and Y.
5
Ans:
(3) If X and Y are two variables, then how
many regression lines can we have? Explain briefly. 1+2=3
Ans:
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
Why do we generally have two regression
equations?
Two regression lines: We know that 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.)
Or
(b) (1) Define correlation analysis. Discuss
different types of correlation. 3
Ans: Correlation
analysis is simply the degree of the relationship between two or more variables
under consideration. If two or more quantities vary in such a way that movements
in one are accompanied by movement in the other quantity, these quantities are
said to be correlated. For example, there exist some relationship between
prices of the product and quantity demanded, rainfall and crops etc.
Correlation analysis measures the degree of relationship the variables under
consideration.
In
the words of Simpson & Kafka “Correlation analysis deals with the
association between two or more variables.”
Various Types of
Correlation
Kinds of correlation may be studied on the basis
of:
A. On the Basis of change in proportion: There
are two important correlations on the basis of change in proportion. They are:
(a) Linear correlation: Correlation is said to
be linear when one variable moves with the other variable in fixed proportion
(b) Nonlinear correlation: Correlation is
said to be nonlinear when one variable moves with the other variable in
changing proportion.
B. On the basis of number of variables: On the
basis of number of variables, correlation may be:
(a) Simple correlation: When only two
variables are studied it is a simple correlation.
(b) Partial correlation: When more than two
variables are studied keeping other variables constant, it is called partial
correlation.
(c) Multiple correlations: When at least three
variables are studied and their relationships are simultaneously worked out, it
is a case of multiple correlations.
C. On the basis of Change in direction: On the
basis of Chang in direction, correlation may be
(a) Positive Correlation: Correlation is said
to be positive when two variables move in same direction.
(b) Negative Correlation: Correlation is said
to be negative when two variables move in opposite direction.
(2) What do you mean by regression analysis?
Write the four properties of regression coefficients. 1+4=5
Ans: Regression
is the measure of the average relationship between two or more variable in
terms of the original units of the data. It is a statistical tool with the help
of which the unknown values of one variable can be estimated from known values
of another variable.
In
the words of Ya Lum Chou, “Regression analysis attempts to establish the nature
of the relationship between variables – that is, to study the functional
relationship between the variable and thereby provide a mechanism for prediction,
or forecasting.”
Characteristics of regression coefficients:
1. Both regression coefficients will
have the same sign.
2. If one regression coefficient is
above unity, then the other regression coefficient should be below unity.
3. If both the regression coefficient
are negative, correlation coefficient should be negative
4. Regression coefficients are
independent of change of origin but not of scale.
(3) Compute the coefficient of correlation
from the following results: 5
Ans:
5. (a) (1) Define index numbers. What are
different types of index numbers? Name each of them. 2+3=5
Ans: Index
number is simply
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 offers a device of estimating the relative changes of a variable when
measurement of actual changes is not possible. It is also an
indicator of inflationary or deflationary tendencies.
In
the words of Croxton and Cowden, “Index number is devices for measuring
differences in the magnitude of a group of related variables.”
Index number is of three types: Price index,
quantity index and value index.
Price Index: Price
index is 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.
Quantity Index: Quantity index is 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.
Value Index: Value indeed is a measure reflecting the
average of the proportionate changes in the value of a specified set of goods
and services between base year and current year. Value of goods and services is
obtained by multiplying prices and quantities. Usually a value 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 values of compared with the base period.
(2) From the following data, calculate the
quantity index number by using Laspeyre’s formula: 5

Base Year 
Current Year 

Items 
Price (in Rs.) 
Quantity 
Price (in Rs.) 
Quantity 
A B C D E 
5 3 4 11 7 
50 100 60 30 40 
10 4 6 14 10 
56 120 60 24 36 
Ans:
(3) What is the importance of consumer price
index? 3
Ans: Uses of consumer price index:
a) CLI numbers are used for adjustment of
dearness allowance to maintain the same standard of living.
b) It is used in fixing various economic
policies.
c) Its helps in measuring purchasing power of
money.
d) Real wages can be obtained with the help of
CLI numbers.
Or
(b) (1) Which index number is considered as
the ideal one and why? 1+2=3
Ans: Fishers ideal index number: Fishers index is an ideal index number because
it considers both current and base year’s prices and quantities. It is the
geometric mean of Laspeyre’s and Paasche’s index and calculated as:
Fisher’s index is regarded as ideal index
because:
i) It considers both base year and current
year’s price and quantity.
ii) It satisfies both time reversal and factor
reversal test.
iii) It is based on Geometric mean which is
theoretically considered to be the best average of constructing index number.
iv) It is free from bias as it considers both
current year and base year price and qty.
(2) Calculate the cost of living index from
the given data: 5
Group 
Index Number 
Weights 
Clothing Food Fuel and lighting House rent Miscellaneous 
360 298 287 110 315 
60 5 7 8 20 
Ans:
(3) What are the tests to check the adequacy
of index numbers? Explain. 1+4=5
Ans: Test of adequacy of Index number
formulae:
There are various formulas for construction of
index number and the problem is that the selection of most appropriate formula
for a given situation. In order to find the most appropriate formula, the
following tests are suggested:
a)
Time reversal
b)
Factor reversal
test
c)
Unit test
d)
Circular test
a) Time Reversal
Test: Time reversal test is
a test to determine whether a given period method will work both ways in time,
forward and backward. In the words of Fisher, “The test is that the formula for
calculating the index number should be such that it will give the same ratio
between one point of comparison and the other, no matter which of the two is
taken as base.” Only Fisher’s ideal index satisfied time reversal test.
Symbolically time reversal test can be written as: P_{01} * P_{10} =
1
b) Factor
Reversal Test: Factor reversal
test holds that the product of a price index and the quantity index should be
equal to the corresponding value index. In the words of Fisher, “Just as each
formula should permit the interchange of the two items without giving
inconsistent results so it ought to permit interchanging prices and quantities
without giving inconsistent results, i.e. the two results multiply together
should give the true value ratio. “ In other words the change in price
multiplied by change in quantity should be equal the total change in value. Only
Fisher’s ideal index satisfied time reversal test. Symbolically factor reversal
test can be written as: P_{01} * Q_{01} =∑P_{1}Q_{1} /
∑P_{0}Q_{0}
c) Unit
Test: Unit test
requires that the formula for construction an index number should be
independent of the units in which, or for which, prices and quantities are
quoted. This formula is satisfied by all the index number formulas except the
simple aggregative index method.
d) Circular
Test: This formula is
similar to time reversal test method. This test is done where there is a
frequent shift in the base on which index number is calculated. If comparison
of more than two years is to be made, it is always desirable to shift the
original base to the previous year which enables us to adjust the index values
from period to period. A test of this shift ability of base is called the
circular test. Symbolically circular test can be written as: P_{01}* P_{12}*P_{20}=
1
6. (a) (1) What is time series? Explain
briefly its main components. 1+4=5
Ans: One
of the most important tasks of any businessman is to make estimates of future
demand of his product so that he can adjust his production according to the
future demand. For this purpose, it is necessary to gather information from the
past. In this connection one usually deals with statistical data which are
collected, observed or recorded at successive intervals of time. Such data are
generally referred to as Time series.
In the words of Morris Hamburg, “A time series
is a set of statistical observations arranged is chronological order.”
Components of Time Series Analysis
The four components of time series are:
(FACTORS RESPONSIBLE FOR TREND IN TIMES SERIES)
1. Secular trend
2. Seasonal variation
3. Cyclical variation
4. Irregular variation
Secular trend: A time series data may show upward trend
or downward trend for a period of years and this may be due to factors like
increase in population, change in technological progress, large scale shifts in
consumer’s demands etc. For example, population increases over a period of
time, price increases over a period of years, production of goods on the
capital market of the country increases over a period of years. These are the
examples of upward trend. The sales of a commodity may decrease over a period
of time because of better products coming to the market. This is an example of
declining trend or downward trend. The increase or decrease in the movements of
a time series is called Secular trend. Examples of Trend or secular trend:
Increase in demand of two wheelers, decrease on death rate due to advancement
of medical science, increase in food production due to increase in population.
Seasonal variation: Seasonal variations are shortterm
fluctuation in a time series which occur periodically in a year. This continues
to repeat year after year. The major factors that are responsible for the
repetitive pattern of seasonal variations are weather conditions and customs of
people. More woolen clothes are sold in winter than in the season of summer.
Regardless of the trend we can observe that in each year more ice creams are
sold in summer and very little in winter season. The sales in the departmental
stores are more during festive seasons that in the normal days. Examples
of seasonal variation: sale of woolen clothes during winter, decline in
icecream sales during winter, demand of TV during international games.
Cyclical variations: Cyclical variations are recurrent upward
or downward movements in a time series but the period of cycle is greater than
a year. Also these variations are not regular as seasonal variation. There are
different types of cycles of varying in length and size. The ups and downs in
business activities are the effects of cyclical variation. A business cycle
showing these oscillatory movements has to pass through four phasesprosperity,
recession, depression and recovery. In a business, these four phases are
completed by passing one to another in this order. It has four important
characteristics: i) Prosperity ii) Decline iii)
Depression iv) Improvement. Examples of cyclical variation:
Recession, Boom, Depression, Recovery, balancing of demand and supply.
Irregular variation: Irregular variations are fluctuations in
time series that are short in duration, erratic in nature and follow no
regularity in the occurrence pattern. These variations are also referred to as
residual variations since by definition they represent what is left out in a
time series after trend, cyclical and seasonal variations. Irregular
fluctuations result due to the occurrence of unforeseen events like floods,
earthquakes, wars, famines, etc. Examples of irregular variation: Flood,
fire, strike, lockout, earthquake, hot wave in winter, rain in desert.
(2) Calculate the trend values by using 3
yearly moving averages for the following data: 3
Year: 
2008 
2009 
2010 
2011 
2012 
2013 
Production: 
77 
88 
94 
85 
91 
98 
Ans:
(3) Define seasonal index. What are the
methods to construct seasonal indices? 1+2=3
Ans:
Not all products and services are in high demand all year and at all times.
Changes in the seasonal trend of demand is a concept called seasonality.
When this is the case, a numerical value is identified called
the seasonality index or seasonality indices. Once this index is
identified, an organization can better make predictions, plan, and analyze by
removing seasonal fluctuations.
Methods
are commonly used for measuring seasonal variation:
a) Method of simple averages
b) Ratio to trend method
c) Ratio to moving average
d) Link relative method.
Or
(b) (1) What do you mean by a seasonal
variation? Give a reason why we should remove the seasonal effects from a given
time series. 1+2=3
Ans: Seasonal variation: Seasonal variations are shortterm
fluctuation in a time series which occur periodically in a year. This continues
to repeat year after year. The major factors that are responsible for the
repetitive pattern of seasonal variations are weather conditions and customs of
people. More woolen clothes are sold in winter than in the season of summer.
Regardless of the trend we can observe that in each year more ice creams are
sold in summer and very little in winter season. The sales in the departmental
stores are more during festive seasons that in the normal days. Examples
of seasonal variation: sale of woolen clothes during winter, decline in
icecream sales during winter, demand of TV during international games.
Seasonal adjustment is the process of
estimating and then removing from a time series influences that are systematic
and calendar related. Observed data needs to be seasonally adjusted as seasonal
effects can conceal both the true underlying movement in the series, as well as
certain nonseasonal characteristics which may be of interest to analysts.
(2) Following table gives the figures of
production (in thousand quintals) of a sugar factory:
Year: 
2014 
2015 
2016 
2017 
2018 
2019 
2020 
Production: 
80 
90 
92 
83 
94 
99 
92 
Fit a straight line trend to the given data.
Plot the data points on graph and show the trend line. Also find the production
for the year 2021. 4+2+2=8
Ans:
Ans: Principles of sampling
1) Principle or Law of statistical regularity: This law is based
upon mathematical theory of probability.
2)
Principle of inertia of large numbers: It is based upon the concept that as the
sample size increases the better results we will get.
3)
Principle of validity: If valid tests are derived only then sampling design is
termed as valid.
(2) Write a short note on one of the
nonrandom sampling methods. 3
Ans: Quota Sampling: This method of study is not much
used. In this method entire data is spilt into as many as there are
investigators and each investigator is asked to select certain items from his
block and study. The success of this method depends upon the
integrity and professional competence of investigators. If some
investigators are competent and others are not so competent, serious
discrepancies will appear in the study.
Or
(b) (1) Mention any two drawbacks of simple
random sampling. 2
Ans:
Disadvantages:
1. The selector has no control over the
selection of units. The researcher cannot contact the far situated units.
2. He cannot prepare the whole field when the
universe is vast.
3. If units have no homogeneity, the method is
not appropriate.
(2) Distinguish between stratified random
sampling and cluster sampling. 3
Ans: Stratified Sampling: This method of selecting samples
is a mixture of both purposive and random sampling techniques. In this all the
data in a domain is spilt into various classes on the basis of their
characteristics and immediately thereafter certain items are selected from
these classes by the random sampling technique. This technique is
suitable in those cases in which the data has sub data and having special
characteristics.
Cluster Sampling: In this method of sampling, the population is
divided into clusters or groups and then Random Sampling is done for each
cluster. In some instances, the sampling unit consists of a group or cluster of
smaller units that we call elements or subunits. Cluster Sampling is different
Stratified sampling. In the case of stratified sampling the elements
of each stratum are homogeneous while in cluster sampling each cluster is
heterogeneous within and a representative of the population.
***
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