Business Statistics
Solved Question Paper June 2025
Dibrugarh
University BCOM 4th SEM NEP Syllabus
COMMERCE (Core)
Paper:
COM(FIN/MKT/BNI/HRM)
Full Marks: 60 |
Time: 2 hours
The figures in the
margin indicate full marks for the questions.
1. Answer any six questions of the following: (2×6=12)
(a) What is the arithmetic mean of 2, 6, 8, 10 and
15?
Ans: AM =
(2+6+8+10+15)/5 = 41/5 = 8.20
(b) Write two differences between sample survey and
complete enumeration.
Ans: Difference between Census and Sample survey:
(a) Under Census method, each and
every individual item is investigated whereas under sample survey only a part
of universe is investigated.
(b) There is no chance of
sampling error in census survey whereas sampling error cannot be avoided under
sample survey.
(c) Large number of enumerators
is required in census whereas less number of enumerators is required in sample
survey.
(d) Census survey is more time
consuming and costly as compared to sample survey.
(e) Census survey is an old method
and it less systematic than the sample survey.
(c) If the correlation coefficient between two
variables x and y is +1 and byx=0.5,
then find the value of bxy.
Ans: The correlation coefficient is the geometric mean of the two regression coefficients:
(d) Write two properties of correlation
coefficient.
Ans:
Properties of r:
i)
The
coefficient of correlation lies between -1 and +1.
ii)
The
co-efficient of correlation is independent to the unit of measurement of
variable.
iii) The co-efficient of correlation is
independent the change of origin and scale.
iv) If two variables are independent to
each other, then the value of r is zero.
(e) Write the mathematical definition of probability.
Ans:
Classical definition of probability = Total number of favourable cases/Total
number of possible outcomes
(f) If P(A)=1/3, P(B)=1/4 and P(A∩B) = 1/6, then
find the value of P(A∪B).
Ans: P(AUB)
= P(A) + P(B) – P(A∩B) = 1/3 + ¼ - 1/6 = 5/12
(g) Write the multiplication model of time series
analysis.
Ans: Multiplicative Model: 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.
(h) What do you mean by 'base year' in the
construction of index numbers?
Ans: Base year is the reference year with which comparisons
of relative changes are made. It may be a year, a month or a day. The indeed
for base period is always taken as 100.
2. (a) (i) Prove
that for any two values AM≥GM≥HM.
Ans:
(ii) Calculate
median from the following distribution: (4)
|
Marks |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
|
No. of Students |
20 |
30 |
25 |
45 |
35 |
40 |
15 |
Ans:
(iii) Calculate
standard deviation for the following distribution: (5)
|
Marks |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
|
No. of Students |
8 |
13 |
16 |
8 |
5 |
Ans:
OR
(b) (i) Write the
advantages of sample survey over complete enumeration. (3)
Ans: Merits of sample survey:
(a)
Since data are limited, time and labour can be saved in sample survey method.
(b)
Sample survey is suitable where highly trained personnel or specialised
equipment is required for collection of data.
(c)
Detailed information can be obtained under sample survey method since data are
collected from a small group of respondents.
(d)
Above all time, energy and money can be saved without sacrificing accuracy of
result.
(ii) Write a note
on random sampling and judgemental sampling. (4)
Ans: Simple Random Sampling: Off all the methods of selecting sample,
random sampling technique is made maximum use of and it is considered as the
best method of sample selection. Random sampling is made in
following ways:
(i)
Lottery Method: In this the numbers of data are written on sheet of paper and
they are thrown into a box. Now a casual observer selects the number
of item required in the sample. For this method it is necessary that
sheet of paper should be of equal dimensions.
(ii)
By Rotating the Drum: In this method, piece of wood, tin or
cardboard of equal length and breadth, with number 0,1 or 2 printed on them,
are used. The pieces are rotated in a drum and then requisite numbers are drawn
by an impartial person.
(iii)
Selecting from Sequential List: In this procedure units are broken
up in numerical, alphabetical or geography sequence. Now we may
decide to choose 1, 5, 10 and so on, if the division is alphabetical order we
decide to choose every item starting from a, b, c and so on.
(iv)
Tippet’s Number: On the basis of population statistics, Tippet has
constructed a random list of four digits each of 10, 400
institutions. These numbers are the result of combining 41,600
population statistics reports.
Purposive or Judgmental or Selective sampling: In this method the investigator
has complete freedom to choose his sample according to his wishes and
desire. To choose or leave an item for the purpose of study depends
entirely upon the wishes of investigator and he will choose items or units
which in his judgment are representative of the whole data. This is
a very simple technique of choosing the samples and is useful in cases where
the whole data is homogeneous and the investigator has full knowledge of the
various aspects of the problem.
(iii) Write a note on any one of the
following: (5)
- Simple random
sampling
- Systematic
sampling
Ans: Simple Random Sampling: Off all the methods of selecting sample,
random sampling technique is made maximum use of and it is considered as the
best method of sample selection. Random sampling is made in
following ways:
(i)
Lottery Method: In this the numbers of data are written on sheet of paper and
they are thrown into a box. Now a casual observer selects the number
of item required in the sample. For this method it is necessary that
sheet of paper should be of equal dimensions.
(ii)
By Rotating the Drum: In this method, piece of wood, tin or
cardboard of equal length and breadth, with number 0,1 or 2 printed on them,
are used. The pieces are rotated in a drum and then requisite numbers are drawn
by an impartial person.
(iii)
Selecting from Sequential List: In this procedure units are broken
up in numerical, alphabetical or geography sequence. Now we may
decide to choose 1, 5, 10 and so on, if the division is alphabetical order we
decide to choose every item starting from a, b, c and so on.
(iv)
Tippet’s Number: On the basis of population statistics, Tippet has
constructed a random list of four digits each of 10, 400
institutions. These numbers are the result of combining 41,600
population statistics reports.
Systematic Sampling: This method of sampling is at first glance
very different from random sampling. In practice, it is a variant of
simple random sampling that involves some listing of elements. In systematic
sampling each element has an equal chance of being selected, but each sample
does not have the same chance of being selected. Here, the first element of the
population is randomly selected to begin the sampling. But thereafter the
elements are selected according to a systematic plan. Systematic sampling
proceeds by picking up one element after a fixed interval.
3. (a) (i) Prove
the relationship between coefficient of correlation and the two regression
coefficients. (3)
Ans:
(ii) If the two
regression lines are 2x - 3y = 0 and 4y - 5x - 8 = 0, then find the arithmetic
mean of x and y. (4)
Ans:
(iii) Calculate the
coefficient of rank correlation from the data given below: (5)
|
X: |
44 |
33 |
40 |
9 |
16 |
65 |
24 |
18 |
44 |
20 |
|
Y: |
13 |
10 |
24 |
6 |
15 |
4 |
20 |
9 |
10 |
19 |
Ans:
OR
(b) (i) What is Karl Pearson's coefficient of correlation? What is its
range? (3)
Ans: Karl Pearson’s Coefficient of correlation: Correlation coefficient is a mathematical
and most popular method of calculating correlation. Arithmetic mean and
standard deviation are the basis for its calculation. The Correlation
coefficient (r), also called as the linear correlation coefficient measures the
strength and direction of a linear relationship between two variables. The
value of r lies between -1 to +1.
(ii) Prove that
Karl Pearson's coefficient of correlation is independent of the change of
origin and scale of measurement. (4)
Ans:
(iii) From the data given below, find the regression equation of y on x:
(5)
|
x: |
52 |
63 |
45 |
36 |
72 |
65 |
47 |
25 |
|
y: |
62 |
53 |
51 |
25 |
79 |
43 |
60 |
33 |
Ans:
4. (a) (i)
If P(A) = 1/2, P(B) = 1/3 and P(AB) = 1/4, then find P(A/B) and P(B/A). (3)
(ii) Find
the probability of getting 53 Sundays in a randomly selected leap year. (4)
Ans:
(iii) Two
bags contain 5 white and 3 black balls. Another contains 4 white and 5 black
balls. Two balls are selected randomly from any one of the urn. Find the
probability that the selected two balls are white. (5)
Ans:
OR
(b) (i) Write three
properties of mathematical expectation. (3)
Ans: Properties of Mathematical Expectations
1)
The expected value of a constant is
the constant itself i.e.,E(a) = a.
2)
E(a+bx) = a + bE (x), where a, b are
constants.
3)
If x and y are random variables then:
E(x + y) = E(x) + E(y).
4)
If x and y be two independent random
variable then: E(xy) = E(x).E(y).
(ii) From the
following data, find the mathematical expectation and variance: (4)
|
X: |
1 |
2 |
3 |
|
P(X=x): |
1/6 |
1/3 |
1/2 |
Ans:
(iii) Mean and
variance of a binomial variate X are 6 and 4. Find the value of n and P. Write
the probability distribution of X. (5)
Ans:
5. (a) (i) Prove
that Fisher's index number satisfies time reversal test. (3)
(ii) Discuss the
limitations of index numbers. (4)
Ans: Limitations of index number: Index number suffers from various
limitations some of which are listed below:
1. Not
completely true: Index number not fully true. The index number simply
indicate arithmetical tendency of the temporal changes in the variable.
2. International
comparison is not possible: Different countries have different bass of
index numbers; these do not help international comparisons.
3. Difference
of time: With the passage of time, it is difficult to make comparison
of index number with the changing time man’s habits.
4. Limited
use: Index numbers are prepared with certain specific objective. If
they are used for another purpose they may lead to wrong conclusion.
5. 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.
(iii) Calculate
cost of living index number from the data given below: (5)
|
Group |
Index No. |
Weight |
|
Clothing |
360 |
40 |
|
Food |
300 |
25 |
|
Fuel and Lighting |
267 |
7 |
|
House Rent |
120 |
8 |
|
Others |
320 |
20 |
Ans:
OR
(b) (i) Write a
short note on seasonal variation in a time series. (3)
Ans: Seasonal variation: Seasonal variations are short-term
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 ice-cream
sales during winter, demand of TV during international games.
(ii) Estimate the
trend values by using 3 yearly moving average for the following data: (4)
|
Year |
2011 |
2012 |
2013 |
2014 |
2015 |
2016 |
2017 |
2018 |
|
Sale |
60 |
46 |
53 |
54 |
48 |
48 |
42 |
51 |
Ans:
(iii) Using the
method of least squares, calculate the trend values for the following data: (5)
|
Year |
2008 |
2009 |
2010 |
2011 |
2012 |
|
Production |
100 |
140 |
150 |
180 |
200 |
Ans:
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