Business Statistics Solved Question Paper 2020
[Dibrugarh University BCOM 3rd SEM CBCS Pattern]
2020 (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: 2 x 8 = 16
a) Calculate AM from
the following data: 10, 20, 15, 18, 30.
Ans:
b) Define mutually
exclusive events.
Ans: Mutually
Exclusive Events: Two events are said to be mutually exclusive
or incompatible when both cannot happen simultaneously in a single trial or, in
other words, the occurrence of any one of them make impossible the occurrence
of the other. For example, if a single coin is tossed either head can be up or
tail can be up, both cannot be up at the same time. To take another example, if
we toss a die and observe 3, we cannot expect 5 also in the same toss of die.
Symbolically, if A and B are mutually exclusive events, P (Aê“µB) = 0.
c) What are the
probabilities of an impossible event and a certain event?
Ans: The probability of an impossible event is zero and the probability of a certain event is one.
d) Mention the two
properties of correlation coefficient.
Ans: Properties of r:
 The coefficient of correlation lies between
1 and +1.
 The coefficient of correlation is
independent to the unit of measurement of variable.
Ans:
f) 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.
g) What do you mean
by cost of living index number?
Ans: Cost of living index numbers generally
represent the average change in prices over a period of time, paid by a
consumer for a fixed set of goods and services. It measures the relative
changes over time in the cost level require to maintain similar standard of
living. Items contributing to consumer price index are generally:
a)
Food
b)
Clothing
c)
Fuel and Lighting
d)
Housing
e)
Miscellaneous.
h) Write the main
objectives of time series analysis.
Ans: The main aim of time series analysis is to helps in
understanding past behaviors. By observing data
over a period of time one can easily understanding what changes have taken
place in the past, such analysis will be extremely helpful in predicting future
behavior.
i) What do you mean
by seasonal variations in time series analysis?
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.
j) Define parameter.
Ans: Parameters are numbers that
describe the properties of entire populations. Statistics are numbers that
describe the properties of samples.
k) In what situations
stratified sampling is used to draw a sample from a population?
Ans:Stratified random sampling is
often used when researchers want to know about different subgroups or
strata based on the entire population being studied—for instance, if one is
interested in differences among groups based on race, gender, or education.
2. (a) (1) Prove that for any two nonzero
numbers GM^{2} = AM X HM. 3
Ans:
(2) Find standard deviation and coefficient
for variation from the following series: 6
Class Interval: 
5 – 15 
15 – 25 
25 – 35 
35 – 45 
45 – 55 
Frequency: 
8 
12 
15 
9 
6 
Ans:
Or(b) (1) Why is standard deviation
considered to be the best measures of dispersion? 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:

It is based on each and every item of the data and it is rigidly defined.

It is capable of further algebraic treatment. Combined SD of two or more groups
can be calculated.

It is less affected by fluctuations of sampling than most other measures of
dispersion.

For comparing variability of two or more series, coefficient of variation is
considered as most appropriate and this is based on SD and Mean.
(2) Calculate median from the following
distribution: 6
Marks: 
0 10 
10 – 20 
20 – 30 
30 – 40 
40 – 50 
50 – 60 
No. of Students: 
4 
6 
10 
15 
12 
6 
Ans:
3. (a) (1) Two coins are tossed
simultaneously. Find the probability of getting same face on both the coins. 3
Ans:
(2) A problem is given to three students A,
B and C. The probability of solving the problem by A, B and C are 1/2, 1/3 and
1/4 respectively. Find the probability that the problem will be solved. 5
Ans:
Ans:
Or
(b) (1) Find the probability that a leap
year selected at random will contain 53 Sundays. 3
Ans:
(2) A random variable X has the following
probability distribution:
X = x 
1 
4 
7 
P(X): 
¼ 
1/2 
1/4 
Find E(X). 3
(3) A binomial variable X has mean 6 and
variance 4. Find the probability distribution of X. 3
Ans:
(4) A normal variate X has a mean 50 and
standard deviation 5. Find the probability that X lies between 40 and 60. 4
Ans:
4. (a) (1) Prove that coefficient of
correlation is the geometric mean of the two regression coefficients. 3
Ans:
Ans:
(3) Find the correlation coefficient from the data given below: 6
x: 
105 
120 
95 
150 
130 
y: 
100 
115 
110 
135 
115 
Or
(b) (1) What is meant by correlation?
Distinguish between positive, negative and zero correlations. 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.”
Difference
between positive, negative and zero correlations
1. Positive Correlation: If increase (or
decrease) in one variable corresponds to an increase (or decrease) in the
other, the correlation is said to be positive correlation.
2. Negative Correlation: If increase (or
decrease) in one variable corresponds to a decrease (or increase) in the other,
the correlation is said to be negative correlation.
3. Zero or No Correlation: If change in one
variable does not affect other variable, then there is no or zero correlation.
(2) Calculate the coefficient of rank
correlation from the data given below: 5
x: 
92 
89 
86 
87 
83 
71 
77 
63 
53 
50 
y: 
86 
83 
77 
91 
68 
52 
85 
82 
57 
57 
Ans:
(3) Derive the regression line of X and Y
from the following data: 5
Ans:
5. (a) (1) What are BSE SENSEX and NSE
NIFTY? 3
Ans: Sensex is the stock market
app index indicator for the BSE. It is also sometimes referred to
as BSE Sensex. It was first published in 1986 and is based on the
marketweighted stock index of 30 companies based on the financial performance.
The large, established companies that represent various
industrial sectors are a part of this.
National Stock Exchange Fifty or Nifty is the
market indicator of NSE. It ideally is a collection of 50 stocks but presently
has 51 listed in it. It is also referred to as Nifty 50 and CNX Nifty by some
as it is owned and managed by India Index Services and Products Ltd. (IISL).
(2) From the data given below, prove that
Fisher index number satisfies time reversal test: 5
Items 
p0 
q0 
p1 
q1 
A 
4 
20 
6 
10 
B 
3 
15 
5 
23 
C 
2 
25 
3 
15 
D 
5 
10 
4 
40 
Ans:
(3) The following table gives the index
number of different groups of items with their respective weights for 2020
(base year 2010):
Group 
Group Index No. 
Weight 
Food 
525 
40 
Closing 
325 
16 
Fuel 
240 
15 
Rent 
180 
20 
Others 
200 
9 
Calculate the overall cost of living index
number and interpret the results. 4+1=5
Ans:
Or
(b) (1) Write the three uses of cost of
living index number. 3
Uses of cost of living index:
 CLI numbers are used for adjustment of
dearness allowance to maintain the same standard of living.
 It is used in fixing various economic
policies.
 Its helps in measuring purchasing power of
money.
 Real wages can be obtained with the help of
CLI numbers.
(2) Prove that Fisher index number
satisfies time reversal test and factor reversal test. 5
Ans:
(3) Find the price index number from the
following data using Paasche and Laspeyres index: 5

Base Year 
Current Year 

Item 
Price 
Quantity 
Price 
Quantity 
A 
6 
50 
6 
72 
B 
7 
84 
10 
80 
C 
10 
80 
12 
96 
D 
4 
20 
5 
30 
Ans:
6. (a) (1) Write the two models used for studying
time series analysis. 3
Ans:
Times series model are of two types. One is multiplicative model and other one
is additive model.
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.
Additive
Model: Another approach is to treat each observation of a time series as the
sum of these four components Symbolically, Y=T + S+ C + I
(2) From the data given below, find the
straight line trend by using the method of least squares:
Year: 
1968 
1969 
1970 
1971 
1972 
1973 
1974 
1975 
1976 
Value: 
80 
90 
92 
83 
94 
99 
92 
110 
100 
Also estimate the value for the year 1980. 8
Ans:
Or
(b) (1) What are the components of time
series? Discuss any one of them. 2+3=5
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
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 seasonal index for the
following data by using the method of simple average (assuming that the trend
is absent): 6
Year 
Q_{1} 
Q_{2} 
Q_{3} 
Q_{4} 
1991 
72 
68 
80 
70 
1992 
76 
70 
82 
74 
1993 
74 
66 
84 
80 
1994 
76 
74 
84 
78 
1995 
78 
74 
86 
82 
Ans:
7. (a) What is simple random sampling?
Explain lottery method used to draw a simple random sample from a population. 5
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.Simple random
sampling selects a smaller group (the sample) from a larger group of the
total number of participants (the population).Researchers can create a simple
random sample using methods like lotteries or random draws.A sampling error can
occur with a simple random sample if the sample does not end up accurately
reflecting the population it is supposed to represent.
Lottery method: This is very popular method of selecting a random sample under which all items of the population are numbered or named on separate slips of paper. These slips of paper should be of identical size, color and shape. These slips are then folded and mixed up in a container or box or drum. A blind fold selection is then made of the number of slips required to constitute the desired size of sample. The selection of items thus depends entirely on chance. For example, if we want to select n candidates out of N. We assign the numbers 1 to N. One number to each candidate and write these numbers (1 to N) on N slips which are made as homogeneous as possible. These slips are then put in bag and thoroughly shuffled and then n slips are drawn one by one. Then the n candidates corresponding to numbers on the slip drawn will constitute a random sample.
Or
(b) What do you mean by the standard error
of a statistic? A random sample of size 100 has mean 15, the population
variance is 25. Find the interval estimate of the population mean with
confidence level of (1) 99% and (2) 95%. 2+3=5
Ans: With the help of regression equations, perfect
prediction of values is not possible. In order to measure the accuracy of
estimated figures, a statistical tool is used which is known as standard error
of estimate. Calculation of standard error of estimate, symbolized as S_{xy }similar
to standard deviation. Standard deviation measures the dispersion about an
average, such as mean. The standard measure of estimate measures the dispersion
about an average line, called the regression line. The formula for calculating
the standard error of estimate is:
The standard error of estimate measures the accuracy of the estimated figures. The smaller the values of standard error of estimate, the closer will the actual value and estimated value. If standard error of estimate is zero, then there is no variation.
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