Business Statistics Solved Question Paper 2022
[Dibrugarh University BCOM 3rd SEM CBCS Pattern]
2022 (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) State two important objects of measures
of central value.
Ans: Two important objectives of measures of central value:
- It help in reducing the data to a single value that is used for doing comparative studies
- It helps to identify the location of the center of various distributions
(b) Define seasonal variation in time
series with example.
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.
(c) Mention two limitations of classical
definition of 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) The arithmetic means of runs scored by
two batsmen X and Y in a series of 10 innings are 20 and 25 respectively. The
standard deviations of their runs are 4 and 8 respectively. Who is the most
consistent of the two?
Ans:
(e) What do you mean by coefficient of
correlation between two variables?
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.
(f) Distinguish between standard deviation
and standard error.
Ans: 1. Standard Deviation is a descriptive statistic, whereas the
standard error is an inferential statistic.
2. Standard Deviation is the measure which assesses the
amount of variation in the set of observations. Standard Error gauges the
accuracy of an estimate, i.e. it is the measure of variability of the
theoretical distribution of a statistic.
(g) Mention two uses of consumer price
index number.
Ans:
a) It’s 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.
(h) Define chronological data with an
example.
Ans: Chronological classification means
classification on the basis of time, like months, years etc. Example: Profits
of a company from 2015 to 2023.
(i) A binomial variable X has mean 6 and
variance 4. Find the values of n and p.
Ans:
(j) Define stratified random sampling.
Ans: 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.
(k) Show that Fisher’s formula satisfies
factor reversal test.
Ans:
2. (a) (1) The arithmetic mean and geometric mean of two observations are 5 and 4 respectively. Find the observations.
(2) Calculate mode from the following
frequency distribution: 3,
4
|
Class: |
20 – 29 |
30 – 39 |
40 – 49 |
50 – 59 |
60 – 69 |
|
Frequency: |
8 |
12 |
4 |
15 |
9 |
(3) Distinguish between absolute and
relative measures of dispersion. 2
Ans: Difference between absolute and relative
measure of dispersion:
1. Absolute measures
are dependent on the unit of the variable under consideration whereas the
relative measures of dispersion are unit free.
2. For comparing two or more distributions, relative measures and not absolute measures of dispersion are considered.
Or
(b) (1) The average salary of male
employees in a factory was Rs. 5,200 and that of females was Rs. 4,200. The
mean salary of all the employees was Rs. 5,000. Find the ratio of male and female
employees in the factory. 3
Ans:
(2) Calculate variance from the following
data: 4
|
Class: |
10 – 20 |
20 – 30 |
30 – 40 |
40 – 50 |
50 – 60 |
|
Frequency: |
8 |
12 |
9 |
11 |
10 |
(3) Define skewness. For a frequency
distribution if Mean = 25, Mode = 30 and Variance = 25, find the coefficient of
skewness. 1+1=2
3. (a) (1) Explain the meaning of the
statement – “The probability of occurrence of an event A is 1/5”. 2
(2) A problem is given to three students X,
Y and Z. The probability of solving the problem by X, Y and Z are 1/2, 1/3 and
1/4 respectively. Find the probability that the problem will be solved. 3
(3) Under what conditions binomial
probability distribution can be used? 3
Ans: This distribution has been used to describe a wide variety of processes in business and the social sciences as well as other areas. The type of process which gives rise to this distribution is usually referred to as Bernoulli trial or as a Bernoulli process. The mathematical model for a Bernoulli process is developed under a very specific set of assumption involving the concept of a series of experimental trials. These assumptions are:
1) An experiment is performed under the same conditions for a fixed number or trials, say, n.
2) In each trial, there are only two possible outcomes of the experiment.
3) 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.
4) The trials are statistically independent,
i.e., the outcomes of any trial or sequence of trials do not affect the
outcomes of subsequent trials.
(4) Define a random
variable. A random variable X has the following probability distribution:
|
X: |
0 |
1 |
2 |
3 |
|
P (X): |
1/8 |
K |
1/4 |
1/8 |
Find the value of K. 2+3=5
Ans: Properties of the Normal Distribution
1) The height of the normal curve is at its maximum at the mean. Hence the mean and mode of the normal distribution coincide. Thus for a normal distribution mean, median and mode are all equal.
2) Since there is only one maximum point, the normal curve is Unimodal, i.e., it has only one mode.
3) The points of inflexion, i.e., the points where the change in curvature occurs are mean + sd.
4) The first and third quartiles are equidistant from the median.
5) The mean deviation is 4th or more precisely 0.7979 of the standard deviation.
(2) Define Poisson probability distribution
with an example. 2
Ans: Poisson distribution is a discrete probability distribution and is very widely used in statistical work. It was developed by a French mathematician, Simeon Denis Poisson (1781-1840), in 1837. Poisson distribution may be expected in cases where the chance of any individual event being a success is small. The distribution is used to describe the behaviour of rare events such as the number of accidents on road, no. of printing mistake in a book, etc., and has been called “the law of improbable events”. In recent years the statisticians have had a renewed interest in the occurrence of comparatively rare events, such as serious floods, accidental release of radiation from a nuclear reactor, and the like.
(3) A die is thrown. If X denotes the point
on the uppermost face, find E(X). 4
(4) A coin is tossed six times. Find the probability of getting at least four heads. 4
Ans:
There are some basis difference between correlation and regression:
(1) Nature of relationship: Correlation
explains the degree of relationship, whereas regression explains the nature of
the relationship.
(2) Causal relationship: Correlation does not
explain the cause behind the relationship whereas regression studies the cause
and effect relationship.
(3) Prediction: Correlation does not help in
making prediction whereas regression enables us to make prediction.
(4) Origin and scale: Correlation coefficient
is independent of the change of origin and scale, whereas regression
coefficient is independent of change of origin but not of scale.
(2) From the following data, find the two regression equations: 3+3=6
|
X: |
70 |
75 |
81 |
84 |
90 |
|
Y: |
100 |
105 |
95 |
110 |
115 |
(3) Why are there two lines of regression? 3
Ans:
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) Show that coefficient of
correlation ranges from – 1 to + 1. 4
(2) The regression lines have the equations x + 2y = 5 and 2x +3y =8. Find Arithmetic Mean x, Arithmetic Mean y and coefficient of correlation. 2+4=6
(3) What is Spearman’s rank correlation? 3
Ans: Spearman’s rank Coefficient of
correlation: This is a
qualitative method of measuring correlation co-efficient. Qualities such as
beauty, honesty, ability, etc. cannot be measured in quantitative terms. So,
ranks are used to determine the correlation coefficient.
Features
of Spearman’s rank correlation:
i)
The sum of the differences of ranks between two variables shall be zero.
ii)
Spearmen’s correlation coefficient is distribution-free or non-parametric.
5. (a) (1) What is time reversal test? Show
that Fisher’s formula satisfies time reversal test. 4
Ans: 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: P01 * P10 = 1
(2) Calculate Fisher’s price and quantity
index number from the following data: 3+3=6
|
|
Base Year |
Current Year |
||
|
Items |
Price (Rs.) |
Quantity |
Price (Rs.) |
Quantity |
|
A B C D |
10 12 8 4 |
4 5 2 6 |
15 20 10 5 |
6 8 5 10 |
(3) What are the limitations of index number? 3
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.
Or
(b) (1) Why is index number called economic
barometer? 3
Ans:
Use of Index Number (why index number
is called economic barometer)
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.
3. 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.
(2) Calculate Cost of living index number
from the given data: 6
|
|
Price |
|
|
|
Items |
Base year |
Current year |
Weight |
|
A B C D |
10 15 9 20 |
18 30 12 32 |
3 2 4 1 |
(3) Write the differences between
chain-base index number and fixed-based index number. 4
Ans:
Difference between chain base method and fixed base method:
|
CHAIN BASE
MEHTOD |
FIXED BASED
MEHTOD |
|
No fixed base is there. |
Base Period is fixed. |
|
Immediately preceding period is taken as
base. |
Base period is arbitrarily chosen. |
|
Calculation is too long. |
Calculation is easy. |
|
During Calculation if there is any error
then the Entire calculation is wrong. |
This is not so in this method. |
|
If data for any period is missing then
subsequent chain indices cannot be computed. |
This problem does not arise here. |
6. (a) (1) What do you understand by
analysis of time series? What is the need to analyze a time series? 1+3=4
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.
Utility of Time
Series Analysis
a) It
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
producing future behavior.
b) It
helps in planning future operations: Plans
for the future cannot be made without forecasting events and relationship they
will have.
c) It helps in evaluating current accomplishments: The performance can be compared with the expected performance and the cause of variation analyzed.
(2) Calculate trend values by the method of
least squares from the data given below: 4
|
Year: |
2000 |
2001 |
2002 |
2003 |
2004 |
2005 |
|
Sales: |
45 |
50 |
48 |
52 |
55 |
60 |
(3) What are the models used in 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
Or
(b) (1) Explain cyclical variations in a
time series. How do seasonal variations differ from them?2+2=4
Ans: 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 phases-prosperity,
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.
Difference between seasonal variations and cyclical variations:
Seasonal variation are short-term fluctuation in a time series which occur periodically in a year.
Cyclical variations are recurrent upward or downward movements in a time series but the period of cycle is greater than a year.
(2) Calculate 3- yearly moving average from
the data given below: 4
|
Year: |
T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
T8 |
T9 |
|
Values: |
8 |
4 |
9 |
6 |
10 |
12 |
7 |
15 |
11 |
Ans:
Demerits of moving average method:
i)
Trend values cannot be computed for all years.
ii)
No there is no hard and fast rule for selecting the period of moving average.
iii)
This method is not appropriate if the trend situation is not linear.
7. (a) (1) Write a note on sampling error. 3
Ans: Sampling Errors: The errors caused by drawing inference about the population on the basis of samples are termed as sampling errors. The sampling errors result from the bias in the selection of sample units. These errors occur because the study is based on a part of the population. If the whole population is taken, sampling error can be eliminated. If two or more sample units are taken from a population by random sampling method, their results need not be identical and the results of both of them may be different from the result of the population. This is due to the fact that the selected two sample items will not be identical. Thus, sampling error means precisely the difference between the sample result and that of the population when both the results are obtained by using the same procedure or method of calculation. The exact amount of sampling error will differ from sample to sample. The sampling errors are inevitable even if utmost care is taken in selecting the sample.
(2) What is simple random sampling? 2
Ans: 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.
(b) (1) What do you mean by sampling
distribution? 2
Ans: Sampling distribution is a statistic that
determines the probability of an event based on data from a small group within
a large population. Its primary purpose is to establish representative results
of small samples of a comparatively larger population.
(2) What are the merits of stratified
random sampling? 3
Ans: Advantage of stratified random sampling:
a) Neither group nor class of importance is
totally neglected as units of each are represented in the sample.
b) If different classes are divided properly,
selection of few units represents the whole group.
c) On the classification of regional basis,
units are not in contact easily. This leads to economy of time and money.
d) There is a facility in substitution of units. If someone is not contacted easily, the other person of the same class can be substituted for him. Such inclusion result will not show any contradicting.
***
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