# Business Statistics Solved Question Paper Dec 2021 [Dibrugarh University BCOM 3rd SEM CBCS Pattern]

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 cross-sectional data. Give an example.

Ans: Cross-sectional 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 cross-sectional 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 = (H-L)/(H+L) = (15-4)/(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 non-random sampling.

Ans: Non-random 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 high-value end of the curve (the right-hand 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, co-efficient 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:

and,

,

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) Non-linear correlation: Correlation is said to be non-linear 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 co-efficients will have the same sign.

2. If one regression co-efficient is above unity, then the other regression co-efficient should be below unity.

3. If both the regression co-efficient are negative, correlation co-efficient should be negative

4. Regression co-efficients 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: P01 * P10 = 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: P01 * Q01 =∑P1Q1 / ∑P0Q0

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: P01* P12*P20= 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 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.

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.

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

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 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.

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 non-seasonal 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:

7. (a) (1) What are the principles of sampling?             2

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 non-random 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

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 sub-units. 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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