# Business Statistics Solved Question Paper Dec 2020 [Dibrugarh University BCOM 3rd SEM CBCS Pattern]

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.

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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 co-efficient of correlation is independent to the unit of measurement of variable.

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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 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.

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 non-zero numbers GM2 = AM X HM.         3

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(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

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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, co-efficient 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

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3. (a) (1) Two coins are tossed simultaneously. Find the probability of getting same face on both the coins.           3

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(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

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Or

(b) (1) Find the probability that a leap year selected at random will contain 53 Sundays.              3

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(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

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(4) A normal variate X has a mean 50 and standard deviation 5. Find the probability that X lies between 40 and 60. 4

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4. (a) (1) Prove that coefficient of correlation is the geometric mean of the two regression coefficients.          3

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(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

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(3) Derive the regression line of X and Y from the following data:        5

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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 market-weighted 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

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(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

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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

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(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

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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

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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 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 seasonal index for the following data by using the method of simple average (assuming that the trend is absent):               6

 Year Q1 Q2 Q3 Q4 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

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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 Sxy 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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