2014 (November)
COMMERCE
(General / Speciality)
(Business Statistics)
The figures in the margin indicate full marks for the questions.
1. Answer any five questions: 2x5=10
(a) Prove that the correlation coefficient is the geometric mean of the two regression coefficients.
Ans: (a linear relation of correlation and regression coefficients)
So we find the correlation coefficient is G.M. of regression coefficients.
(b) State the limitations of Laspeyres’ formula for the construction of index numbers.
Ans: 1. It considered only base year’s price or quantity. 2. It does not satisfied time reversal and factor reversal test.
(c) Calculate AM and HM of 2, 4 and 8.
Ans: AM of
(d) Write the two models used for the study of time series.
Ans: Ans:  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.
Another approach is to treat each observation of a time series as the sum of these four components Symbolically, Y=T + S+ C + I
(e) “The correlation coefficient between two variables X and Y is r and =0.65,” What can be concluded from this statement?
Ans: It means there is a positive relationship between the two variable x and y to the extent of 0.65.
(f) What do you mean by business forecasting?
(g) What is the difference between Karl Pearson’s coefficient of correlation and Spearman’s coefficient of correlation?
Ans: Karl Pearson’s coefficient of correlation is used in case of quantitative data while Spearman’s coefficient of correlation is used in qualitative data.
(h) Define price index number and quantity index number.
Ans: Price index and Quantity index
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. A quantity index is built up from information on prices of various commodities.
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.
2. (a) (i) which measures of dispersion is regarded as the best and why? 3
Ans: Sd is the best measure of dispersion because:
 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.
 SD is most prominently used in further statistical work.
(ii) The AM of the following distribution is 1.46, find the missing frequencies: 4
No. of Students :

0

1

2

3

4

5

Total

No. of Days :

46

25

10

5

200

Ans: Calculation of Missing Frequencies
No. of Students

Frequency
 
0
1
2
3
4
5

46
25
10
5

0
75
40
25

(iii) Calculate the standard deviation for the following data: 7
Wages in (Rs.)

No. of Men

O and above
20 and above
40 and above
60 and above
80 and above
100 and above

50
45
34
16
6
0

Ans: Calculation of Standard deviation
(50)
 
0 – 20
20 – 40
40 – 60
60 – 80
80 – 100
100 – 120

10
30
50
70
90
110

5
11
18
10
6
0

50
330
900
700
540
0

– 40
– 20
0
20
40
60

1,600
400
0
400
1,600
3,600

– 200
– 220
0
200
240
0

8,000
4,400
0
4,000
9,600
0

50

2,520

20

26,000

2. (b) (i) For any two values, prove that 3
Ans:
(ii) The mean, median and mode of a group of 25 observations are 143, 144 and 147. A set of 6 observations is added to this data with values132, 125,130,160,165 and 157. Find the mean and median for the combined group of 31 observations. 4
Ans:
Among the new observations (132, 125, 130) are less than median and (160, 165, 157) are greater then median.
The median is unaffected.
Lastly new observations are distinct (i.e. not repeated) as such there will be no effect on the original observations i.e. on 147 which occurs most frequently.
Mode is also unaltered.
(iii) Calculate AM and SD for the following Data: 7
Midpoint :

15

20

25

30

35

40

45

50

55

Frequency :

2

22

19

14

3

4

6

1

1

Ans: Ans: Calculation of Mean and Standard deviation
X

F

fx

d = x  35

d2

fd

fd2

15

2

30

20

400

40

800

20

22

440

15

225

330

4950

25

19

475

10

100

190

1900

30

14

420

5

25

70

350

35

3

105

0

0

0

0

40

4

160

5

25

20

100

45

6

270

10

100

60

600

50

1

50

15

225

15

225

55

1

55

20

400

20

400

Sum

N = 72

2005

0

1500

515

9325

3. (a) (i) Define Karl Pearson’s Coefficient of correlation. 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) discuss the uses of regression analysis. 4
Ans: The following are main Advantages of regression analysis:
(1) Helpful to statisticians: The study of regression helps the statisticians to estimate the most probable value of one variable of a series for the given values of the other related variables of the series.
(2) Nature of relationship:  Regression is useful in describing the nature of the relationship between two variables.
(3) Estimation of relationship:  Regression analysis is widely used for the measurement and estimation of relationship among economic variables.
(4) Predictions:  Regression analysis is helpful in making quantitative predictions on the basis of estimated relationship among variables.
(5) Policy formulation:  The predictions made on the basis of estimated relationship are used in policy making.
(iii) From the following data, find the two regression lines:
Ans:
Or
(b) (i) The correlation coefficient of two variables X and Y is r = 0.60, variance of X and Y are respectively 2.25and 4.00; =10, =20. From the above data, find the regression equation of X on Y. 3
Ans:
(ii) Calculate Spearman’s rank correlation coefficient from the data given below: 4
X :

11

12

13

14

18

15

Y :

13

12

15

14

16

11

Ans: Calculation of Spearmen’s Rank Correlation
X

Y
 
11
12
13
14
18
15

13
12
15
14
16
11

6
5
4
3
1
2

4
5
2
3
1
6

2
0
2
0
0
4

4
0
4
0
0
16

24

(iii) Find the value of the coefficient of correlation from the data given below: 7
Income :

46

54

56

56

58

60

62

Expenditure :

36

40

44

54

42

58

54

Ans: Calculation of Karl Pearson’s Coefficient of Correlation using Assumed mean method
X

Y
 
46
54
56
56
58
60
62

36
40
44
54
42
58
54

– 10
– 2
0
0
2
4
6

100
4
0
0
4
16
36

– 18
– 14
– 10
0
– 12
4
0

324
196
100
0
144
16
0

– 180
– 28
0
0
– 24
16
0

392

328

0

160

– 50

780

– 216

Assumed mean for X = 56
Assumed mean for Y = 54
4. (a) (i) Write the differences between chain base index and fixed base index. 3
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.

(ii) Prove that Fisher’s index number satisfies time reversal test. 4
Ans: Weighted aggregate price relatives according to Fisher’s formula are given by (omitting the factor 100)
Hence Fisher’s Index Number satisfied “Time Reversal Test.”
(iii) Find the quantity index number from the following data using Paasche’s and Laspeyre’s index:
Items

Base year

Current year
 
Price (in Rs.)

Quantity

Price (in Rs.)

Quantity
 
A
B
C

4
6
8

10
15
15

6
4
10

15
20
4

Ans: Calculation of Index Number
Base Year

Current Year
 
Comm.
 
A
B
C

4
6
8

10
15
15

6
4
10

15
20
4

40
90
120

60
120
32

60
60
150

90
80
40

250

212

270

210

Or
(b) (i) What is cost of living index? How does it help in policy formulation by the Government? 3
Ans: Cost of living index number (CLI): 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 measure the relative changes over time in the cost level require to maintain similar standard of living. Items contributing to consumer price index are generally:
 Food
 Clothing
 Fuel and Lighting
 Housing
 Miscellaneous.
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.
(ii) Why Fisher’s index number is regarded as an ideal index number? 4
Ans: Fisher’s index is regarded as ideal index because:
 It considers both base year and current year’s price and quantity.
 It satisfies both time reversal and factor reversal test.
 It is based on Geometric mean which is theoretically considered to be the best average of constructing index number.
 It is free from bias as it considers both current year and base year price and qty.
(iii) From the following data, prove that Fisher’s Index number satisfies time reversal test and factor reversal test:
Items

P0

Q0

P1

Q0

A
B
C
D
E

4
5
2
1
3

20
15
30
50
25

6
6
3
1
5

18
12
30
60
28

Ans: Time reversal and factor reversal test of fisher’s ideal index number
Comm.
 
A
B
C
D
E

4
5
2
1
3

20
15
30
50
25

6
6
3
1
5

18
12
30
60
28

80
75
60
50
75

72
60
60
60
84

120
90
90
50
125

108
72
90
60
140

340

336

475

470

5. (a) (i) Discuss the uses of studying time series. 3
Ans: Utility of Time Series Analysis
The analysis of Time Series is of great significance not only to the economist and businessman but also to the scientist, geologist, biologist, research worker, etc., for the reasons given below:
 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.
 It helps in planning future operations: Plans for the future cannot be made without forecasting events and relationship they will have. Statistical techniques have been evolved which enable time series to be analyzed in such a way that the influences which have determined the form of that series to be analyzed in such a way that the influences which have determined the form of that series may be ascertained.
 It helps in evaluating current accomplishments: The performance can be compared with the expected performance and the cause of variation analyzed. For example, if expected sale for 1995 was 10,000 refrigerators and the actual sale was only 9,000, one can investigate the cause for the shortfall in achievement. Time series analysis will enable us to apply the scientific procedure for such analysis.
 It facilitates comparison: Different time series are often compared and important conclusions drawn there from. However, one should not be led to believe that by time series analysis one can foretell with 100percnet accuracy the course of future events.
(ii) What is trend in a time series? State the factors responsible for trend in a time series. 4
Ans: 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 shift in consumer’s demands etc.
The four components of time series are:
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 shift 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.
Seasonal variation: Seasonal variation 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.
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
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 results due to the occurrence of unforeseen events like floods, earthquakes, wars, famines, etc.
(iii) Calculate trend values by using the method of least squares from the data given below: 7
Year :

2001

2002

2003

2004

2005

2006

Values :

101

107

113

121

136

148

Ans: Calculation of Trend values using method of least square
Year

Value

Trend Value
 
2001
2002
2003
2004
2005
2006

101
107
113
121
136
148

– 5
– 3
– 1
1
3
5

25
9
1
1
9
25

– 505
– 321
– 113
121
408
740

121 + 4.71 (– 5) = 97.45
121 + 4.71 (– 3) = 106.87
121 + 4.71 (– 1) = 116.29
121 + 4.71 (1) = 125.71
121 + 4.71 (3) = 135.13
121 + 4.71 (5) = 144.55

726

0

70

330

Or
(b) (i) The trend equation for publicity cost (Rs. In’ 000) of a company is =20.2 – 0.8t. Origin 2001 (1st July), t unit = 1 year, Y unit = yearly cost. Shift the origin to 2010. 3
(ii) Calculate 3 yearly moving average from the following data: 4
Year :

1

2

3

4

5

6

7

8

9

10

Profit: (in Rs.)

20

21

23

22

25

24

27

26

28

30

Ans: 3 Yearly moving averages
Year

Profit

3 yearly moving total

3 yearly moving average

1
2
3
4
5
6
7
8
9
10

20
21
23
22
25
24
27
26
28
30


64
66
70
71
76
77
81
84



64/3 = 21.33
66/3 = 22
70/3 = 23.33
71/3 = 23.77
76/3 = 25.33
77/3 = 25.77
81/3 = 27
84/3 = 28


(iii) Fit a straight line trend to the following data and estimate the profit for the year 2010: 7
Year:

2001

2002

2003

2004

2005

2006

2007

Profit: (‘000 Rs.)

60

72

75

65

80

85

90

Ans: Calculation of Trend values using method of least square:
Year

(Values)
 
2001
2002
2003
2004
2005
2006
2007

60
72
75
65
80
85
90

– 3
– 2
– 1
0
1
2
3

9
4
1
0
1
4
9

– 180
– 144
– 75
0
80
170
270

527

0

28

121

Origin = 2004
Unit = 1 year
The value of trend for the year 2010 is
6. (a) (i) Discuss about demand forecasting. 5
(ii) Discuss the limitations of business forecasting. 9
Or
(b) (i) Discuss the factors of a good forecasting. 5
(ii) Discuss the steps for forecasting. 9