2016 (November)
COMMERCE
(General/Speciality)
Course: 303
(Business Statistics)
The figures in the margin indicate full marks for the questions
(New Course)
Full Marks: 80
Pass Marks: 24
Time: 3 hours
1. Answer any eight questions: 2x8=16
 What do you mean by statistical unit?
Ans: A statistical unit is the unit of observation or measurement for which data are collected or derived.
 Write one advantage of sampling method and one disadvantage of complete enumeration method.
Ans: Census: Since all the individuals of the universe are investigated, highest degree of accuracy is obtained.
Sample: While using secondary data, time and labour are saved.
Ans: Given,
 If the coefficient of correlation betweenandis 0.67, then what will be the coefficient of correlation betweenand?
 If the correlation coefficient between two variablesand is +1 and, then find the value of.
Ans: Given,
 Given the annual trend equation of a company is (unit = 1 year), estimate the monthly trend equation of the company.
Ans: Annual Trend
Monthly Trend
 Write the multiplication model of time series analysis.
Ans: Multiplicative model: T.C.S.I, Here T = Secular Trend, C = Cyclical trend, S = Seasonal variation and I = Irregular variation.
 Define covariance between two variables.
Ans: Covariance is a measure of how much two random variables vary together. It’s similar to variance, but where variance tells you how a single variable varies, co variance tells you how two variables vary together.
 If the price index number for the year 2016 compared to 2006 is 210 and monthly income of a person in 2006 be Rs. 10,500, then what should be his monthly income in 2016?
Ans:
Year

Index No.

Income

2006
2016

100
210

10,500

 Write the formula for Fisher’s ideal index number.
Ans:
 If the two regression lines areand, then find the arithmetic mean ofand.
Ans:
 What do you mean by quantity index number?
Ans: 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) SD is regarded as the best measure of dispersion. (Fill up the blank) 1
(ii) In a moderately asymmetrical distribution mode and mean are 32.1 and 35.4 respectively. Find the median. 3
Ans: 3 Median = 2 Mean + Mode
3 Median = 2x35.4 + 32.1
3 Median = 70.8 + 32.1
3 Median = 102.9
Median = 102.9/3
= 34.3
(iii) Find the mean deviation from mean for the following data: 5
(Marks):

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

60 – 70

(No. of Students):

20

25

32

40

42

35

10

Ans:
Calculation for MD from Mean
Frequency

Mid Value
 
010
1020
2030
3040
4050
5060
6070

20
25
32
40
42
35
10

5
15
25
35
45
55
65

100
375
800
1,400
1,890
1,925
650

30
20
10
0
10
20
30

600
500
320
0
420
700
300

= 204

= 7,140

= 2,840

(iv) Calculate the coefficient of variation for the following data: 7
(Weight):

0 – 10

0 – 20

0 – 30

0 – 40

0 – 50

0 – 60

0 – 70

0 – 80

(No. of Persons):

15

30

53

75

100

110

115

125

Ans:
Calculation of Coefficient of variation
(35)
 
010
1020
2030
3040
4050
5060
6070
7080

5
15
25
35
45
55
65
75

15
15
23
22
25
10
5
10

75
225
575
770
1125
550
325
750

– 30
– 20
– 10
0
10
20
30
40

300
400
100
0
100
400
900
1,600

 450
 300
 230
0
250
200
150
400

4,500
6,000
2,300
0
2,500
4,000
4,500
16,000

125

4,395

20

39,800

Or
(b) (i) for a symmetrical distribution value of mean, median and mode are same (Equal). (Fill up the blank) 1
(ii) Prove that 3
Ans:
(iii) Calculate quartile deviation for the following data: 5
(Class):

5 – 10

10 – 15

15 – 20

20 – 25

25 – 30

30 – 35

35 – 40

(Frequency):

10

15

25

40

35

20

5

Ans: Calculation of QD
Midvalue (X)

Frequency
 
510
1015
1520
2025
2530
3035
3540

7.5
12.5
17.5
22.5
27.5
32.5
37.5

10
15
25
40
35
20
5

10
25
50
90
125
145
150

N=150

(iv) Calculate mean and median for the following distribution: 7
(No. of Firms):

10 – 19

20 – 29

30 – 39

40 – 49

50 – 59

60 – 69

70 – 79

(Production):

3

61

223

137

53

19

14

Ans:
Calculation of Mean
Frequency

MidValue
 
1019
2029
3039
4049
5059
6069
7079

3
61
223
137
53
19
14

14.5
24.5
34.5
44.5
54.5
64.5
74.5

43.5
1,494.5
7,693.5
6,096.5
2,888.5
1,225.5
1,043

= 510

= 20,485

Calculation of Median
C.B.

Frequency
 
1019
2029
3039
4049
5059
6069
7079

9.519.5
19.529.5
29.539.5
39.549.5
49.559.5
59.569.5
69.579.5

3
61
223
137
53
19
14

3
64
287
424
477
496
510

N=510

3. (a) (i) What is the range of coefficient of correlation? 1
Ans: + 1 to  1
(ii) Write the properties of coefficient of correlation. 3
Ans: Properties of r:
 r is the independent to the unit of measurement of variable.
 r does not depend on the change of origin and scale.
 If two variables are independent to each other, then the value of r is zero.
(iii) Given the two regression equationsand, find the coefficient of correlation betweenand. 5
Ans:
Assuming 1st equation is of Y on X and 2nd Equation is of X on Y:
(iv) Find the two regression equations from the data given below: 7
Ans:
Given,
Or
(b) (i) Whenthere is one regression equation. (Fill up the blank) 1
(ii) In a Bivariate data the sum of squares of the differences between the ranks of observed values is 231 and the rank correlation coefficient is – 0.4, find the number of pairs of items. 3
Ans: Given,
(iii) For a Bivariate data ofand, variance ofandare respectively 2.25 and 4.00, and, find the regression equation ofand. 5
Ans: Given,
Now,
Regression equation ofon
(iv) Calculate coefficient of correlation betweenandfrom the following data: 7
Ans:
(ii) Write the chief features of index number. 3
Ans: feature of index number:
1. Measures of relative changes: Index number measure relative or percentage changes in the variable over time.
2. Quantitative expression: Index numbers offer a precise measurement of the quantitative change in the concerned variable over time.
3. Average: Index number show changes in terms of average.
(iii) From the data given below, calculate 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:
CALCULATION OF LASPEYRE’S QUANTITY INDEX NUMBER
 
Commodity

Base Year

Current year

QOPO

Q1P0

QOP1

Q1P1
 
PO

QO

P1

Q1
 
A

5

50

10

56

250

280

500

560

B

3

100

4

120

300

360

400

480

C

4

60

6

60

240

240

360

360

D

11

30

14

24

330

264

420

336

E

7

40

10

36

280

252

400

360

SUM

1,400

1,396

2,080

2,096

Laspeyer’s Index Number:
(iv) Calculate Fisher’s price index number from the data given below: 7
Base Year

Current Year
 
Items

Price (in Rs.)

Quantity

Price (in Rs.)

Quantity

A
B
C
D
E
F

10
8
12
20
5
2

10
12
12
15
8
10

12
8
15
25
8
4

8
13
8
10
8
10

Ans:
CALCULATION OF FISHER’S INDEX NUMBER
 
Commodity

Base Year

Current year

POQO

P1Q1

POQ1

P1QO
 
PO

QO

P1

Q1
 
A

10

10

12

8

100

80

120

96

B

8

12

8

13

96

104

96

104

C

12

12

15

8

144

96

180

120

D

20

15

25

10

300

200

375

250

E

5

8

8

8

40

40

64

64

F

2

10

4

10

20

20

40

40

SUM

700

540

875

674

Or
(b) (i) GM is regarded as the best measure for the construction of index number. (Fill up the blank) 1
(ii) Discuss why Fisher’s index number is regarded as an ideal index number. 3
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) Give a comparative study of fixed base and chain base indices. 5
Ans: Difference between chain base method and fixed base method:
CHAIN BASE MEHTOD

FIXED BASED MEHTOD
 
1

No fixed base is there.

Base Period is fixed.

2

Immediately preceding period is taken as base.

Base period is arbitrarily chosen.

3

Calculation is too long.

Calculation is easy.

4

During Calculation if there is any error then the
Entire calculation is wrong.

This is not so in this method.

5

If data for any period is missing then subsequent chain indices cannot be computed.

This problem does not arise here.

(iv) Calculate Cost of living index number from the following data: 7
Items

Price of the Base Year

Price of the Current Year

Weight

Food
Fuel
Clothing
House Rent
Others

30
8
14
22
25

47
12
18
15
30

4
1
3
2
1

Ans:
CALCULATION COST OF LIVING INDEX NUMBER
 
Items

Base Year

Weight

I = Pn/P0 x 100

I.W
 
PO

Pn
 
Food

30

47

4

156.6

626.4

Fuel

8

12

1

100

100

Clothing

14

18

3

128.5

385.5

House Rent

22

15

2

68.18

136.36

Others

25

30

1

120

120

SUM

11

1368.26

5. (a) (i) Continuous price rise is an example of secular trend in a time series. (Fill up the blank) 1
(ii) Write a short note on graphic method of measuring trend in a time series. 3
Ans: Graphic method:  This is the simplest method of studying trend. The procedure of obtaining a straight line trend is:
a) Plot the time series on a Graph.
b) Examine the direction of the trend based on the plotted information.
c) Draw a straight line which shows the direction of the trend.
The trend line thus obtained can be extended to predict future values.
Merits:
i) This method is simplest method of measuring trend.
ii) This method is very flexible. I can be used regardless of whether the trend is a straight line or curve.
Demerits:
 This method is highly subjective because it depends on the personal judgement of the investigator.
 Since this method is subjective in nature it cannot be used for predictions.
(iii) Write how trends in a time series are measured by the method of moving averages. 5
Ans:
(iv) Calculate trend values for the data given below by using the method of least squares: 7
(Year):

1997

1998

1999

2000

2001

2002

2003

(Values):

30

45

39

41

42

46

49

Ans:
CALCULATION FOR STRAIGHT LINE TREND
 
YEAR

VALUE (Y)

t

t2

Yt
 
1997

30

3

9

90

= 41.71 + 0.214 (3) = 41.07

1998

45

2

4

90

= 41.71 + 0.214 (2) = 41.28

1999

39

1

1

39

= 41.71 + 0.214 (1) = 41.5

2000

41

0

0

0

= 41.71 + 0.214 (0) = 41.71

2001

42

1

1

42

= 41.71 + 0.214 (1) = 41.93

2002

46

2

4

92

= 41.71 + 0.214 (2) = 42.15

2003

49

3

9

147

= 41.71 + 0.214 (2) = 42.36

SUM

292

0

28

62

292

(iv) Calculate trend values for the data given below by using the method of least squares: 7
(Year):

1997

1998

1999

2000

2001

2002

2003

(Values):

30

45

39

41

42

46

49

Ans:
CALCULATION FOR STRAIGHT LINE TREND
 
YEAR

VALUE (Y)

t

t2

Yt
 
1997

30

3

9

90

= 41.71 + 2.214 (3) = 41.07

1998

45

2

4

90

= 41.71 + 2.214 (2) = 41.28

1999

39

1

1

39

= 41.71 + 2.214 (1) = 41.5

2000

41

0

0

0

= 41.71 + 2.214 (0) = 41.71

2001

42

1

1

42

= 41.71 + 2.214 (1) = 41.93

2002

46

2

4

92

= 41.71 + 2.214 (2) = 42.15

2003

49

3

9

147

= 41.71 + 2.214 (2) = 42.36

SUM

292

0

28

62

292

Or
(b) (i) Give an example of random fluctuations in a time series. 1
Ans: Irregular variations for example strike, lock out, flood.
(ii) Write a short note on trends in a time series. 3
Ans: 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.
(iii) Calculate trends by the method of 3 yearly moving averages from the data given below: 5
(Year):

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

(Production):

52

79

76

66

68

93

87

79

90

95

Ans: b. (iii) Calculate trend by the method 3 yearly moving average from the data given below.
Year

Production

3 yearly moving total

3 yearly moving average

1995
1996
1997
1998
1999
2000
2001
2002
2003
2004

52
79
76
66
68
93
87
79
90
95


207
221
210
227
248
259
256
264



69
73.66
70
75.66
82.66
86.33
85.33
88


(iv) Fit a straight line trend by the method of least squares and hence find the probable sale for the year 1988:
(Year):

1980

1981

1982

1983

1984

1985

1986

1987

(Sales):

12

13

13

16

19

23

21

23

Ans: Calculation of Straight Line Trend
Year

Production
 
1980
1981
1982
1983
1984
1985
1986
1987

12
13
13
16
19
23
21
23

– 7
– 5
– 3
– 1
1
3
5
7

49
25
9
1
1
9
25
49

– 84
– 65
– 39
– 16
19
69
105
161

140

0

168

150
