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
a) 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.
b)
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.
d) If the coefficient of
correlation between
and
is 0.67, then what will be the coefficient of correlation
between 2x and 5y?
Ans: Since
Correlation coefficient is independent of the origin and scale, so it is not
affected by addition or subtraction or multiplication or division. In the given
question, value of correlation coefficient will be same in each case.
h) 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.
l)
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
C.I.

Frequency

Mid Value

fx

d=xMean

fd

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
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
(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

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:
i)
r is the independent to the unit of
measurement of variable.
ii)
r does not depend on the change of origin and
scale.
iii)
If two variables are independent to each
other, then the value of r is zero.
Or
(b)
(i) When r = + 1, there 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
4.
(a) (i) Fisher’s index number is the GM mean of Laspeyres and Paasche’s
indices. (Fill up the blank) 1
(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

(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:
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.
(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:
i) This
method is highly subjective because it depends on the personal judgement of the
investigator.
ii) 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: Method of moving average: Under this
method the average value for a certain time span is secured and this average is
taken as the trend value for the unit of time falling at the middle of the
period covered in the calculation of the average. While using this method it is
necessary to select a period for moving average.
The following steps must be followed to
calculate moving average:
a) First of all select the period for
moving average.
b) Find the average of the period
selected. Average will be placed in the middle of the given period.
c) Thereafter, calculate the average
after leaving one year.
d) This process will be continued till
the end.
(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

t^{2}

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: Calculation of Three yearly moving
average:
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
