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|=|x-Mean|
|
f|d|
|
0-10
10-20
20-30
30-40
40-50
50-60
60-70
|
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 Co-efficient
of variation
0-10
10-20
20-30
30-40
40-50
50-60
60-70
70-80
|
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
Mid-value (X)
|
Frequency
|
||
5-10
10-15
15-20
20-25
25-30
30-35
35-40
|
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
|
Mid-Value
|
||
10-19
20-29
30-39
40-49
50-59
60-69
70-79
|
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
|
||
10-19
20-29
30-39
40-49
50-59
60-69
70-79
|
9.5-19.5
19.5-29.5
29.5-39.5
39.5-49.5
49.5-59.5
59.5-69.5
69.5-79.5
|
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
|
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: 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
|
