Sunday, November 05, 2017

Business Statistics Solved Question Papers: Nov' 2016

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
  1. What do you mean by statistical unit?
Ans: statistical unit is the unit of observation or measurement for which data are collected or derived. 
  1. 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.
  1. If two variablesandare related asand, then what will be the value of?
Ans: Given,
  1. If the coefficient of correlation betweenandis 0.67, then what will be the coefficient of correlation betweenand?
  2. If the correlation coefficient between two variablesand is +1 and, then find the value of.
Ans: Given,
  1. 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
  1. 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.
  1. 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.
  1. 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


  1. Write the formula for Fisher’s ideal index number.
Ans:
  1. If the two regression lines areand, then find the arithmetic mean ofand.
Ans:
  1. 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
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

(35)
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
(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
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

= 510

= 20,485


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:
  1. r is the independent to the unit of measurement of variable.
  2. r does not depend on the change of origin and scale.
  3. 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:

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
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:-
  1. It considers both base year and current year’s price and quantity.
  2. It satisfies both time reversal and factor reversal test.
  3. It is based on Geometric mean which is theoretically considered to be the best average of constructing index number.
  4. 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:-
  1. This method is highly subjective because it depends on the personal judgement of the investigator.
  2. 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

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