2015 (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 frequency of a class interval?
- Define a discrete variable.
- If
be the mean of
series,
the mean of
series and
, then show that
.
- The coefficient of rank correlation between
pairs of observations is
and the sum of the squares of differences between the ranks is
, then what will be the value of
- Is it possible for two regression equations to be
and
Explain, with reason, why?
- What do you mean by cost of living index number?
- What do you mean by ‘base year’ in the construction of index numbers?
- The trend equation for publicity cost (in ‘000 Rs.) of a company is given by
(Origin 1st July, 2005,
). Shift the origin to 1st July, 2012.
- Name the components of a time series.
- Write the differences between Laspeyres’ and Paasche’s indices.
- Given the annual trend line equation of production of a company as
2. (a) (i) Prove that for any two values
. When do they become equal? 3
. When do they become equal? 3
(ii) The mean and standard deviation of 100 observations are 50 and 5, and that of another 150 observations are 40 and 6 respectively. Find the combined standard deviation of this 250 observations. 5
(iii) Calculate the coefficient of standard deviation for the following data: 8
Marks: (less than)
|
80
|
70
|
60
|
50
|
40
|
30
|
20
|
10
|
No. of students
|
100
|
90
|
80
|
60
|
32
|
20
|
13
|
5
|
Or
(b) (i)Prove that for a variable, the sum of deviations of the observed values from their arithmetic mean is zero. 3
(ii) Find the mean daily wages from the following data: 5
Wages (Rs.)
|
230-240
|
240-250
|
250-260
|
260-270
|
270-280
|
280-290
|
No. of workers
|
8
|
20
|
40
|
18
|
10
|
4
|
(iii) Give the definition of semi-inter-quartile range. Mention one merit and one demerit of. 3
(iv) Why is standard deviation considered to be the best measure of dispersion? Explain. 5
3. (a) (i) State the properties of Karl Pearson’s coefficient of correlation. 3
(ii) What do you mean by regression lines? Explain why there should be two lines of regression. 5
(iii) Calculate Karl Pearson’s coefficient of correlation from the given data: 8
X:
|
35
|
42
|
20
|
50
|
72
|
64
|
Y:
|
40
|
48
|
24
|
60
|
84
|
68
|
Or
(b) (i) Discuss the uses of regression equations. 3
(ii) Calculate the coefficient of rank correlation from the data given below: 5
SL. No.
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
X:
|
48
|
33
|
40
|
9
|
16
|
65
|
24
|
18
|
44
|
20
|
Y:
|
13
|
10
|
24
|
6
|
15
|
4
|
20
|
9
|
10
|
19
|
(iii) Derive the equations of two regression lines for the following data: 8
X:
|
35
|
42
|
20
|
50
|
72
|
64
|
Y:
|
40
|
48
|
24
|
60
|
84
|
68
|
4. (a) (i) What are the different types of index numbers? Name each of them. 3
(ii) Calculate CLI from the given data: 5
Group
|
Index No.
|
Weight
|
Clothing
|
360
|
60
|
Food
|
298
|
5
|
Fuel and Lighting
|
287
|
7
|
House Rent
|
110
|
8
|
Miscellaneous
|
315
|
20
|
(iii) Using the following data prove that Fisher’s index number satisfy (1) time reversal test and (2) factor reversal test: 8
Commodity
|
Base Year
|
Current Year
| ||
Price (in Rs.)
|
Quantity.
|
Price (in Rs.)
|
Quantity
| |
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
|
Or
(b) (i) The monthly salary of an employee in the year 2010 was Rs. 18,500 and then the cost of living index was 160. In the year 2012 the cost of living index became 200. What should be the monthly salary of the employee in the year 2012 to rightly compensate him for price rise?
(ii) Discuss the limitations of index numbers.
(iv) What is Laspeyres’ price index number? Using the following data, show that Laspeyer’s formula does not satisfy the time reversal test: 8
Commodity
|
Base Year
|
Current Year
| ||
Price (in Rs.)
|
Quantity.
|
Price (in Rs.)
|
Quantity
| |
A
B
C
D
E
|
6
2
4
10
8
|
50
100
60
30
40
|
10
2
6
12
12
|
56
120
60
24
36
|
5. (a) (i) Write the two mathematical models used for analysis of time-series data. 3
(ii) Calculate the trend value by using 3 yearly moving averages for the following data: 5
Year:
|
2008
|
2009
|
2010
|
2011
|
2012
|
2013
|
Production:
|
77
|
88
|
94
|
85
|
91
|
98
|
(iii) Fit a straight line trend for the following data and estimate the production for the year 2010: 8
Year:
|
2002
|
2003
|
2004
|
2005
|
2006
|
2007
|
2008
|
Production (‘000 ton):
|
200
|
225
|
175
|
165
|
200
|
265
|
285
|
Or
(b) (i) What do you mean by seasonal variations? What are the uses of studying it? 3
(ii) Calculate 3 yearly weighted moving average with weights 2, 2, 1, 2, 2, 1, 2 for the following data: 5
Year:
|
1995
|
1996
|
1997
|
1998
|
1999
|
2000
|
2001
|
Value:
|
45
|
43
|
47
|
49
|
46
|
45
|
42
|
(iii) Using the method of simple averages, calculate the seasonal indices for the following data: 8
Quarter
|
2010
|
2011
|
2012
|
I
|
72
|
74
|
84
|
II
|
50
|
68
|
60
|
III
|
78
|
74
|
62
|
IV
|
92
|
90
|
76
|
(Old Course)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
1. (a) (i) Prove that the sum of the variate values from their arithmetic mean is zero. 2
(ii) If mean and mode of a moderately skewed distribution are 36.4 and 34.6 respectively, find its median. 3
(iii) What is standard deviation? Why is it regarded as the best measure of dispersion? 4
(iv) If arithmetic mean of the following distribution is 72.5, find the missing frequency and also calculate the mode for the distribution: 4+3=7
Marks:
|
30-39
|
40-49
|
50-59
|
60-69
|
70-79
|
80-89
|
90-99
|
No. of students:
|
2
|
3
|
11
|
20
|
--
|
25
|
7
|
Or
(b) (i) Discuss about the relationship between mean, median and mode. 2
(ii) Calculate the mode and median for the following data: 3
6, 4, 3, 6, 5, 3, 3, 2, 4, 3, 4, 3, 3, 4, 3, 4, 2, 2, 4, 3, 5, 4, 3, 4, 3, 3, 4, 1, 2, 3.
(iii) Calculate the coefficient of quartile deviation for the following data: 4
Wages (Rs.):
|
58-62
|
62-66
|
66-70
|
70-74
|
74-78
|
78-82
|
82-86
|
No. of Persons
|
4
|
7
|
11
|
16
|
12
|
8
|
2
|
(iv) Calculate the standard deviation for the following distribution: 7
Marks (more than):
|
0
|
10
|
20
|
30
|
40
|
50
|
60
|
No. of students:
|
80
|
77
|
72
|
68
|
53
|
40
|
0
|
2. (a) (i) Prove that, coefficient of correlation is the geometric mean of the two regression coefficients. 2
(ii) Find the value of
from the following data: 3
from the following data: 3
and
denotes the derivations of
and
series from their AM. 3
(iii) What do you mean by regression lines? Explain why there should be two lines of regression. 4
(iv) Find the appropriate regression equation from the data given below: 7
Age:
|
56
|
42
|
72
|
36
|
63
|
47
|
55
|
49
|
38
|
Blood Pressure:
|
147
|
125
|
160
|
118
|
149
|
128
|
150
|
145
|
115
|
Or
(b) (i) What are the properties of two regression coefficients? 2
(ii) If two regression equations are 
and
find the values of
and
. 3
andfind the values ofand. 3
(iii) From the following data, find the two regression equations: 4
X
|
Y
| |
Arithmetic Mean
Standard Deviation
|
20
5
|
25
4
|
Correlation coefficient 0.6
(iv) Calculate the value of coefficient of correlation from the following data: 7
X:
|
43
|
44
|
46
|
40
|
44
|
42
|
45
|
42
|
38
|
40
|
Y:
|
29
|
31
|
19
|
18
|
19
|
27
|
27
|
29
|
41
|
30
|
3. (a) (i) “The wholesale price index of the year 2015 taking 2010 as base year is 165.” What information can you derive from this statement about the rise in wholesale prices? 2
(ii) Write a short note on factor reversal test. 3
(iii) Calculate the cost of living index number from the following data: 4
Commodity
|
Weight
|
Index
|
Food
|
50
|
241
|
Clothing
|
2
|
221
|
Fuel
|
3
|
204
|
House Rent
|
16
|
256
|
Others
|
29
|
179
|
(iv) Calculate Fisher’s price index number from the data given below: 7
Commodity
|
Base Year
|
Current Year
| ||
Price (in Rs.)
|
Quantity.
|
Price (in Rs.)
|
Quantity
| |
A
B
C
D
E
|
25
20
30
12
90
|
30
22
54
20
15
|
30
22
33
15
19
|
35
25
64
25
18
|
Or
(b) (i) Write the differences between chain base and fixed base index numbers. 2
(ii) What are ‘weights’ in the construction of index numbers? Discuss the importance of weights. 3
(iii) Discuss the limitations of index numbers. 4
(iv) Calculate Fisher’s index from the data given below, and hence prove that Fisher’s index number satisfies factor reversal test: 7
Commodity
|
Base Year
|
Current Year
| ||
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
|
4. (a) (i) Write the two mathematical models used for the analysis of time series data. 2
(ii) Write a short note about the uses of studying time series. 3
(iii) Calculate 3 yearly moving averages from the data given below: 4
Year:
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
Value:
|
150
|
200
|
225
|
175
|
165
|
200
|
265
|
285
|
350
|
(iv) Using the principle of least squares, calculate the trend line equation for the data given below: 7
Year:
|
1985
|
1986
|
1987
|
1988
|
1989
|
1990
|
1991
|
Production:
|
23
|
20
|
18
|
18
|
14
|
13
|
13
|
Or
(b) (i) Given the annual trend equation
What is its monthly trend equation? 2
What is its monthly trend equation? 2
(ii) In a multiplicative model of time series analysis T = 100, S = 1.2, C = 1.04 and I = 0.9, find the value of y. 3
(iii) Discuss the limitations of time series analysis. 4
(iv) Using the method of 4 yearly moving averages, calculate the trend values for the following data: 7
Year:
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
Price:
|
20
|
21
|
23
|
22
|
25
|
24
|
27
|
26
|
28
|
30
|
5. (a) (i) What do you mean by business forecasting? 2
(ii) Write the differences between forecasting and estimation. 3
(iii) Discuss the assumptions of forecasting. 4
(iv) Write short notes on sales forecasting and demand forecasting. 7
Or
(b) (i) Discuss the uses of extrapolation as a method of forecasting. 2
(ii) What precautions are to be taken while forecasting? 3
(iii) Discuss the limitations of forecasting. 4
(iv) Discuss how regression analysis may be used for forecasting. 7