Year – 2013 November (New Syllabus)
1. (a) Answer the following questions: 1x5=5
(i) Which average is considered to be best for the construction of index number?
Ans. GM is considered to be the best average for the construction of index number.
(ii) Which is the GM of 5, 10, 20, 0 and 100?
Ans. Calculation of GM is not done by using ‘0’. It is applied only for + ve value.
(iii) Write the relationship among AM, GM and HM.
Ans.
(iv) When rank correlation used?
Ans. When Qualitative information is given, then spearmen’s rank correlation is used.
(v) Write the relationship among fisher’s index, Laspeyre’s index and Paasches’ index.
Ans.
(b) Fill up the blanks: 1x3=3
- The index number for the base year is taken as 100.
- When r = ± 1, the number of regression line is one.
- Flood in Assam is an Example of Irregular variation in time series.
2. (a) (i) State the features of a good measure of average. 3
Ans: The following are the important properties which a good average should satisfy
- It should be easy to understand.
- It should be simple to compute.
- It should be based on all the items.
- It should not be affected by extreme values.
- It should be rigidly defined.
- It should be capable of further algebraic treatment.
(ii) If the AM of the following distributions is 67.45, find the value of the missing frequency: 5
Height
|
Frequency
|
60-62
|
5
|
63-65
|
54
|
66-68
|
----
|
69-71
|
81
|
72-74
|
24
|
Ans: Given, AM = 67.45
Height
|
Frequency
(
 ) |
Mid Value
(
 ) |  |  |  |
60-62
63-65
66-68
69-71
72-74
|
5
54
81
24
|
61
64
67
70
73
|
– 6
– 3
0
3
6
|
– 2
– 1
0
1
2
|
– 10
– 54
0
81
48
|
Total
|
164 +
|
65
|
(iii) Calculate the coefficient of variation of the following data: 5+2=7
Weight
|
No. of persons
|
115-125
|
4
|
125-135
|
5
|
135-145
|
6
|
145-155
|
3
|
155-165
|
1
|
165-175
|
1
|
Ans: Calculate the Co-efficient of variation of the following data.
Class interval
|
Frequency
(
) |
(
) |  |  |  |  |
115-125
125-135
135-145
145-155
155-165
165-175
|
4
5
6
3
1
1
|
120
130
140
150
160
170
|
– 3
– 2
– 1
0
1
2
|
9
4
1
0
1
4
|
– 12
– 10
6
0
1
2
|
36
20
6
0
1
4
|
Total
|
20
|
– 25
|
67
|
Coefficient of Variation = sd/Mean*100 = (13.3/137.5)*100 = 9.67%
Or
(b) (i) For any two values, prove that AM≥GM≥HM. 3
Ans: 
(ii) Calculate mode and median for the data given below: 5
Marks Less than
|
(No. of students)
|
10
|
15
|
20
|
35
|
30
|
60
|
40
|
100
|
50
|
150
|
60
|
220
|
70
|
245
|
80
|
270
|
Ans: Calculation of Median and Mode
Marks
|  |  |
0-10
10-20
20-30
30-40
40-50
50-60
60-70
70-80
|
15
20
25
40
50
70
25
25
|
15
35
60
100
150
220
245
270
|
(iii) An analysis of the monthly wages paid to the works in two departments A and B f a company gave the following result. Find the combined standard deviation of the wages of the workers of the company as a whole: 7
Department A
|
Department B
| |
No. of persons
|
60
|
20
|
Average wages
|
Rs. 648
|
Rs. 584
|
Standard deviation
|
4
|
5
|
Ans:
3. (a) (i) Prove that the correlation coefficient is the GM of the two regression coefficients. 3
Ans: 
(a linear relation of correlation and regression coefficients). So we find the correlation co-efficient is G.M. of regression coefficients.
(a linear relation of correlation and regression coefficients). So we find the correlation co-efficient is G.M. of regression coefficients.
(ii) Explain why there should be two lines of regression. 5
Ans: Two regression lines: - We know that there are two lines of regression: - x on y and y on x. For these lines, the sum of the square of the deviations between the given values and their corresponding estimated values obtained from the line is least as compared to other line. One regression line cannot minimise the sum of squares for both the variables that is why we are getting two regression lines. (We get one regression line when r = +1 and Two regression lines will be at right angles when r = 0.)
(iii) Calculate the coefficient of correlation from the following data: 7
∑X = 125, ∑Y= 100, ∑X2 =650, ∑Y2 =460, ∑XY =508, N=25.
Ans:
Or
(b) (i) Write the two regression equations. 3
Ans: There are two regression lines:
(ii) Regression equations of two correlated variables X and Y are 5X – 6Y + 90=0 and 15X – 8Y – 130=0. Find which equation is the regression equation of Y on X and Which one is for X on Y. Also find means of X and Y. 5
Ans:
For finding correlation, we are find the value of byx and bxy. Again in the given two equation which one is rents for y on x is not show let us assume the equation (i) is mean for x on y and (ii) is mean for y on x.
(iii) Find out the value of Y when X = 36 from the data given below: 7
X
|
Y
| |
Mean
Standard Deviation
|
30
4
|
45
10
|
Correlation coefficient = +0.8
| ||
Ans:
4. (a) (i) Discuss the relative merits and demerits of Laspeyre’s and Paasche’s indices. 3
Ans: Merits:
1. Easy to calculate.
2. It can be easier and cheaper to produce since the only quantities required are for the base period in case of laspeyre’s method and current period in case of paasche’s method.
Demerits:
1. Both methods does not satisfy time reversal and factor reversal test.
2. Both method cannot be used if quantities are unobtainable.
(ii) During a certain period when the cost of living index goes up from 110 to 200, the dearness allowance of an employee was also increased from Rs.325 to Rs.500. Does the worker really gain? If so, by how much? 4
Ans:
Year
|
(Exp)
CLI
|
Salary
|
1st
2nd
|
110
200
|
325
500
|
(iii) Using Fisher’s formula, calculate price index number from the data given below: 7
2005
|
2012
| |||
Items
|
Price
|
Quantity
|
Price
|
Quantity
|
A
B
C
D
|
12
18
21
25
|
5
4
3
2
|
15
22
18
20
|
6
5
4
3
|
Ans:
CALCULATION OF INDEX NUMBER
| ||||||||
PO
|
QO
|
P1
|
Q1
|
PO QO
|
PO Q1
|
P1 QO
|
P1 Q1
| |
A
|
12
|
5
|
15
|
6
|
60
|
72
|
75
|
90
|
B
|
18
|
4
|
22
|
5
|
72
|
90
|
88
|
110
|
C
|
21
|
3
|
18
|
4
|
63
|
84
|
54
|
72
|
D
|
25
|
2
|
20
|
3
|
50
|
75
|
40
|
60
|
SUM
|
245
|
322
|
257
|
332
| ||||
Fisher’s Price Index Number:
Or
(b) (i) Describe the various types of Index numbers. 3
Ans: There are three types of index number:
a) Price Index Number: A measure reflecting the average of the proportionate changes in the prices of a specified set of goods and services between two periods of time. Usually a price 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 prices compared with the base period. A quantity index is built up from information on prices of various commodities.
b) Quantity Index Number: 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.
c) Value Index Number: An index number formed from the ratio of aggregate values in the given period to the aggregate values in the base period is called value index number.
(ii) The following series of index numbers were constructed with the year 2000 as base year. Form a new set of index number with the year 2005 as base year: 4
Year
|
2001
|
2002
|
2003
|
2004
|
2005
|
2006
|
Index No.
|
105
|
118
|
125
|
130
|
150
|
156
|
Ans:
Calculation of index number by shifting base year from 200 to 2005
Year
|  |  |  |
2001
2002
2003
2004
2005
2006
|
105
118
125
130
150
156
|
150
150
150
150
150
150
|
105/150 x 100 = 70
118/150 x 100 = 78.67
125/150 x 100 = 83.33
130/150 x 100 = 86.67
150/150 x 100 = 100
156/150 x 100 = 104
|
(iii) Calculate Cost of Living Index number from the data given below and hence suggest what should be the salary of a person whose salary in the base year was Rs.500 to maintain his living status: 5+2=7
Items
|
Index No.
|
Weight
|
Food
Clothing
Fuel and lighting
House Rent
Miscellaneous
|
360
295
287
110
315
|
60
5
7
8
20
|
Ans: Calculation of Cost of Living index
Items
|
Index No.
(P)
|
Weight
(W)
|
I.P
|
Food
Clothing
Fuel & lighting
House Rent
Miscellaneous
|
360
295
287
110
315
|
60
5
7
8
20
|
21,600
1,475
2,009
880
6,300
|
Total
|  = 100 |  = 32,264 |
Now, Actual salary of the person should be = (500*322.64)/100 = 1,613.20
5. (a) (i) Discuss the uses of studying time series. 3
Ans: Utility of Time Series Analysis
The analysis of Time Series is of great significance not only to the economist and businessman but also to the scientist, geologist, biologist, research worker, etc., for the reasons given below:
- It helps in understanding past behaviors: By observing data over a period of time one can easily understanding what changes have taken place in the past, Such analysis will be extremely helpful in producing future behavior.
- It helps in planning future operations: Plans for the future cannot be made without forecasting events and relationship they will have. Statistical techniques have been evolved which enable time series to be analyzed in such a way that the influences which have determined the form of that series to be analyzed in such a way that the influences which have determined the form of that series may be ascertained.
- It helps in evaluating current accomplishments: The performance can be compared with the expected performance and the cause of variation analyzed. For example, if expected sale for 1995 was 10,000 refrigerators and the actual sale was only 9,000, one can investigate the cause for the shortfall in achievement. Time series analysis will enable us to apply the scientific procedure for such analysis.
- It facilitates comparison: Different time series are often compared and important conclusions drawn there from. However, one should not be led to believe that by time series analysis one can foretell with 100percnet accuracy the course of future events.
(ii) From the following data, calculate trend values by using the method of 3-yearly moving averages: 4
Year
|
2001
|
2002
|
2003
|
2004
|
2005
|
2006
|
2007
|
Production
|
100
|
120
|
95
|
105
|
108
|
110
|
120
|
Ans: Calculation of 3 – yearly Moving Average
Year
|
Value
|
3 yearly moving total
|
3 yearly moving average
|
2001
2002
2003
2004
2005
2006
2007
|
100
120
95
105
108
110
120
|
-
100 + 120 + 95 = 315
120 + 95 + 105 = 320
95 + 105 + 108 = 308
105 + 108 + 110 =323
108 + 110 + 120 = 338
-
|
-
315/3 = 105
320/3 = 106.67
308/3 = 102.67
323/3 = 107.67
338/3 = 112.67
-
|
(iii) What do you mean by trends in a time-series analysis? What are the factors responsible for the occurrence of trends? What are the uses of studying trends? 7
Ans: A time series is a set of statistical observations arranged is chronological order. Time series may be defined as collection of magnitudes of some variables belonging to different time periods. It is commonly used for forecasting.
The four components of time series are:
1. Secular trend
2. Seasonal variation
3. Cyclical variation
4. Irregular variation
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.
Seasonal variation: Seasonal variation are short-term fluctuation in a time series which occur periodically in a year. This continues to repeat year after year. The major factors that are responsible for the repetitive pattern of seasonal variations are weather conditions and customs of people. More woolen clothes are sold in winter than in the season of summer .Regardless of the trend we can observe that in each year more ice creams are sold in summer and very little in Winter season. The sales in the departmental stores are more during festive seasons that in the normal days.
Cyclical variations: Cyclical variations are recurrent upward or downward movements in a time series but the period of cycle is greater than a year. Also these variations are not regular as seasonal variation. There are different types of cycles of varying in length and size. The ups and downs in business activities are the effects of cyclical variation. A business cycle showing these oscillatory movements has to pass through four phases-prosperity, recession, depression and recovery. In a business, these four phases are completed by passing one to another in this order. It has four important characteristics: i) Prosperity ii) Decline iii) Depression iv) Improvement
Irregular variation: Irregular variations are fluctuations in time series that are short in duration, erratic in nature and follow no regularity in the occurrence pattern. These variations are also referred to as residual variations since by definition they represent what is left out in a time series after trend, cyclical and seasonal variations. Irregular fluctuations results due to the occurrence of unforeseen events like floods, earthquakes, wars, famines, etc.
Utility of Time Series Analysis
The analysis of Time Series is of great significance not only to the economist and businessman but also to the scientist, geologist, biologist, research worker, etc., for the reasons given below:
- It helps in understanding past behaviors: By observing data over a period of time one can easily understanding what changes have taken place in the past, Such analysis will be extremely helpful in producing future behavior.
- It helps in planning future operations: Plans for the future cannot be made without forecasting events and relationship they will have. Statistical techniques have been evolved which enable time series to be analyzed in such a way that the influences which have determined the form of that series to be analyzed in such a way that the influences which have determined the form of that series may be ascertained.
- It helps in evaluating current accomplishments: The performance can be compared with the expected performance and the cause of variation analyzed. For example, if expected sale for 1995 was 10,000 refrigerators and the actual sale was only 9,000, one can investigate the cause for the shortfall in achievement. Time series analysis will enable us to apply the scientific procedure for such analysis.
- It facilitates comparison: Different time series are often compared and important conclusions drawn there from. However, one should not be led to believe that by time series analysis one can foretell with 100percnet accuracy the course of future events.
Or
(b) (i) Write the two models used for analysis of time series. 3
Ans: Ans: - In Traditional time series analysis, it is ordinarily assumed that there is a multiplicative relationship between the components of time series. Symbolically, Y=T X S X C X I
Where T= Trend
S= Seasonal component
C= Cyclical component
I= Irregular component
Y= Result of four components.
Another approach is to treat each observation of a time series as the sum of these four components Symbolically, Y=T + S+ C + I
(ii) What is seasonal variation in a time series? Discuss the uses of studying seasonal variation in business. 4
Ans: Seasonal variation: Seasonal variation are short-term fluctuation in a time series which occur periodically in a year. This continues to repeat year after year. The major factors that are responsible for the repetitive pattern of seasonal variations are weather conditions and customs of people. More woolen clothes are sold in winter than in the season of summer .Regardless of the trend we can observe that in each year more ice creams are sold in summer and very little in Winter season. The sales in the departmental stores are more during festive seasons that in the normal days.
(iii) Using the method of least squares, find the trend values for the following data: 7
Year
|
2001
|
2002
|
2003
|
2004
|
2005
|
2006
|
2007
|
Income
|
67
|
53
|
43
|
61
|
56
|
79
|
58
|
Ans:
(iii) Using the method of least square find the trend value of the following:
YEAR
|
VALUE (Y)
|
X(t)
|
X2
|
XY
|
Trend Values
|
2001
|
67
|
-3
|
9
|
-201
| |
2002
|
53
|
-2
|
4
|
-106
| |
2003
|
43
|
-1
|
1
|
-43
| |
2004
|
61
|
0
|
0
|
0
| |
2005
|
56
|
1
|
1
|
56
| |
2006
|
79
|
2
|
4
|
158
| |
2007
|
58
|
3
|
9
|
174
| |
417
|  = 0 |
28
|
36
|
6. (a) (i) State the assumptions under which business forecasting is carried out. 3
(ii) Discuss how forecasting is done by regression analysis method. 4
(iii) Prepare a note why a business manager should use forecasting methods. 7
Or
(b) (i) Discuss the limitations of business forecasting. 3
(ii) Discuss the economic models of business forecasting. 4
(iii) Discuss the qualities of a good method of forecasting. 7