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## Friday, November 03, 2017

### Business Statistics Solved Question Papers: November' 2013

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
1. The index number for the base year is taken as 100.
2. When r = ± 1, the number of regression line is one.
3. 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
1. It should be easy to understand.
2. It should be simple to compute.
3. It should be based on all the items.
4. It should not be affected by extreme values.
5. It should be rigidly defined.
6. 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.
(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:
1. 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.
2. 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.
3. 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.
4. 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:
1. 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.
2. 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.
3. 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.
4. 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