# Business Statistics Solved Question Papers: Nov' 2016

2016 (November)
COMMERCE
(General/Speciality)
Course: 303
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: 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
 (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 (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:
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 