SLM_(1)

Old Element Row Intersecting Element Replaced Key Row New Element (2) (3) (4) (2) – (3) × (4) 18 10 18–3/10×10=15 3/l0 1 3/l0 – 3/10 × 1 = 0 1 3/10 0 1 – 3/l0 × 0 = 1 0 5/24 0 – 3/10 × 5/24 = –1/16 1/10 –1/12 1/10 – 3/10 (–1/12) = 1/8 Using the respective rules obtained in steps 7 and 8, he have (Table 4) as follows: Cj 45 55 0 0 P.M. Sol X XS S X1 10 X2 15 1 21 2 Zj 1275 1 0 5/24 –l/12 Zj – Cj 1275 0 45 1 –1/16 1/8 0 55 225/24 – 55/16 -45/12 + 55/8 0 9.38 – 3.44 = 5.94 6.88 – 3.75 =3.13 We get X1 = 10, X2 = 15 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 145 Therefore the solution is Profits = 45 (10) + 55 (15) = 450 + 825 = 1275 7.3.2 Problem: Solve by Simplex and Graphical Method. Maximise Z = 5x1 + 4x2 Subject to 3xI + 2x2 d 50 x1 + x2 d 22 Simplex Method Z = 5x1 + 4x2 + 0.S1 + 0.S2 3x1 + 2x2 + 1.S1 + 0.S2 = 50 X1 + X2 + 0.S1 + 1.S2 = 22 where, sl and s2 are constants-slack variables Basis x x s s bi ai/bij 50 50/33 = 1.66 1 2 1 2 22 22 s10 3* 2 1 0 s20 1 1 0 1 cj 5 4 0 0

soln 0 0 50 22 'j 5 4 0 0 n c1 = 5 z1 = 0 '1=5 s c2 = 4 z2 = 0 '2=4 c3 = 0 z3 = 0 '3=0 1 c4 = 0 z4 = 0 '4=0 1/3 Basis x1 x –1/3 s ai/bij x10 1 25 s20 0 2 0 2 bi m 16 cj 5 0 0 50/3 = 16.66 soln 25 2/3 –5/3 1 16/3 = 5.33 'j 0 1/3 * 0 4 16 0 2/3 0 CU IDOL SELF LEARNING MATERIAL (SLM)

146 Business Mathematics and Statistics Old row – row element in pivot column 1 – 1 – 1× 1 = 0 = 1/3 0 – 2/3 × 1 = -1/3 – = 1 1 – 1/3 × 1 = 16/3 22 z1 = 5 0 x 1 s bi z2 = 10/3 50/3 x 1 6 '1 =0 2 16 z '2 =2/3 x s '3= –5/3 –2 1 3 = 5/3 '4 =0 2 1 3 0 z4 = 0 0 1 x1 0 1 0 -2 Basis 1 1 –1 –2 6 x15 0 4 0 x24 5 16 0 cj 6 0 –1 soln 0 'j n Old row - row element in pivot column 1 2/3 -0 × 2/3 = 1/3 0 -1 × 2/3 = - –1 × 2/3 = 50/3 -3 × 2/3 = - 16 × 2/3 =

z1 = 5 D1=0 z2 = 4 D2=0 z3 = 1 D3=–1 z4 = 2 D4=–2 x1 = 6 x2 = 16 Z = Profit= 5 (6) + 4 (16) = 94 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 147 7.3.3 Graphical Method Intercept for Coordinates Constraints x2 axis x1 axis x2 axis xl axis [x, = 0] [x2 = 0] 3x1 + 2x2 = 50 x, + x2 = 22 25 50/3 = 16.66 (25,0) (0,50/3) (22,0) (0,22) 22 22 The line drawn with the above terminal points intersect at x1 = 6 x2 = 16 Z = 5(6) + 4(16) Z = 94 is the maximum value. 26 Scale: x-axis l cm = 2 units 24 y-axis 1 cm = 2 units 22 20 18 16 x2 14 12 10 8

6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 x 1 Fig. 7.2 CU IDOL SELF LEARNING MATERIAL (SLM)

148 Business Mathematics and Statistics Problem 6: A firm makes two types of furniture: Chairs and tables. The profit contribution from each product as calculated by the accounting department is ` 20 per chair and ` 30 per table. Both products are processed on the three machines M1 , M2 and M3 The time required by each product and total time available per week on each machine are as follows. Machine Chair Table Available Hours Per Week M 3 36 2 50 13 6 60 M2 5 M3 2 How should manufacture schedule his production in order to maximise profit. Solution: We formulate L.P.P. with the objective function and the constraint inequalities as under: Let X be the number of chairs to be manufactured. Y be the number of tables to be manufactured The L.P.P. can be formulated as maximise P = 20x + 30y Subject to 3x + 3y d 36

5x + 2y d 50 2x + 6y d 60 x,yt0 Let SI, S2, S3 be the slack variables them we have 3x + 3y + 1.51S1 + 0.5S2 + 0.5S3 = 36 5x + 2y + 0.5S1 + 1.5S2 + 0.5S3 = 50 2x + 6y + 0.5S1 + 0.5S2 + 1.5S3 = 60 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 149 Simplex table is: Cj Qty 20 30 0 00 xy S1 S2 S3 0 S1 36 33 1 00 0 S2 50 52 0 10 0 S3 60 26 0 01 Zj 0 00 0 00 Cj – Zj 20 30 0 00 Key Column Out going variable is S3 and in coming is X2 the row corresponding to X2 will be as under on dividing the key row by 6. (the key element) 30 X2 10 1/3 1 0 0 1/6 Also the row corresponding to Sl , will have the elements obtained as under: New number = Old no minus (Corresponding element in the key column x Corresponding number in key row) Row Sl Row S2 36–(3×10)=6 50–(2×10)=30 3–(3×1/3)=2 5 –(2×1/3)=13/3 3–(3×1)=0 2–(2×1)=0 1–(3×0)=1 0 –(2×0)=0

0–(3×0)=0 1–(2×0)=1 0–(3×1/6)=–1/2 0 –(2×1/6)=–1/3 Cj Prod Mix Qty Xl X2 S1 S2 S3 0 6 0 S1 30 20 I 0 –112 30 10 Zj S2 300 13/3 0 0 1 –1/3 Cj – Zj X 1/3 I 0 0 1/6 2 10 30 0 05 10 0 0 0 –5 n Key Column CU IDOL SELF LEARNING MATERIAL (SLM)

150 Business Mathematics and Statistics Cj Prod Mix Qty X1 X2 S1 S2 S3 20 Xl 3 1 -1/4 30 X2 9 0 0 Y2 0 1I4 0 S2 17 0 3/4 1 -1/6 0 0 -13/6 Zj 330 20 30 5 0 5/2 Cj – Zj 0 0 -5 0 -5/2 ? X1 = 3 units X2 = 9 units and profit P = 20x + 30y = 20(3) + 30(9) 60 + 270 = 330 7. Profits = ` 330 Special Cases in Linear Programming  Four special cases and difficulties arise at times when using the graphical approach to solving LP problems  = Infeasibility = Unboundedness = Redundancy = Alternate Optimal Solutions No feasible solution

 No feasible solution  Exists when there is no solution to the problem that satisfies all the constraint equations No feasible solution region exists This is a common occurrence in the real world Generally one or more constraints are relaxed until a solution is found CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 151 Example: Unboundness Case  Unboundedness  Sometimes a linear program will not have a finite solution

In a maximization problem, one or more solution variables, and the profit, can be made infinitely large without violating any constraints In a graphical solution, the feasible region will be open ended This usually means the problem has been formulated improperly CU IDOL SELF LEARNING MATERIAL (SLM)

152 Business Mathematics and Statistics Example of Unboundness case Redundancy Case  Redundancy 

A redundant constraint is one that does not affect the feasible solution region One or more constraints may be more binding This is a very common occurrence in the real world It causes no particular problems, but eliminating redundant constraints simplifies the model CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 153 Example of Redundancy Alternate Optimal Solutions  Alternate Optimal Solutions 

Occasionally two or more optimal solutions may exist Graphically this occurs when the objective function’s isoprofit or isocost line runs perfectly parallel to one of the constraints This actually allows management great flexibility in deciding which combination to select as the profit is the same at each alternate solution CU IDOL SELF LEARNING MATERIAL (SLM)

154 Business Mathematics and Statistics Example of Alternate Solutions (2) Multiple Solution Multiple solutions of a linear programming problem are solutions each of which maximize or minimize the objective function under Simplex Method. Under Simplex Method, the existence of multiple optimal solutions is indicated by a situation under which a non-basic variable in the final simplex table showing optimal solution to a problem, has a net zero contribution. In other

words, if at least one of the non-basic variable in the (Ci±=j) row of the final simplex table has a zero value, it indicates that there is more than one optimal solution. = Unbounded Solution Sample If we consider Maximize (x + y) Subject to x–yet1 x+yt2 x, y t 0 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 155 The feasible region is as follows: Objective Function 2 Feasible Region 1 23 In this case, you can see we can move as much as we want the objective function in the growing sense of x and y coordinates without leaving the feasible region Therefore, objective function can grow too into feasible region, so we are in an unbounded solution case for this problem. 4. Redundant Constraints

A redundant constraint is a constraint that can be removed from a system of linear constraints without changing the feasible region. Consider the following system of nonnegative linear inequality constraints and n variables (m t n): AX d b, X t O ZKHUH$H Rmxn, b H Rm, X H Rn, and O H Rn /HW$iX d bi be the ith constant of the system (2.1) and let S = {X HRn/ AiX d bi, X t O} be the feasible region associated with system (2.1). /HW6k = {X H Rn / AiX d bi, X t O, i = k} be the feasible region associated with the system of equations AiX d bi, i = 1, 2…, m, i = k. The kth constraint AkX d bk (1 < k < m) is redundant for the system (2.1) if and only if S = Sk. CU IDOL SELF LEARNING MATERIAL (SLM)

156 Business Mathematics and Statistics Definition 2.1.Redundant constraints can be classified as weakly and strongly redundant constraints. Weakly Redundant Constraints The constraint AiX < biLVZHDNO\\UHGXQGDQWLILWLVUHGXQGDQWDQG$iX = biIRUVRPH;HS. Strongly Redundant Constraints The constraint AiX < biLVVWURQJO\\UHGXQGDQWLILWLVUHGXQGDQWDQG$iX < biIRUDOO;HS. Binding Constraint Binding constraint is the one which passes through the optimal solution point. It is also called a relevant constraint. Nonbinding Constraint Nonbinding constraint is the one which does not pass through the optimal solution point. But it can determine the boundary of the feasible region. Example 2.2:Consider the following linear inequality constraints: z 2x1 + 1x2 d 8, z 4x1 + 0x2 d 15, z 1x1 + 3x2 d 9, z 1x1 + 2x2 d 14,

z 1x2 d 4 z 1x1 + 1x2 d 5, ZKHUH In Figure 1, the region OABCD is the feasible region and the vertex C is the optimal point. The constraints (1), (2), (3) and (6) are binding, (4) and (5) are strictly redundant. The 2nd constraint is non-binding. Among the binding constraints, (6) is weakly redundant. CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 157 Figure 1 7.3.5 The Concept of Duality: Given a linear programming problem it is possible to find its replica which is also a linear programming problem. The given L.P is called the ‘Primal’ and its replica is known as Dual. As the two problems are replicas of each other, it follows that when the primal problem is of the maximization type, the dual would be of the minimmisation type and vice versa. Interpretation of the Primal Problem:

Given each z = unit values, of the output Cj and also given the upper limit in regard to the availability of each input bj, then we have to determine how much of each output Xj be produced so as to maximize the value of the total output. Interpretation of the Dual problem: If the values of each input (bi) are given along with a lower limit of the unit value of each output Cj then we are required to determine the unit values that should be assigned to each input yJ so as to minimize that value of the total input. 7.3.6 Problem: Maximise P = 6x1 + 7x2 Subject to 2x1 + 3x2 d 12 CU IDOL SELF LEARNING MATERIAL (SLM)

158 Business Mathematics and Statistics 2x1 + x2 d 8 x1 + x2 t 0 As explained above the dual of this L.P is: D = 12y1 + 8y2 2y1 + 2y2 t 6 3y1 + y2 t 7 y1, y2 t 0 7.3.7 The Primal Dual Relationship can be shown as follows: Primal Dual Maximise Minimise P = 6x1 + 7x2 C = 12y1 + 8Y2 Subject to 2x1 + 3x2 d 12 2y1 – 2y2 t 6 2x1 + 1x2 d 8 3y1 + 1 y2 t 7

X1, X2 t 0 Y1, Y2 t 0 The above can be represented in a matrix form as under: Primal Dual §X1· §Y1· Minimise = [6, 7] ¨ ¸ Minimise D = [12, 8] ¨ ¸ ©X2¹ ©Y2¹ Subject to [2, 3] [x1] < [12] [2 2] [y1] > [6] [8] [3, 1] [y2] > [7] [2, 1] [x2] < 7.4 Summary linear Programming is a method used by the managers/authorities to make decision in business/ enterprise. the subject of Operation Research is concerned with the analysis, condensation and presentation of data for the process of decision-making. LP is one of the mathematical techniques involves the optimisation of a function which is called the objective function.

CU IDOL SELF LEARNING MATERIAL (SLM) Linear Programming 159 LP problems can be solved by using one of the following methods: (i) The Graphic Method (ii) The simplex Method The graphic method is to solve the problem by identifying it and draw the graph with decision variables and the objective function along the constraints. The optimal solution can be obtained. The simplex method: It is a systematic procedure (analgorithm) for solving LP Problems. 7.5 Key Words/Abbreviations Simplex Procedure Terminology as shown in Steps 1 to 17, given in 7.3. 7.6 Learning Activity Work out examples given in 7.7 after studying the related illustrative examples. .................................................................................................................................................. ..................................................................................................................................................

7.7 Unit End Questions (MCQ and Descriptive) (A) and (B) Descriptive Type: Short Answer Type Questions 1. What is meant by ‘Linear Programming’ ? Explain its uses and limitations. 2. Explain the major assumption in ‘Linear Programming’. 3. Describe the various steps involved in the formulation a L.P. problem. 4. Explain the following terms : (a) Objective function (b) Constraints (c) Non-negativity condition (d) Feasible solution 5 Explain the various steps in solving a L.P. problem involving two decision variables, by the graphical method.

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160 Business Mathematics and Statistics 6. What are the steps involved in the simplex method for solving linear programming problems? 7. A home decorator manufactures two types of lamps, Alpha and Delta. Both these lamps require the services of a cutter and a finisher. Alpha requires 3 hours of cutter’s time and 2 hours of finisher’s time. Delta requires 2 hours of cutter’s time and 1 hour of finisher’s time. The cutter has 180 hours and finisher 110 hours time each month. If one Alpha gives a profit of `10 and a Delta a profit of ` 7, formulate this as a L.P. problem. 8. A firm makes two types of furniture : chairs and tables. The contribution for each product as calculaed by accounting department are ` 20 per chair and ` 30 per table. Both products are processed on three machines M1, M2 and M3. The time required for each product and total time available per week on each machine are as follows: Machine Chair Table Available hours M1 3 3 36 M2 5 2 50 M3 2 6 60 9. A multiplant company has three manufacturing plants, A, B and C are two markets X and Y. Production cost per piece in A, B and C is ` 1,500, ` 1,600 and ` 1,700 respectively. Selling price of X and Y are ` 4,700 respectively. Demand for X and Y is 3500 and 3600 pieces respectively. Unit transportation costs are as follows : From To XY A 1000 1500 B 2000 3000 C 1500 2500

Formulate a Linear Programming model. 10. Solve the following I.P.P. by graphical method. Minimize C = x1 + x2 Subject to x1 + x2 t 12 5x1 + 8x2 t 74 x1 + 6x2 t 24 x1 , x2 t 0 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 161 ? Two products L and K are processed on 3 machines m1, m2 and m3. The processing time per unit, machine availability and profit per unit are as follows: Machine Processing Time in Hrs Availability in Hrs L K m1 2 3 1500 m2 3 2 1500 m3 1 1 1000 Profit per unit in ` 10 12 Formulae a L.P.P. and solve it by samplex method 12. Minimize C = 2x + y, by graphical method. Subject to 3x + y = 3 4x + 3y t 6 x + 2y d 3 x,yd0 13. State the dual of: Minimize Z = 50x1 – 80x2 + 140x3 Subject to x1 – x2 – 3x3 t 4 5x1 – 2x2 – 232 t 3 x1 , x2 , x3 t 0 = A company manufacures two products. The basic time data, machine capacity and profit contribution is given in the table below :

Machine Machine hrs. Required Machine hrs. Product I per unit Available per week Lathes Product II Milling 11 80 12 120 PP Profit per unit (`) 6 8 Formulate the problem as a linear programming problem and solve it graphically to determine the quantity of each product to be manufactured to maximize profit. CU IDOL SELF LEARNING MATERIAL (SLM)

162 Business Mathematics and Statistics ? A company makes two kinds of leather belts. Belt A is of high quality and belt B is of lower quality. The respective profits are ` 4 and ` 3 per belt. Each belt of type A requires twice as much time as a belt of type B and if all the belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts (both A and B combined). Belt A requires a fancy buckle and only 400 such buckles are available per day. There are only 700 buckles a day available for type B. Determine the number of bels to be purchased for each type so as to maximize profit. Solve graphically. 7.8 References References of this unit have been given at the end of the book. ˆˆˆ

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Introduction to Statistics 163 UNIT 8 INTRODUCTION TO STATISTICS Structure 8.0 Learning Objectives 8.1 Introduction 8.2 Definition and Scope of Statistics 8.3 Statistical Methods and Applied Statistics 8.4 Statistics and Mathematics 8.5 Statistics and Economics 8.6 Statistics and Econometrics 8.7 Use of Statistics in Trade, Commerce, Industry and Decision-making

8.8 Uses and Limitations of Statistics 8.9 Statistics, Statisticians and Citizens 8.10 Summary 8.11 Key Words/Abbreviations 8.12 Learning Activity 8.13 Unit End Questions (MCQ and Descriptive) 8.14 References 8.0 Learning Objectives After studying this unit, you will be able to:  Explain the meaning, significance and the historical background.   Grasp the essentials with reference to some definition and scope of the subject and as well as the distinction between statistical methods and applied statistics. CU IDOL SELF LEARNING MATERIAL (SLM)

164 Business Mathematics and Statistics  Differentiate the relationship between statistics and other subjects: Mathematics, Economics etc.  Analyse the use of statistics in trade, commerce, industry and decision-making.   Describe the uses and limitations of statistics.   Judge the role of statistics, statisticians and citizens.   On the basis of the overall study of this unit, students can prepare a brief summary. 8.1 Introduction to Statistics In a broad sense, statistics was born when man first began to count and express his ideas and sensible facts in quantitative terms — that is, in numbers. Civilization’s progress could have been held at bay but for the discovery of numbers, for indisputably its growth through the centuries has been nurtured by numbers. The use of statistics dates back to ancient times when the Pharaohs and the Hebrews took censuses of population and wealth from time to time. Ancient kings maintained records on population, wealth, area of land, crop yields, livestock, births and deaths etc. in their kingdoms. As a result, statistics came to be known as the ‘Science of Kings’. Statistics were collected even in ancient India, during the reign of the Mauryan and the Gupta kings. According to Kautilya’s treatise on statecraft, the Arthashastra; the Mauryan kings undertook the task of collecting statistics on population, agriculture etc. from time to time. The Mughal emperors, too, maintained statistics of population, land, agriculture etc.; the Ain-i Akbari, for instance, provides statistical details during the reign of Emperor Akbar (1556-1605). The Latin word ‘Status’ and the Italian word “Statista” both mean a political state and it appears that the word statistics has been derived from both of them. The definition of statistics

became broader and broader as it developed from century to century. The emergence or statistics in the modern form is largely due to the influence, keen interest and work of great mathematicians of the eighteenth and nineteenth centuries. At present, statistics is used in all fields of human activity and has come to be regarded as indispensable to the study of many sciences, especially the social sciences. CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Statistics 165 8.2 Definition and Scope of Statistics When used in the singular, the word ‘statistics’ means the subject of statistics. In the plural, it means quantitative information or numerical data. Various authors defined statistics in different ways, emphasizing various viewpoints. A.L Bowley, for instance, has given four definitions: “Statistics is the science of the measurement of social organisms guarded as whole in all its manifestations.” “Statistics may be called the science of counting.” “Statistics are numerical statements of facts in any department of enquiry, placed in relation to each other.” “Statistics may rightly be called the science of averages.” Each of these definitions stresses only one viewpoint or aspect of the variety and richness of statistics, for it is not only concerned with applications to sociology, averages, counting or figures but with many other things. Hence, these definitions have their own limitations. Bowley also defines statistics as the ‘Science of estimates and probabilities.” While there’s no doubt that in a large number of cases it deals with estimates and probabilities, but again these are not the only methods or aspects of statistics. Webster (Dictionary Definition) defines statistics as “Classified facts respecting the condition of people in the state — especially those facts which can be stated in a table or tables of numbers or in any tabular or classified arrangement.” This definition is inadequate as it stresses only on one aspect of statistics: classification of facts — in a table or tables of numbers-regarding the condition of people in the state. It does not mention its applications in other fields like biology, physics, sociology, economics, meteorology or astronomy.

According to Udney Yule: “By statistics we mean quantitative data affected to a marked extent by multiplicity of causes.” This definition, too, emphasizes on only one aspect of statistics, omitting its other important characteristics. W.I. King defines it thus: “The science of statistics is the method of judging collective natural or social phenomena from the results obtained by the analysis of an enumeration or collection of estimates.” Though this definition is a remarkable one, it could have emphasized the all-pervading or all-embracing aspects of statistics. CU IDOL SELF LEARNING MATERIAL (SLM)

166 Business Mathematics and Statistics According to Horace Secrist: “By statistics we mean aggregates of facts affected to a marked extent by multiplicity of causes numerically expressed, enumerated or estimated according to reasonable standards of accuracy, collected in a systematic manner for a predetermined purpose and placed in relation to each other.” This definition is comprehensive and exhaustive and covers most of the aspects of statistics. It’s certainly very close to the definition of modem statistics but yet, it could have been broader. In this modern scientific, technological, industrial, space-conquering, planet-probing and atomic age, statistics and statistical techniques have entered a new epoch and are being increasingly and advantageously used in every sphere of human knowledge. Just as a few mathematical equations symbolize the unknown depths of hidden facts, so also statistical facts provide a good deal of information. And with the growing realization of the indispensability of statistical knowledge — it being all-pervading and all-embracing — its definition will broaden and glitter at the hands of capable statisticians. However, it must be accepted that it is not easy for any capable statistician to come up with a definition of statistics that will remain relevant for ever. But talented authors and statisticians are coming up with more elegant, refined, comprehensive and exhaustive definitions from time to time. The modem view of statistics is that it is not just the art of collecting, sorting, classifying, grouping, summing up and using numerical facts. It is also ‘a science that provides the tools and techniques that can be used advantageously to measure and analyse facts in any sphere of human knowledge’. Though reputed authorities on the subject have given remarkable definitions, one more may be added to them: “Statistics, in the singular, is the subject that explains all the devices of collection, presentation, analysis and inference of numerical facts and figures in any field of knowledge, and in the plural, stand for any sensible statements of numerical facts which can be analysed, interpreted, compared and related if possible.” 8.3 Statistical Methods and Applied Statistics

Statistics is divided into two main divisions: statistical methods and applied statistics. Statistical methods or the theory of statistics is also known as mathematical statistics as it developed mainly through mathematics. The theory of statistics has the mathematical theory of probability as its CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Statistics 167 basis. Statistical methods deal with the formulation of devices or general rules of procedure that can be employed in handling different types or forms of data. For instance, the rules for the collection of data, classification, tabulation, analysis, comparison by averages and the methods of finding the coefficient of correlation, regression equations and coefficients and index number construction are all statistical methods. The theory of statistics is further divided into statistical inference and the design of experiments. Statistical inference involves drawing inferences or valid conclusions from a sample of any population. These conclusions are concerned with the specification of the population from the information supplied by the sample. Design of experiments is concerned with designs for the collection and analysis of data. Experiments are conducted on the basis of these to test and verify the validity of hypotheses. Therefore, selecting the design is of the utmost importance: a wrong one inevitably leads to fallacious and absurd conclusions in the course of experimentation. Applied statistics deals with practical applications to business and economic data by statistical methods to tackle concrete problems like population, agriculture, industry, trade and transport, wages and prices of commodities. Economic statistics, which is a branch of applied statistics, deals with statistics or prices, production, income, expenditure, investment, sales profits etc. Applied statistics has several branches. Some or these are: ( Business and financial statistics: These are statistics of public finance, business money, banking, currency and exchange. ( Population statistics: These form a part of demography, which is a collective and detailed statistical study of human life with reference to population growth and vital and social aspects. ( Agricultural statistics: These deal with area, yield of crops, livestock etc. ( Industrial statistics. ( Labour and employment statistics.

( Administrative statistics. ( Social statistics. ( Trade and transport statistics. ( Price statistics. ( National income statistics. CU IDOL SELF LEARNING MATERIAL (SLM)

168 Business Mathematics and Statistics Statistics of consumption, production, exports, imports, prices and wages help in framing economic policies. Economic growth is determined to a considerable extent by certain statistical measure like index numbers etc. 8.4 Statistics and Mathematics The science of statistics owes its development and progress to the mathematical theory of probability. Statistical methods or mathematical statistics have their origins in mathematics and owe a great deal to mathematicians. One of the first to sow the seeds of the idea of probability was Jerome Cardano (1501-1536), a gambler and mathematician, who came up with rules to minimize risks in gambling and precautions against cheating in his book ‘Liber De Ludo Aleae’. Another step in this direction was taken in the year 1654, when another gambler and amateur mathematician, Chevelier-de-Mere, communicated a problem in a game of dice to the French mathematician Pascal Pascal in turn passed it on to Fermat, another French mathematician. Later, these ideas were taken forward by mathematician. Jacob Bernoulli (1654-1705), who first explained the law of large numbers in his book, ‘Ars Conjectandi’ and another mathematician Daniel Bernoulli (1700-1782), who expressed the idea of ‘mortal expectation’. They laid the first foundations of the modem theory of probability. Years later, mathematicians like Laplace (1749-1827), Gauss (1777-1855) and S. Poisson (1781-1840) made significant contributions to probability and statistical theory. In this manner, eminent men — mathematicians, statisticians, social scientists and research workers in diverse fields-contributed to the development, enrichment and progress of the theory of statistics over the generations. The following are some of the notable men who contributed to the growth and development of statistics: Euler, Lagrange. De Moivre. Knapp, Lexis, Morgan, Chrystal, Bayes, Charlier, August Mcitzen, Davenport Francis, Edgeworth, Francis Galton, Karl Pearson, Udney Yule, W.W. Gossett, A.L. Bowley, Adams, W.I. King, W. Pearson and R.A. Fisher.

8.5 Statistics and Economics There is a great deal of inter-relationship and interdependence between statistics and economics. Economists have to use mathematical methods and statistical procedures to make more intensive, accurate and detailed studies of economic problems. According to economist Alfred Marshall: “Statistics are the straw out of which I, like every other economist, have to make bricks.” CU IDOL SELF LEARNING MATERIAL (SLM)