 So, we will continue our discussion on relationship between different economic variables given a quantification or through different methods graph or the mathematical equation. So, if you remember in the last class we have started discussion about the derivative of various function, how to solve the various function then we introduce the optimization technique where we did two type of optimization one is the maximization of revenue or maximization of profit and second one is the maximization of the revenue or maximization of the profit and second one is the minimization of the cost. So, whenever we are doing this optimization technique either it is a maximization or it is a minimization problem, we did not consider the case of a constraint and we just optimize the maximization of a profit function or we just optimize the minimization of a cost function. So, today we will discuss the optimization technique with a constraint either in the form of the income or in the form of the cost when it comes to a cost and when it comes to the revenue either it is maximization or it is a minimization of the cost. So, in case of constant optimization this is a technique used for achieving a target under constant situation or condition is called constant optimization. So, maybe the motivation for optimization is remain same achieving a target either to maximize the profit or to minimize the cost, but here the difference is that there is a constraint along with the objective function and how to do this constraint optimization we generally discussed two type of technology, we will talk about the substitution technique and later on we will take the Lagrangian multiplier method. So, taking the substitution technique it can be applied to the problem of profit maximization or it can be for the cost minimization. For a profit maximization one of the variable express in term of the other variables and solve the constraint equation for obtaining the value of one variable. Suppose there are two variables x and y, so the best way for solving it through the substitution technique is represent one variable with the other variable and then you solve for that variable and finally, you substitute the value of one variable in term of what you have solved to the other variable. And here the value is obtained is substituted in the objective function which is maximized or solve for obtaining the value of the other variables. So, whatever the value is obtained by substituted it will be again substituted back in the objective function which is maximized and solve for obtaining the value for the other variable. So, we will see how we use this substitution technique in case of a profit optimization problem and in case of a cost minimization. And how this is different for cost minimization? Maybe the method again remains same, the constraint equation is expressed in term of any one of these two goods of the variables and the equation is obtained from step one is substituted in the objective function. So, whether it is a cost function, whether it is a profit function, the basic rule for the substitution technique is that we express one variable in term of the other variables, we get the value of one variable and finally, again substitute back to the objective function. So, we will just take an example how generally we do the constant optimization along with the constant with the objective function, whether the objective function is a profit maximization or whether the objective function is a cost minimization. So, we will take a case of profit maximization first and in case of profit maximization we will maximize the profit. So, here profit is equal to 100 x minus 2 x square minus x y plus 180 y minus 4 y square. This is the profit function and the profit maximization, here the optimization problem is the profit maximization. Since we are saying that this is the case of a constant optimization, there is also a constant attached to this and the constant is in the form of x plus y is equal to 30. So, now what is the optimization problem? The optimization problem is maximization of profit function with respect to the constant that is x plus y which is equal to 30. Now, how we will do this? The first step is we will express x in term of y or we can express y in term of x and after the getting the value of x or y, again we will substitute this value of x and y in the profit function. So, now we will see let us substitute the value of x and y before it converting into another term or may be the profit maximization problem. So, suppose x plus y is equal to 30. So, this can be rewritten as x is equal to 30 minus y. So, in this case x we are representing in term of y or y can be 30 minus x. So, substituting the value of x and y in the profit equation what we can get? Pi is equal to 100, it was 100 x. So, x we are representing in term of y. So, this is 30 minus y plus 2, 30 minus y square because it was 2 x square minus 30 minus y y because it was x y plus 180 y minus 4 y square. So, if you look at the profit function now all the terms in term of y there is no x over here in the case of the profit function. Now, again if you will simplify this then this comes to 1200 plus 170 y minus 5 y square. So, what we did the first step is this where we represent x in term of y. Now, substituting the value of x in the form of y in the profit equation which gives the profit equation which is equal to 1200 plus 170 y plus minus 5 y square. Now, to find out the value of y what we have to do? We have to take the derivative of pi with respect to y and which we need to set equal to 0. So, if you are taking this then this comes as the first order derivative because for any maximization minimization rule in order to get the value always the first order derivative has to be equal to 0. So, del pi by del y is equal to 0 which is like 1200 plus 170 y minus 5 y square which is equal to 0. Now, solving this will give you 170 minus 10 y which is equal to 0 or may be minus 10 y is equal to minus 170 and y is equal to 17. Now, what is our x? x is equal to 30 minus y. So, this is equal to 30 minus 17 which is equal to 13. So, we get a value y which is equal to 17, we get a value of x which is equal to 13. Now, putting the value of y and x in our profit equation, we get profit which is equal to 2800. So, here how we maximize the profit with respect to a cost constant or with respect to a value of x and y? The first step is always to represent one variable in term of the other variable. So, in this case what we did? We represented x in term of y and after represent the value of one variable in term of the other variables, then we put the value in the objective function. So, if you remember in the previous slide what I am showing that we represent the profit function only in term of variable y. Then after getting the profit function, we took the first order derivative equal to 0 in order to get the value of y and through that we got the value of y which is equal to 17 and from there we got the value of x which is equal to 13. Now, putting the value of x and y in the original profit function, we get a profit which is equal to 2800. So, by substitution technique following the two step, we got the profit, we got the value of x and we got the value of y. Now, we will see through substitution technique how we can do a cost minimization problem. So, here what is the optimization problem? The optimization problem is to minimization of the total cost. Now, what is total cost? Let us take total cost is equal to 2 x square minus x y plus 3 y square. Now, the firm here what is the constant? The firm has to get 36 units of x and y as the combined order. Now, what is the optimum combination? Optimum combination is to what should be the minimum cost to produce this 36 unit of x and y. So, in this case what should be the constant? The constant is again if you look at x plus y is equal to 36. So, optimization problem is to minimize the total cost with respect to or may be subject to x plus y is equal to 36. Now, following the substitution technique what is the first step? The first step we have to represent one variable in term of the other variable. So, x is equal to 36 minus y because if you remember the first step substitution technique is always representing one variable in term of the other variable. So, here x is equal to 36 minus y. Now, putting the value of x in the cost equation 2 x square. So, this is 2 36 minus y square x and y. So, this is 36 minus y y plus 3 y square. So, this is 3 y square. So, if you again simplify this, this comes to 2592 minus 252 minus 252 minus 180 y plus 6 y square. So, after putting the value of x in the cost function in term of y, we get a total cost function which is equal to 2592 180 y plus 6 y square. Now, in order to get the value of y and in order to get the optimum combination or the optimum cost, what we have to do? We have to take the first order derivative of total cost function with respect to y and we have to set it equal to 0 in order to get the value of y. So, now we have to take a derivative of the first order derivative of 2592 minus 180 y plus 6 y square and this has to be equal to 0. So, if you do this, then we get the value 180 plus 12 y which is equal to 0 which you further simplify, then it is minus 12 y is equal to minus 180 and y is equal to 15. And if y is equal to 15, then x is equal to 36 minus y which is equal to 21. So, y is equal to 15, x is equal to 21. Now, this is the optimum combination. Now, this is the optimum combination. The firm should produce 15 unit of y and 21 unit of x and this is the optimum combination for the firm. Now, what is the next best task for us? The next best task for us is to whether producing this combination, the firm is incurring the minimum cost of production or what should be the minimum cost to produce this combination. So, for that what we need to do? We need to put the value of y, we need to put the value of x in the cost equation and we need to find out the minimum cost. So, what was your cost equation? The cost equation is 2 x square minus x y plus 3 y square. So, putting the value of x is equal to 21 and y is equal to 15, this comes to 882 minus 315 plus 675 which is equal to 1 to 4 2. So, this is the minimum cost what the firm incurs in order to produce 15 unit of y and 21 unit of x. So, what is the optimization problem here? The optimization problem here is to minimize the cost with a constraint that at any cost the firm has to produce 36 unit of both the goods that is x and y. So, this is the optimum combination for the firm and this is the minimum cost to produce the optimum combination of the firm. Next, we will see the second method too for this constant optimization and that is the Lagrangian multiplier method. So, apart from substitution technique the most popular or may be the most commonly used technique to do a constant optimization is always a Lagrangian multiplier method. So, what is Lagrangian multiplier method? It is again one kind of method to solve the constant optimization and it involves combining of both the objective function and the constant equation and solving by using the partial derivative methods. Basically, it takes the partial derivative with respect to both the variables and then it gets the value of x and y and you know by getting the value of x and y it maximizes or the profit or minimizes the cost. So, we will see how it works for the Lagrangian methods. Let us take a profit maximization case. Suppose the profit equation is 100 x minus 2 x square minus x y plus 1, 180 y minus 4 y square. Again, the profit maximization of the constant subject to x plus y is equal to 30. The same profit equation what we took for the substitution technique and the same constant what we took for by using the substitution technique method. So, x plus y is 30 that is constant and profit is what we take for the substitution method. Now, how it is different from the other method? In case of other method, we are substituting the value of x and x for y or y for x. Here we will not do that, but rather we will use partial derivative method to solve this profit maximization problem. In this case what we do? So, x plus y is equal to 30. So, we will find another variable that here that is x plus y minus 30 is equal to 0 and lambda x plus y minus 30. So, now, we will reframe the objective function using the adding a Lagrangian multiplier over here. And what is the Lagrangian multiplier here? That is lambda x plus y minus 30. This is the another term what we are getting here. So, what is our new profit function? New profit function is 100 x minus 2 x square minus x y plus 180 y minus 4 y square minus 4 y square. This is our original profit function. Along with that we add a Lagrangian multiplier that is lambda x plus y minus 30. So, if you look at now the constant also we have added in the objective function. So, this is our Lagrangian function. Lagrangian function comes from the constant and what we add in the objective function in order to maximize the profit. Now, this is the profit function now. Now, we have to find out the value of unknowns over here. What are the unknowns over here? The unknown is x, the unknown is y and the unknown is lambda. So, we need to solve for the value of x. We need to solve for the value of y. We need to solve for the value of lambda. Now, how we will do that? We will take a partial derivative of the profit function with respect to x, we will take a partial derivative with respect to profit with respect to y and we will take a partial derivative with with respect to lambda, which is the Lagrangian function or which one is the Lagrangian multiplier. So, we will take the first one that is l may be first derivative of this with respect to x. So, this we will get as 100 minus 4 x minus y plus lambda, which is equal to 0. And let us call it the equation 1. Similarly, for the second one, we have to take the derivative with respect to y. So, this comes as lambda. So, this as x plus 180 minus 8 y plus lambda, which is equal to 0 and this is equation 2. The third unknown is with respect to lambda. So, this is del l pi with respect to lambda that gives you x plus y minus 30 and this is equation 3. Now, if you make a summation and if you can make it 2 equation, then it comes to 100 minus 4 x minus y plus lambda is equal to 0. And if you add the second 2 equation, this is 180 minus x minus 8 y plus lambda is equal to 0. So, if you do a subtraction from 1 to 2 over here, then you would get minus 80 minus 3 x plus 7 y plus 0, which is equal to 0. So, if you look at what we did over here, we basically in order to get the value of x and y, we got a got 2 equation, may be 2 joint equation in order to get the value of the unknown. So, in the first case, this is 100 minus 4 x minus y plus lambda and second case, it is 180 minus x minus 8 y plus lambda, which is equal to 0. If you subtract the second one from the first one, we get minus 80 minus 3 x plus 7 y plus 0, which is equal to 0. Now, if you look at the equation 3, x plus y minus 30, this is our equation 3, we will multiply 3 with this equation. So, this is x plus y minus 30, which is equal to 0. So, if you multiply 3 in equation 3, this comes to 3 x plus 3 y minus 90 equal to 0. And what was your previous equation 1 we did for this, this is minus 3 x plus 7 y and minus 80. And if you take this again, this comes to minus 3 x plus 7 y minus 80, which is equal to 0. From these 2 equations, if you sum it, then it comes to 10 y minus 170, which is equal to 0 and y is equal to 17. So, we got the first unknown value, that is y is equal to 17. Now, we can get the value of x from here, because x plus y is equal to 13. So, from that we can get the value x, which is equal to 13. Now, we can get a value from the, so this is our first unknown, this is our second unknown or third unknown is lambda. So, from the value of x and y, we can get the value of lambda and lambda will come to minus 31. So, we have the, we know the value of all these 3 unknown. Once we put it in the value in the profit equation, we get the profit and we can see whether the profit is maximum or not. Similarly, using this Lagrangian method, we can also solve a costumination problem where the optimization problem is to minimize the cost. So, let us look at to the cost minimization problem. Now, the cost function over here is 100 x square plus 150 y square and this is subject to x plus y, which is equal to 500. Now, Lagrangian method, what is the first step? The first step is to get the Lagrangian function from the constant equation, then again form a cost function adding the Lagrangian multiplier or the Lagrangian function. So, in this case the Lagrangian function is lambda 500 minus x minus y. So, this is our Lagrangian function. Taking this, what is our Lagrangian cost function? That is 100 x square plus 150 y square plus lambda 500 minus x minus y. So, if you look at the Lagrangian function, now how we have again three unknowns that is x, y and lambda. In order to find that what we will do, we will follow the same maybe formula what we did for the profit maximization. We will find out del L c with respect to x, we will find out del L c with respect to y and we will find out the del L c with respect to lambda. After getting the equation, again we can get the value of x as 1.5 y. So, that comes to y is equal to 200 and x is equal to 300 and from there we can get the value of the cost and we can get the value of the Lagrangian multiplier. So, what is the essential difference between the substitution technique and the Lagrangian technique? We use both the methods to solve the constant optimization problem and what is a constant optimization problem? A constant optimization problem is 1, where we maximize the profit function or the minimize the cost function with a constant that is in the form of the other variable. So, in case of substitution technique what we do? We substitute the, we represent the value of one variable in term of the other variables and then we substitute that value in term of the other variables in the objective function, whether it is a profit function or whether it is a cost function. Then we solve for it and we get, when we get one value, we represent that in term of others and get the final value of the other variable also and in case of Lagrangian multiplier method, we form a Lagrangian function on the basis of a constant. Add that to our objective function and solve the objective function through the partial derivative method in order to know the value of unknowns and what are the unknowns over here? Unknowns always in term of the variable 2, variable those are in the objective function, here typically in the case of the x and y. So, we started our discussion for this typical topic, we started our discussion with the relationship between different variables, whether it is linear, collinear, linear, maybe non-linear or curvilinear. Then we discussed different function attached to different kind of function, linear, non-linear and the curvilinear. Then we show, then I think we discussed the method of how to solve the different functions and then we talked about the optimization problem, where we maximize the profit and minimize the cost and today's class we have talked about the optimization with a constant using both the method that is Lagrangian method and the substitution technique. So, now we will move to a new kind of technique, regression technique that is generally used again to understand the relationship between the two variables, two economic variables and what are the things what we are going to discuss in case of a regression technique. We will talk about a nature of managerial problem and here there again the use of function, how we use the regression technique. Then we see how to formulate a function so that we can estimate with the regression technique. Then we do an estimation of a linear function using the regression technique and also we will see the multivariate regression and few test in the multivariate regression estimate. So, before going to the regression techniques and method to solve or method to get the value through regression technique, now we will understand what is regression. So, regression is a statistical technique used to qualify the relationship between the interrelated economic variable. So, we know that economic variables are interrelated and there are number of methods may be through graph, through function or through may be the mathematical relationship. We always explain the value of the economic variable. So, regression technique is a mathematical technique what basically quantify the relationship between the two variables. So, in general sense we always say that two variables they are positively related or the negatively related. So, regression technique is one step ahead of this and it gives the exact magnitude of relationship between two variables that how they are related even if they are related positively or negatively, what is the extent of relationship, what is the magnitude of relationship. So, regression technique that is where regression is a statistical technique or the mathematical technique used to quantify the relationship between the interrelated economic variable. It is generally used in physical and social studies where the problem of specifying the relationship between two or more variable is involved. So, either it used in the physical or it is used in the social studies and particularly in this case where the relationship is whether the may be the problem is to specifying the relationship or specifying the magnitude of relationship between two or more variables. Now, here in the regression technique what we do? We do the estimation of coefficient of the independent variable and also we do the measurement of the reliability of the estimated coefficient. So, using the regression technique the first step is to estimate the coefficient of the independent variables and the second one is the measurement of the reliability of the estimated coefficient. So, before getting into the regression technique let us understand may be in what kind of managerial problem we need to use the regression technique or where is the case where we use the regression technique in order to understand the relationship between the two variables. Let us take an example, suppose a manager spending money on advertisement to promote sales of his firm's product. So, manager is spending money on advertisement on promotional activity to promote the sale of his firm's product. Reacting to this spending money on advertisement sales has been increasing, but not continuously. Sometimes it is increasing more, sometimes it is increasing less and sometimes it is constant. Now, here what is the managerial decision problem or what is the nature of the managerial problem? The managerial problem here is to find an answer to is there a positive relationship between the advertisement expenditure and the total sales. Because the manager is spending a good amount of money on the advertising expenditure the first question always comes whether it is affecting positively to the total sales. Second one if at all it is affecting positively to the total sales, then what is the major or relationship or what is the quantitative response to the sales to an increase in the advertising expenditure. So, first one is how they are related and that always we have discussed and always we can represent in the different form. Here the focus is the second one what is the major of the relationship or what is the quantitative response of sales to an increase in the advertisement expenditure. So, this how we can do maybe we take the data from the in a specific time period or maybe we take a time series data and there we can say how the advertisement expenditure and the sales they are related. So, if you look at in the table the first one gives a timeline in term of the year second one is the advertising expenditure and third one is the sales. So, if you look at in different years the advertising expenditure there is some amount of money it is going on increasing or sometimes it is if it is look at it is going on increasing from 5 to 8, 8 to 10, 10 to 12 and so on and so forth. And the sales also it is going on increasing, but if you look at the increase in the sales is not maybe on a increasing manner sometimes it is increasing, sometimes it is increasing less, sometimes it is increasing high or sometimes it is increasing maybe it is just constant. So, if you look at between the year 1998 and 1999 the sales is remain constant even if the maybe the expenditure has increased or the expenditure has decreased. So, in this case in order to understand this may be how to plot it or how to may be presented this kind of relationship. So, may be a more clear and certain answer to this question can be find out by plotting a sales start against the advertisement expenditure. So, whatever we have shown in the previous graph may be previous table may be we can represent that in a graph and we can say that whether there is a clarity that what kind of relationship between the advertisement expenditure and the sales whether it is positive whether it is negative that is the that is getting address the first question. And the second one is how they are related like whether what is the magnitude of relationship whether the advertisement expenditure is increasing and along corresponding with that there is more increase in the sales less increase in the sale or what is the percentage change in the sales. So, will if you plot this in a scatter diagram. So, may be we take sales on the x axis and advertisement expenditure on the y axis if you can plot it then it is 2, 4, 6, may be 8, 10, 12, 14, 16. Similarly, we can say 40 may be 45, 50, 55, 60, 65. So, now if you look at may be you take a specific year specific year may be the sales is 6, sales is 8, sales is 10 or sales is that and this is there is a small change over here may be we are taking the ad expenditure over here and we are taking the sales over here. So, if you can plot it then may be this is a point where the advertisement expenditure is 4 and the sales is 40. Similarly, may be we are taking one point here that is 8 and 50, one point over here that is 10 and 55. So, if you look at each combination gives us a combination between the or each point gives us a combination between the advertisement expenditure and the sales. So, similarly may be this is a point where it is 10 is 55 or we get a combination over here that is 14 and 60. So, if you draw a line over here this is basically the regression line. So, regression line is used for what may be this regression line what we can say or this is the line where there are different combination of sales and advertisement expenditure. It is not clear that all the point will be on the same line there is possibilities that there may be one combination where the advertisement expenditure is 12 and the sales is 60. So, in that case this is the combination that is suppose 12 and 60. So, if you look at this combination is this point is not line on the regression line. So, if you put it in a scatter diagram. So, this is what this representation is generally called as a scatter diagram and if you put this in a scatter diagram it shows a relationship between two variable and again it is not sure that when the advertisement expenditure moves from 2 to 4 or 4 to 6 or 8 to 10 what happens to the exactly the change in the sales from 40 to 45, 45 to 50 or 55 to 55. What is the exact nature of change between the advertisement expenditure and the sales. So, scatter diagram with just we are doing a graphical representation of what we presented on the table and it gives the different may be combination and each combination gives you a data on the advertisement expenditure and the sales. So, the primarily how the manager solve this problem primarily manager solve this problem by plot ticket in a scatter diagram and then they generally say that how they are related. But the question comes here that whether through the scatter diagram or whether through this table or whether through this real data real information we get the exact nature or exact relationship between the two variable and in this case typically the advertisement expenditure and the sales. May be the answer is no because it is not shows the exact relationship between the advertisement expenditure and the sales of future planning. Neither scatter diagram not the real life examples or the not the real life data generally gives this answer. So, this question can be answered by the regression technique and particularly in this case we use the regression technique and regression technique gives the exact magnitude of relationship, the exact relationship between the two variables that those are in question may be it is in this case may be this the advertisement expenditure and the sales. Now, for the regression technique again there are two steps. So, we have already understand what is a regression technique. So, regression technique is one which gives a quantification to the relationship between two economic variable and how they do that they follow two steps for that one. Initially they formulate a hypothesis and it is done on the basis of the observed relationship between two or more facts or the events of real life and second one they translating the hypothesis into a function and finally, they evaluate the function in order to get the value of the data. So, there are two steps one is formulation of the hypothesis and second one is the transforming or the translating the hypothesis into a function. So, we will see the first one how there is a how generally the hypothesis is being form or how there is a formulation of the hypothesis. So, hypothesis if you look at it is not a may be a relationship between the two economic variable rather it is a estimated relationship between two variable at it is a probability that these two variables can react in a certain way, but the result is not known here the result is unknown here. Suppose, we are saying here for example, advertising expenditure and sales growth they are positively related. So, this is a hypothesis this is a statement on the basis of the observed data. We do not know the what is the outcome or what can be the result whether it gives a positive relationship or whether it gives a negative relationship. So, this hypothesis is a one kind of verbal statement from where generally we formulate a hypothesis we formulate a function we evaluate the function in order to note the exact relationship between these two variables. So, we will start with a formulation of hypothesis generally how we formulate a hypothesis. So, here what we take the hypothesis we take the hypothesis sales data is positively related to advertisement expenditure. This is the hypothesis for the hypothesis for that what the background observation we have to take or what the background may be the information we have to take. Now, what is hypothesis as I was telling it is may be a verbal statement about the relationship between two variables. So, it is a postulate it is a untested proposition we have not yet tested regarding the relationship between two or more variables in the real world phenomena. It shows only the probability of the event and serve as a guide for the future action, but cannot predict the result of an action. So, as we are discussing the outcome is not known over here it can always take what be the probability of the event like in this case if the advertisement expenditure increases what will happen to the sales. So, the outcome of the event is unknown only it can predict that since the both of them they are positively related when you increase the advertisement expenditure the possibility is that that the sales will also increase. So, this is a hypothesis is a postulate untested proposition regarding the relationship between two variable in a real life phenomenon. Now, as we are just formulating the hypothesis suppose you take the hypothesis sales growth is a positive function of the advertisement expenditure. Now, on the basis of that the hypothesis suggest that if a firm spend money on advertisement its sales will most probably increase. Again, it is a probability it is there is no certainty that if the firm is spending money on the advertisement the sales has bound to increase. There is a probability that the sales will increase if the firm is spending money on the advertisement. It does not convey the approximate increase in the sales for a given advertisement expenditure. So, hypothesis need to be converted into mathematical equation or stated in the form of a estimable function. Since it is a probability of event we need to support it through the data we need to support it through the function in order to check whether the hypothesis is validated or not or how the hypothesis goes through or not for that we need to convert it to a mathematical equation or convert it in the form of the estimable function. So, there it comes to the second step and what is the second step? The second step when you translate the hypothesis into a function. So, it is a hypothesis as I was mentioning it is a verbal statement on the basis of some observed relationship between two variables. So, in the second step we translate the hypothesis into a function we translate the verbal hypothesis into the form of a estimable function and to formulate the relationship to formulate the verbal hypothesis into a estimable function we need to identify what kind of relationship is there between the dependent and the independent variable. So, the relationship between the dependent variable and the independent variable need to be specified and it is in the form of an equation. So, the form of equation can be linear it can be non-linear depending on the relationship. So, hypothesis can be translated as suppose in this case we take a equation that is y is equal to a plus b x where y is the sales the advertising expenditure a and b are constant. So, y is here dependent variable x is independent variable. So, y is the sales x is the advertising expenditure a and b are the constant. The constant a over here is the intercept and it gives the quantity of sales without advertisement when x is equal to 0. So, there are two constant one is constant a another is constant b constant a is the intercept it gives the quantity of sales without advertisement when x is equal to 0 and constant b is the coefficient of y in relation to x and it gives the measure to increase in sales due to certain increase in the advertisement expenditure. So, if you look at b is directly related to the value of x it gives the measure to increase in the sales due to certain increase in the advertisement expenditure. So, b is the slope a is the intercept a gives the quantity of sales without advertisement when x is equal to 0 and b is the coefficient of y in relation to x and it gives the measure to increase the sales due to certain increase in the advertisement expenditure. Now, tasks of analysis come here to find out the value of a and b because a is unknown over here b is unknown over here a gives us a value of sales when there is no advertisement expenditure and b gives us the value of the slope which tells us that if advertisement expenditure increases by certain proportion what is the exact proportion change in the sales. So, the task of analyst is to find out what is the value of a and what is the value of b there are two methods to find out the value of a and b one is the rudimentary method or the elementary method and the second one is the mathematical method or the regression technique. So, we will see through using the elementary method or using the rudimentary method how to find the value of a and b and then again we will see the mathematical method or the regression technique using that how to find the value of a and b. So, using in order to find the value of a and b through the elementary method we will use again the same data of the what we are taking from the managerial problem that is the in the form of what we present in the form of the scattered diagram and then we will find the value of a and b over there. So, what we take in the y axis and what we take in the x axis, x axis we take the advertisement expenditure and y axis we takes the sales. So, this is 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 or 20 and here we take maybe 40, 45, 50, 55, 60, 65, 75, 70 and we will have the current 70, 75, 80 and 85 and 90. So, since we have to suppose to the value of may be intercept it starts from here and we take a regression line suppose this is r and suppose this is l. So, if you take this may be r is 1 and l is the other combination, may be there are different combination here of the advertisement expenditure and the sales. Suppose we get a combination that is 10 and 58. So, this goes to here where here suppose this point this is point j where it is 10 and 58 or may be below somewhere that we get a combination k which is again 10 and 55 or may be we get a combination m where it is may be it is somehow between 40. So, this is combination m where we get 10 and 40. Now if you remember just now we discussed that the value of the intercept. So, if you look at when the 0 expenditure on advertisement expenditure the total sales is 14. When the advertising expenditure is 10 may be we have different combination where 10 is equal to 10 is advertisement expenditure 40 is the sales k is the advertisement expenditure 55 is the sale and at the point j 10 is the advertisement expenditure and 58 is the sale. So, when it is equal to 0 we get the value of intercept that is alpha which is equal to 40,000 if you are taking this as the 0 unit. Now what is the next task? Next task is to find out the slope of this because this is the value of a when expenditure is equal to 0 alpha is value of intercept and this is the value of a. Now second what we have to find out through the rudimentary method or through the elementary method we need to find the value of b.