 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 a 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 it 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 why 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 variables 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 may be on a increasing manner, sometimes it is increasing, sometimes it is increasing less, sometimes it is increasing high or sometimes it is increasing may be it is just constant. So, if you look at between the year 1998 and 1999, the sales is remain constant even if the may be the expenditure has increase or the expenditure has decrease. 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 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 or finally, they value 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 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 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 suggests 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 x is 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 analysis 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 scatter 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 take the sales. So, this is 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 or 24 and here we take may be 40, 45, 50, 55, 60, 65, 70, 75, 80 and 85 and 90. So, since we have 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 40. When the advertisement expenditure is 10 may be we have different combination where 10 is equal to 10 is advertisement expenditure on 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 40000 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.