 Hello and welcome to the session. In this session we fit a linear function to the given bivariate data using slope as well as using least square method and when we see which equation is the best fit to the given data we will also find coefficient of determination. Now we are let us consider the following bivariate data showing student to teacher ratio for public schools. Now for this data we will find line of fit using slope then we will find slope and intercept and interpret their meaning. Also we will write equation of regression line using b square method and we will find coefficient of determination and interpret it. So firstly let us find line of fit using slope. Now let us consider the two points where let us take these two points. Now let us consider these two points. Let us denote this first point where there will be x1, y1 and the second point where there is x2, y2. Now slope at line joining these two points is given by m is equal to y2 minus y1 upon x2 minus x1. So putting these values we have slope m is equal to y2 minus y1 that is 16.2 minus 16.8 for the point x2 minus x1 that is 2 minus 1 and on solving this we get m is equal to minus 0.28 so here slope m is equal to minus 0.28. Now we know that equation of line passes through the point x1, y1 and having slope m is given by y minus y1 is equal to m into x minus x1 the whole. Now here we have slope m is equal to minus 0.28 and the point x1, y1 as 1 16.8 so here equation of the line will be y minus y1 that is 16.8 is equal to m that is minus 0.28 into x minus x1 that is 1 the whole. This implies y minus 16.8 is equal to minus 0.28 into x plus 0.28 and on solving this equation we get y is equal to minus 0.28 into x plus 17.08 which is of the form y is equal to mx plus v the way you can see slope m is equal to minus 0.28 and y into sub b is equal to 17.08. Now slope is equal to minus 0.28 tells that there is decrease in student to teacher ratio by 0.28 per year and y into sub b that is v is equal to 17.08 is predicting that initially that is in the year 2005 the student to teacher ratio was 17.08. So we have found this equation as the line of fit. Now let us find the another equation of line of fit by using method of v square. Now the method of v square the regression equation is given by y is equal to a plus vx where a is equal to y bar minus b into x bar and regression coefficient b is given by summation of xy minus m into x bar into y bar whole upon summation of x square minus m into x bar square where x bar is the mean of x observations that is summation x upon n and y bar is the mean of y observations that is summation y upon n and here m is number of observations. Now by putting the values of x bar and y bar we have b is equal to n into summation of xy minus summation x into summation y whole upon n into summation of x square minus summation of x whole square. Now in this table we are given the x values and y values. Now here we have weight to more columns in this column we will find the values of xy and in this column we will find the values of x square. Now let us find the values of xy. Now here x is equal to 1 y is equal to 16.8 so xy will be 1 into 16.8 that is 16.8 similarly 2 into 16.2 is 32.4 3 into 16.6 is 49.8 4 into 15.7 is 62.85 into 15.9 is 79.5 and 6 into 15.4 is 92.4. Now let us find the values of x square. Now here x is 1 so x square will be 1 square that is 1 then 2 square is 4, 3 square is 9, 4 square is 16, 5 square is 25 and 6 square is 36. Now adding all the values of x we get summation x is equal to 1 plus 2 plus 3 plus 4 plus 5 plus 6 that is equal to 21. Similarly summation y is equal to 96.6 then we have summation of xy is equal to 333.7 then summation of x square is equal to 91. Now we will substitute all these values in this formula and we will find the regression coefficient b so we have b is equal to n that is here number of observations is equal to 1 2 3 4 5 and 6 so b is equal to 6 into summation of xy that is 333.7 minus summation of x that is 21 into summation of y that is 96.6 whole upon n that is 6 into summation of x square that is 91 minus summation of x whole square that is 21 whole square and on solving this we get b is equal to minus 0.251. Now we have to find value of a now we know that y that is equal to summation y upon n and xy is equal to summation x upon n so putting these values we can find the value of a so this implies a is equal to summation y that is 96.6 upon n that is 6 minus b that is minus of minus 0.251 into summation x that is 21 upon n that is 6. Now on solving this we get a is equal to 16.9785 therefore e square regression equation is given by y is equal to 16.9785 minus 0.251 into x so this is the line of fetch which we have obtained from the least square method and this is also the equation of line of fit which we have obtained earlier. Now see this diagram of scatter plot here we have drawn both the lines of fit this red line is the line of fit given by the equation y is equal to minus 0.28 into x plus 17.08 and this blue line is the regression line given by the equation y is equal to 16.9785 minus 0.251 into x. Now see that least square regression line is the line of best fit since it means the distance between the predicted and the observed value of y. Now see this point that is the point which coordinates 515.9 is closer to the blue curve that is this blue line then this red line thus least square regression line that is this blue line is the best fit line of the data. Now let us understand the concept of coefficient of determination. Now coefficient of determination determines the strength of correlation between the two variables. Now point of determination is simply squaring the coefficients of correlation and r eliminates the direction of the correlation and gives us a value from 0 to 1 which measures the strength of correlation. Now if r square is equal to 0 then there is no correlation if 0 is less than r square is less than 0.25 then there is very weak correlation and if 0.25 is less than equal to r square is less than 0.50 then there is weak correlation and if 0.50 is less than equal to r square is less than 0.75 then there is moderate correlation if 0.75 is less than equal to r square is less than 0.90 then there is strong correlation and if 0.90 is less than equal to r square is less than 1 then there is very strong correlation and if r square is equal to 1 then there is a perfect correlation. Now here using calculator we have found coefficient of correlation that is r is equal to minus 0.8765 so coefficient of determination that is r square is equal to 0.76825 so there r is negative and r square lies between 0.75 and 0.90 so there is a strong negative correlation between the two variables also you must remember that the coefficient of determination represents the percent of data that is closest to the line of best fit. Here we have r square is equal to 0.76825 which means that about 76% of the total variation in y can be explained by the linear relationship between x and y given by the regression equation. It is the measure that tells how well the regression line represents the data. So in this fashion we have learnt how to fit a linear function to the bivariate data using slope as well as using method of e square also we have learnt coefficient of determination and this completes our fashion Hope you all have enjoyed the fashion.