 Hello and welcome to the session. In this session, we will discuss a question which says that find the line of best fit to the following data using a part x as independent variable, b part x as dependent variable. And this data is given to us. Now before starting the solution of this question, we should know some results. First is the regression line which passes x bar y bar that is the mean. And when independent variable, when the regression equation y minus y bar is equal to b y x minus x bar the whole. Where is the mean value of y? x bar is the mean value of x. The regression coefficient is equal to summation dx into dy over summation dx square. dx is equal to x minus x bar and dy is equal to y minus y bar. That is dx and dy is of the variables x and y from the arithmetic means of the series. And when, that is when y is independent variable, the regression equation is x bar equal to b x y into y minus y bar. The regression coefficient b x y is equal to summation dx into dy over summation dy square. Now these results will work out as a key idea for solving out this question. We will start with the solution of best fit to the following data. We will form a table for the given data. So we have made a table for the given data in the second column, different values of y and dx. Then in the next column we will find dy. In the next column, dx into dy, then in the next column dx square dy square. Now these are the x which is x is equal to 80. Now the number of 7, 8 and 9, that is the mean value of x is equal to number of observations are 9. So x bar will be equal to x minus x bar which is equal to x minus 1. Now on adding the different values of y is equal to, so y bar that is the mean value of y is equal to summation y over m, number of observations are 9. So this will be equal to 7. Now dy is equal to y minus y bar which will be equal to y minus, now we will find the value of dx which is equal to x minus 9. Now the first one, so dx will be equal to 2 minus 9 which is equal to minus 7. Here the value of x is 3, so dx will be equal to 3 minus 9 which is equal to minus 6, minus 2, 12 minus 9 is 3. Now for dy, we will use this formula which is equal to y minus 7. Now the first value of y is 1, so dy will be equal to 1 minus 7 which is equal to minus 6. So we will give minus 5 minus 1 will give minus 3, 11 minus 7 is minus 7 will give 3, 12 minus 7 is minus dx into dy. For this we will multiply these values of dx with dy. So here x will give 42, minus 6 into minus 5 is minus 4 into minus 1 will give 4. Here if we leave 6, 1 into 2, 2, f will be 4, this gives 15 and 6 into 5 will be 30. Now we will find dx square, we will square the values 49, 1 square is 1, here we will give 36. Similarly we will find dy square, minus 6 square will give 36, minus 5 square will give 25, minus 1 square will give 1, minus 3 square will give 9, 2 square is 1, 3 square is 9. Now we will add all these values, dx into dy is equal to 145. The values of dx square, so we get summation dx square is equal to 192, dy square. So summation dy square will be equal to 126. Now using this which is given in the key idea, the regression coefficient is equal to summation dx into dy, for summation dx square. Now summation dx into dy is 145 and summation dx square is 192. So this will be equal to 145 over 192. Now the regression coefficient is equal to summation dx into dy over summation dy square. Now summation dx into dy is 145 and summation dy square is 126. So this will be equal to 145 over 126. Now a regression equation which is given in the key idea, y minus y bar is equal to b y x bar the whole. Now x is equal to 145 by 192. And this implies minus 7 is equal to 145 over 192 minus 9 the whole. On plus multiplying 192 y minus 1344 is equal to 145 x minus 1305 which implies 145 x minus 192 y plus 1344 minus 1305 is equal to 0 Another way is 145 x minus 192 y plus 39 is equal to 0. So y and x is one second part. Now using this result which is given in the key idea, the regression equation dependent variable x bar is equal to b x y into y minus y bar the whole. b x y is equal to 145 by 186 and y bar is equal to 7. Putting all the values here this implies x minus 9 is equal to 145 by 126 into the whole. Now this implies on plus multiplying 156 x minus 1134 is equal to 145 y minus 145 minus 1134 plus 1015 is equal to 0. Which further implies y minus 119 is equal to 0. So this is the regression equation of y minus independent variable of regression. That means we have determined hope you all have enjoyed the session.