 Hello and welcome to the session. In this session we will discuss a question which says that James joins a health club to lose weight. At the end of every month he records his weight. The table shows the record of his weight for first 6 months. Now for this data we have to draw a scatter plot. Then we have to draw a line that appears to fit the data. And next we have to write an equation of line that best represents the data. Now let us start with the solution of the given question. Here we are given this table showing weight in pounds of James at the end of each month. First of all let us draw this scatter plot for this data. Now here on this graph on x-axis we take months and on y-axis we take weight with suitable scale. Now the first order pair that we have to plot is 1 to 37. So this is the point representing this order pair. And let us plot the next order pair. So here is the point representing the order pair 2 to 28. Similarly we will plot all the ordered pairs on this graph. So we have plotted all the ordered pairs on this graph. And this is the required scatter plot for the given data. Now the second part we have to draw a line that appears to fit the data. Now let us find equation of regression line using slope. Here let us take these two points. Here let us remove the first order pair by x1, y1. Now slope of line divided by x1, y1 and x2, y2 is given by m is equal to y2 minus y1. Then we find x2 minus x1. So putting these values we have m is equal to y2 minus y1. That is 216 minus 228. Now we have m x2 minus x1 that is 4 minus 2. And this implies m is equal to minus 12 upon 2 which implies m is equal to minus 6. So we have obtained slope m is equal to minus 6. Now here we have the point x1, y1 and also we have obtained the slope m. And we know that equation of line passing 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. So here equation of line is y minus 228 is equal to minus 6 into x minus 2 the whole. So on solving this we obtain equation of the line as y is equal to minus 6 into x plus 230. So to draw this line we join these three points. That is the point with coordinates 228 and the point with coordinates for 216. So joining these two points we get a straight line which represents this equation. And this is the line that appears to fit the data. Now we will find the regression line using method of least squares. Now using this method regression line is given by y is equal to a plus bx there. a is equal to y bar minus b into x bar where x bar is mean of x observations that is summation x upon n and y bar is equal to mean of y observations that is summation y upon n where n is number of observations. And regression coefficient b is given by n into summation of xy minus summation of x into summation of y will become n into summation of x square minus summation of x 1 square. Now in the given table we have made two more columns. In this column we will find the values of xy and in this column we will find the values of x square. Firstly let us find the values of xy. Now here x is equal to 1, y is equal to 237. So xy will be equal to 1 into 237 that is 237. Then 2 into 228 is 456. Similarly we have found every values of xy. Now we will find the values of x square. Now here x is equal to 1 so x square will be 1 square that is 1. Then 2 square is 4, 3 square is 9, 4 square is 16, i 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 1322. Then summation of xy is equal to 4524 and summation of x square is equal to 91. Now using all these values we can find the regression coefficient b that is even by this formula. So using all these values we have b is equal to n that is 6 into summation of xy that is 4524 minus summation of x that is 21 into summation of y that is 1322 fell 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 obtain b is equal to minus 5.88571. Now we have to find a now y bar is summation y upon n and x bar is summation x upon n. So we have written this equation in this form. Now we will put the values of summation x and summation y and we have a is equal to 1322 upon 6 minus 5.88571 into 21 upon 6 the whole and on solving this we obtain a is equal to 240.933318. So now we have the values of b and a. So the least square regression line is given by y is equal to a that is 240.933318 plus bx that will be plus or minus 5.88571 into x. So this is the least square regression line. Now earlier we have obtained this equation of line let it be equation 1 and this regression line equation be equation 2. Now on comparing equations 1 and 2 we see that equation 2 will give more accurate results than equation 1 because it minimizes the error of distance between the predicted value and the observed value. Thus this equation that is equation 2 is the line of best fit. That is the least square regression line is the best fit line for the given data. So this is the solution of the given question. That's all for this session. Hope you all have enjoyed the session.