 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that for the following data set, this is the given data set where x values are given as 2, 3, 5, 6, 7, 9, 10 and the corresponding y values are given as 1, 4, 6, 5, 9, 8, 11. The line of best fit is given by equation y hat is equal to 1.05769x minus 0.06043 and the first part says calculate the residuals and the second part is draw the residual plot and analyze it. We know that for each data point the residual is given by y minus y hat where y is the observed y value and y hat is the predicted y value of the data point which is predicted by the line of best fit for the x value of the data point. With this key idea we shall proceed to the solution. We are given bivariate data in the form of this table. Also we are given line of best fit which is given by the equation y hat is equal to 1.05769x minus 0.06043. We have to calculate the residuals and we need to draw the residual plot as well as we need to analyze it. So first we calculate residuals first we shall find predicted value of y using given line of fit. Then with the help of the key idea we will find residuals using formula residual is equal to y minus y hat where y is the observed y value and y hat is the predicted y value of the data point. So we form a table to calculate residuals. In the first column we have written x values which are given in the data set. In second column we have written the observed y values which are also given in the data set. Now we calculate the predicted value of y using the given equation and then from this predicted value we will calculate residuals. Now for x is equal to 2 the value of y hat will be equal to 1.05769x minus 0.06043. Using calculator we get this value as 2.05495 and by rounding this value we get the value of y hat as 2.05. For x is equal to 2 the value of y hat is equal to 2.05. Similarly for x is equal to 3 we get the value of y hat as 3.11. For x is equal to 5 y hat is equal to 5.23. For x is equal to 6 y hat is equal to 6.35. For x is equal to 7 y hat is equal to 7.34. For x is equal to 9 y hat is equal to 9.45 and for x is equal to 10 the value of y hat is equal to 10.51. Now we will find out residuals. For x is equal to 2 residual which is given by y minus y hat will be equal to 1 minus 2.05 which is equal to minus of 1.05. For x is equal to 3 residual will be given by 4 minus 3.11 which is equal to 0.89. For x is equal to 5 residual is given by 6 minus 5.23 which is equal to 0.77. For x is equal to 6 residual will be given by 5 minus 6.35 which is equal to minus of 1.35. Similarly for x is equal to 7 residual is 1.66. For x is equal to 9 residual is given by minus of 1.45 and for x is equal to 10 residual is given by 0.49. Now we have got the residuals. Now we shall plot the residuals obtained. We will plot the residuals against the x values to form a residual plot. The residual plot will show how the points vary about the line of best fit. We will draw the x values along x axis and the value of residuals along y axis. For x is equal to 2 the first residual is minus of 1.05. So we place a point at minus of 1.05 when x is equal to 2 since it is negative. So it lies below x axis. The next residual is 0.89 against x is equal to 3. So we place a point at 0.89 when x is equal to 3 since it is positive. So it will lie above the x axis. Similarly we shall plot the other points. Similarly we have plotted all other residual points and we get this residual plot. Now we shall analyze this residual plot and we know that residual plot helps to determine whether it is appropriate to fit a linear model to a data set. If points are randomly scattered about x axis with no obvious pattern then it indicates that the data varies randomly about the line of best fit and the linear model is appropriate for the data and if the points in the residual plot form a pattern like a curve and are not random then it indicates linear model is not appropriate for the data. Now see in this residual plot the points are randomly scattered about x axis not forming any pattern. So linear model is appropriate for the given set of values which is the required answer. This completes our session. Hope you enjoyed this session.