Question

(d) Calculate the sum of the squares of the residuals. 4. Draw by hand a best-fit line, and then find the line's equation for the data below. Finally predict the result of dependent variable when the independent variable is at 45. (It is not necessary to use the best-fit line equations of section 9.2 for this problem) 1800 1600 1400 1200 1000 800 400 t 200 10 15 2025 30 35 40 45

Answer 1

Clearly, data points plotted on the graph show a roughly straight line hence a free hand best fit line should be such a straight line which ensures approximately equal number of point on either sides of the line being drawn.

An example is as shown below :

                      

The best fit line is an equation of the form :

                                  y=a+bx where a and b are constants to be evaluated such that the error sum of squares is minimum.

The approimates of a and b are obtained using the equations :

                                            $a=\overline{y}-b\overline{x}$        ------------   (i)

          and,                           $b=Cov(X,Y)/Var(X)$     ----------   (ii)

From the data set , we calculate the means of X and Y which are $\overline{x}$ and $\overline{y}$ and the using them along with the Covariance and variance of X we can find out a and b.

If the estimated values of a and b be represented by $\widehat{a}$ and $\widehat{b}$ respectively,

Then the equation we obtain is :

                                          $y=\widehat{a}+\widehat{b}x$

Now to approximate the value of the dependent variable y when the dependent variable is 45, we put x=45 and evaluate

                                                             $y=\widehat{a}+\widehat{b}*45$