Q.The linear equation fitted using the data of 7 weeks for a variable $y$ is $\hat{y} = 25.1 - 1.5t$. Estimate the value of $y$ for the eighth week.

The method of least squares fits the straight line $Y_c = a+bX$ that minimises $\Sigma(Y-Y_c)^2$. Coding time as $X$ deviations from a convenient origin so that $\Sigma X = 0$ (the middle year for an odd number of years; the "$2X$" half-year coding for an even number) reduces the normal equations to $a=\Sigma Y/n$ and $b=\Sigma XY/\Sigma X^2$. Because it produces an actual equation, it gives a trend value for every year, includ …