Q.Obtain the linear equation for trend for a time series with $n = 7$, $\sum y = 145.2$, $\sum ty = 637.3$.

For $n=7$ with coded $t$ ($\sum t=0,\ \sum t^2=28$): $a=\frac{\sum y}{n}=20.74$, $b=\frac{\sum ty}{\sum t^2}=22.76$; $\hat y=20.74+22.76t$.

Using the method of least squares with time coded so that $\sum t=0$ (for $n=7$, take $t=-3,-2,-1,0,1,2,3$), the normal equations reduce to

$a=\frac{\sum y}{n},\qquad b=\frac{\sum ty}{\sum t^2}.$

Here $\sum t^2=(-3)^2+\dots+3^2=9+4+1+0+1+4+9=28$. …