Q.Obtain the linear equation for trend for a Time Series with $n = 8$, $\Sigma y = 344$, $\Sigma ty = 1342$.

Since $\Sigma t=0$: $a=\dfrac{\Sigma y}{n}=43$ and $b=\dfrac{\Sigma ty}{\Sigma t^2}=\dfrac{1342}{168}\approx 7.99$, giving $\hat{y}=43+7.99t$.

Given: $n=8$, $\Sigma y=344$, $\Sigma ty=1342$.

Step 1 — coding of time. For even $n=8$, take $t$ in half-year units centred at the middle:

$t=-7,-5,-3,-1,1,3,5,7,\quad \text{so } \Sigma t=0.$

$\Sigma t^2=2(7^2+5^2+3^2+1^2)=2(49+25+9+1)=2(84)=168.$

Step 2 — normal equations (with $\Sigma t=0$). …