Line through the mean point and the origin compared with the regression line for load and extension of a spring

In an experiment, different loads are hung from a spring. For each load of \(x\) kg, the extension of the spring, \(y\) mm, is measured. The seven results are given in the table below.

The results are plotted on the scatter diagram below.

With no load, the extension is \(0\) mm. The experimenter therefore draws a straight line through the origin and the point \((\bar{x},\bar{y})\), where \(\bar{x}\) is the mean of the seven loads and \(\bar{y}\) is the mean of the seven extensions.

1. Find
   1. the value of \(\bar{x}\) and the value of \(\bar{y}\);
   2. the equation of the line drawn through the origin.[3]

The regression line of \(y\) on \(x\) for the seven results has equation \(y=ax+b\), where \(a,b\in\mathbb{R}\).

2. Find the value of \(a\) and the value of \(b\).[2]
3. Use this regression line to estimate the extension of the spring for a load of \(9\) kg.[2]