Question

Fit $y=a+x$ to the data $$\begin{array}{llll}\mathrm{x}& 0& 1& 2\\ \mathrm{y}& 1& 3& 4\end{array}$$ by the method of least squares.

Short answer

The line that best fits the given data by the method of least squares is $y=1+1.67$.

Step-by-step solution

Calculate Summations

First, calculate the summations needed in the following calculations, which are $\sum x$ and $\sum y$. Here, $\sum x=0+1+2=3$ and $\sum y=1+3+4=8$

Find the Mean Values

Find the mean values of x and y with the formulas $\bar{x}=\frac{\sum x}{n}$ and $\bar{y}=\frac{\sum y}{n}$. In this case, $\bar{x}=\frac{3}{3}=1$ and $\bar{y}=\frac{8}{3}\approx 2.67$

Determine the Value of 'a'

As the given line is $y=a+x$, this can be re-written in slope-intercept form as $y=mx+b$, with $m=1$ and $b=a$. By comparing to the general line equation, determine 'a' as the y-intercept which is the mean value of y minus the mean value of x multiplied by the slope. Thus, $a=\bar{y}-m\cdot \bar{x}$, i.e., $a=2.67-1\cdot 1=1.67$