If we want to use linear regression, we need first to identify the pairs of numbers. In this case
[tex](1,40),(1,55),(2,60),(2,70),(3,70),(4,80),(5,90),(6,80),(7,90),(8,100)[/tex]
And we obtain the slope and the y-intercept of the best fit using the next formulas
[tex]\begin{gathered} a=\frac{(\Sigma y)(\Sigma x^2)-(\Sigma x)(\Sigma xy)}{n(\Sigma x^2)-(\Sigma x)^2} \\ b=\frac{n(\Sigma xy)-(\Sigma x)(\Sigma y)}{n(\Sigma x^2)-(\Sigma x)^2} \end{gathered}[/tex]
where n is the number of points. So n=10.
Now, we have to find, x squared, and xy.
[tex]\begin{gathered} x^2 \\ 1,1,4,4,9,16,25,36,49,64 \\ xy \\ 40,55,120,140,210,320,450,480,630,800 \end{gathered}[/tex]
and their sum
[tex]\begin{gathered} \Sigma x \\ 1+1+2+2+3+4+5+6+7+8=39 \\ \Sigma y \\ 40+55+60+70+70+80+90+80+90+100=735 \\ \Sigma x^2 \\ 1+1+4+4+9+16+25+36+49+64=209 \\ \Sigma xy \\ 40+55+120+140+210+320+450+480+630+800=3245 \end{gathered}[/tex]
Then, we replace this results in the formulas
[tex]\begin{gathered} a=\frac{(735)(209)-(39)(3245)}{(10)(209)-(39)^2}=\frac{27060}{569}=47.56 \\ b=\frac{(10)(3245)-(39)(735)}{(10)(209)-(39)^2}=\frac{3785}{569}=6.65 \end{gathered}[/tex]
Then, the best fit is the equation
[tex]y=6.65x+47.56[/tex]
