Chapter 5.8, Problem 42MR

To determine

To show:equation of best fit line and represent it by graph .

Answer to Problem 42MR

  $y=57.868x+5.06$

Explanation of Solution

Given information:The data of the given linear modelin terms of year and price is represented in the table.

Graph:Assuming year to be $x_1$ and price to be $y_1$ to plot the graph and find $f(x)$

  

Interpretation :

Line of best fit is a straight line that is represented on scatter plot, whether it may pass through some of the points or none of the points which best represent it’s relationship with it.

A line of best fit is usually found through simple linear regression that is by r-squared value which represents that if the value of r is nearer to 1 then it is a best fit line.

The equation of best fit line is shown in the above graph and let its equationis supposed to be as

  $y=ax+b$

Where , $\begin{array}{l}a=\frac{{\displaystyle \sum _{i=1}^n{X}_i{Y}_i}-n\overline{XY}}{{\displaystyle \sum _{i=1}^n{X}_i^2}-n\overline{X^2}}\\ b=\overline{Y}-a\overline{X}\end{array}$

  $\begin{array}{l}\begin{array}{cccc}x& y& xy& x^2\\ 1& 5.30& 5.30& 1\\ 2& 5.57& 11.14& 4\\ 3& 5.82& 17.46& 9\end{array}\\ 4\text{ 6}\text{.05 24}\text{.2 16}\end{array}$

  $\begin{array}{l}\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{X}_i}}{n}=2.5\\ \overline{y}=\frac{{\displaystyle \sum _{i=1}^n{Y}_i}}{n}=5.68\end{array}$

  $\overline{x^2}=\frac{{\displaystyle \sum _{i=1}^n{X}^2{}_i}}{n}=7.5$

Therefore on simplifying the value we get

  $\begin{array}{l}b=5.06\\ a=57.868\end{array}$

Hence the required equation is

  $y=57.868x+5.06$