Chapter 5, Problem 16CR

To determine

To show:whether y varies directly with x and write an equation for the direct variation.

Answer to Problem 16CR

Yes, y varies directly with x and the equation is given by $y=3x-3$

Explanation of Solution

Given information:The data of the given linear modelis represented in the table.

graph:Assuming the value of x to find $f(x)$ and plot the graph

  

  

Interpretation :

As observed from the graph y varies directly with respect to x and the equation is defined by the best fit line.

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.

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

  $y=ax+b$

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

  $\begin{array}{l}\begin{array}{cccc}x& y& xy& x^2\\ -1& -6& 6& 1\\ 2& 3& 6& 4\\ 5& 12& 60& 25\end{array}\\ \text{ 9 24 216 81}\end{array}$

  $\begin{array}{l}\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{X}_i}}{n}=8.5\\ \overline{y}=\frac{{\displaystyle \sum _{i=1}^n{Y}_i}}{n}=16.5\end{array}$

  $\overline{x^2}=\frac{{\displaystyle \sum _{i=1}^n{X}^2{}_i}}{n}=55.5$

Therefore on simplifying the value we get

  $\begin{array}{l}b=-3\\ a=3\end{array}$

Hence the required equation is

  $y=3x-3$