Chapter 13.3, Problem 13E

To determine

To find: the least square regression line for the points.

Answer to Problem 13E

The least square regression line formed will be $y=1.6x+7.5$

Explanation of Solution

Given information:

The points are

  $\left(-4,1\right),\left(-3,3\right),\left(-2,4\right),\left(-1,6\right)$

Formula used:

The least square regression line for the set of points $\left(x,y\right)$ can be calculated as $f\left(x\right)=ax+b$

Where $a=\frac{n{\displaystyle \sum _{i=1}^n{x}_i{y}_i}-\left({\displaystyle \sum _{i=1}^n{x}_i}{\displaystyle \sum _{i=1}^n{y}_i}\right)}{n{\displaystyle \sum _{i=1}^n{x}_i{}^2}-{\left({\displaystyle \sum _{i=1}^n{x}_i}\right)}^2}$ and $b=\frac{1}{n}\left({\displaystyle \sum _{i=1}^n{y}_i}-a{\displaystyle \sum _{i=1}^n{x}_i}\right)$

Calculation:

Here number of observations (n) is 4

Construct a table as shown below:

  $\begin{array}{cccc}x& y& xy& x^2\\ -4& 1& -4& 16\\ -3& 3& -9& 9\\ -2& 4& -8& 4\\ -1& 6& -6& 1\\ \begin{array}{l}{\displaystyle \sum _{i=1}^4{x}_i}=\left(-4-3-2-1\right)\\ =-10\end{array}& \begin{array}{l}{\displaystyle \sum _{i=1}^4{y}_i}=\left(1+3+4+6\right)\\ =14\end{array}& \begin{array}{l}{\displaystyle \sum _{i=1}^4{x}_i{y}_i}=\left(-4-9-8-6\right)\\ =-27\end{array}& \begin{array}{l}{\displaystyle \sum _{i=1}^4{x}_i{}^2}=\left(16+9+4+1\right)\\ =30\end{array}\end{array}$

Now for the regression line

  $\begin{array}{c}a=\frac{n{\displaystyle \sum _{i=1}^n{x}_i{y}_i}-\left({\displaystyle \sum _{i=1}^n{x}_i}{\displaystyle \sum _{i=1}^n{y}_i}\right)}{n{\displaystyle \sum _{i=1}^n{x}_i{}^2-{\left({\displaystyle \sum _{i=1}^n{x}_i}\right)}^2}}\\ =\frac{4\left(-27\right)-\left(\left(-10\right)\left(14\right)\right)}{4\left(30\right)-{\left(-10\right)}^2}\\ =\frac{8}{5}\\ =1.6\\ b=\frac{1}{n}\left({\displaystyle \sum _{i=1}^n{y}_i}-a{\displaystyle \sum _{i=1}^n{x}_i}\right)\\ =\frac{1}{4}\left(14-a\left(-10\right)\right)\\ =\frac{1}{4}\left(14-\left(1.6\right)\left(-10\right)\right)\left(\because a=1.6\right)\\ =\frac{1}{4}\left(14+16\right)\\ =\frac{30}{4}\\ =7.5\end{array}$

Hence the least square regression line formed will be $y=1.6x+7.5$