Chapter B.3, Problem 4E

To determine

To calculate:

The least squares regression line and verify with a graph.

Answer to Problem 4E

The least squares regression line is $y=0.808x+0.77$ .

Explanation of Solution

Given information:

Points: $\left(0,-1\right),\left(2,1\right),\left(3,2\right),\left(5,3\right)$

Calculation:

The estimated least squares regression equation for a set of data is:

  $y=a+bc$..........................(1)

The above equation can be written as:

  ${\displaystyle \sum _{i=1}^n{y}_i}=a\left({\displaystyle \sum _{i=1}^n{x}_i}\right)+nb$..........................(2)

  ${\displaystyle \sum _{i=1}^n{x}_i{y}_i}=a\left({\displaystyle \sum _{i=1}^n{x}^2{}_i}\right)+b\left({\displaystyle \sum _{i=1}^n{x}_i}\right)$..........................(3)

Where, $a$ and $b$ are constants.

Consider the points $\left(0,-1\right),\left(2,1\right),\left(3,2\right),\left(5,3\right)$ . To get the least square line construct a table as below:

$x$	$y$	$xy$	$x^2$
$0$	$-1$	$0$	$0$
$2$	$1$	$2$	$4$
$3$	$2$	$6$	$9$
$5$	$3$	$15$	$25$
${\displaystyle \sum _{i=1}^{n}x=10}$	${\displaystyle \sum _{i=1}^{n}y=5}$	${\displaystyle \sum _{i=1}^n{x}_1{y}_1=23}$	${\displaystyle \sum _{i=1}^n{x}^2=38}$

Now, apply the system for the least square regression line with $n=4$ . Put the above calculated values in the equation (2) and (3) the equation is given below:

  $5=a\left(10\right)+4b$..........................(4)

  $23=a\left(28\right)+b\left(10\right)$..........................(5)

Now divide equation (4) by $2$ , then subtract the equation (4) with equation(5):

  $\begin{array}{l}12.5=25a+10b\\ -23=-38a-10b\\ -10.5=-13a\\ a=0.808\end{array}$

Put the value of $a$ into equation (4), the calculation is given below:

  $\begin{array}{l}5=10\times \left(0.808\right)+4b\\ 5=8.08+4b\\ -3.08=4b\\ b=0.77\end{array}$

Put the calculated values $a$ and $b$ into equation (1). Thus, the least square regression equation can be written as:

  $y=0.808x+0.77$

The regression equation by graphing utility as follows,

  

The above graph shows the least square regression equation $y=0.808x+0.77$ .