Chapter B.3, Problem 3E

To determine

To calculate:

The least squares regression line and verify with a graph.

Answer to Problem 3E

The least squares regression line is $y=0.2617x+1.935$ .

Explanation of Solution

Given information:

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

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 $\left(-3,1\right),\left(-1,2\right),\left(1,2\right),\left(4,3\right)$ . To get the least square line construct a table as below:

$x$	$y$	$xy$	$x^2$
$-3$	$1$	$-3$	$9$
$-1$	$2$	$-2$	$1$
$1$	$2$	$2$	$1$
$4$	$3$	$12$	$16$
${\displaystyle \sum _{i=1}^{n}x=1}$	${\displaystyle \sum _{i=1}^{n}y=8}$	${\displaystyle \sum _{i=1}^n{x}_1{y}_1=9}$	${\displaystyle \sum _{i=1}^n{x}^2=27}$

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:

  $8=a\left(1\right)+4b$..........................(4)

  $9=a\left(27\right)+b\left(1\right)$..........................(5)

Now multiply equation (4) by $27$ , then subtract the equation (4) with(5):

  $\begin{array}{l}216=27a+108b\\ -9=-27a-b\\ 207=107b\\ b=1.93\end{array}$

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

  $\begin{array}{l}8=a+4b\\ 8=a+4\times 1.93\\ 8=a+7.73\\ a=0.262\end{array}$

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

  $y=0.2617x+1.935$

The regression equation by graphing utility as follows,

  

The above graph shows the least square regression equation $y=0.2617x+1.935$ .