Chapter B.3, Problem 2E

To determine

To calculate:

The least squares regression line and verify with a graph.

Answer to Problem 2E

The least squares regression line is $y=1.05x-1.4$ .

Explanation of Solution

Given information:

Points: $\left(0,-1\right),\left(2,0\right),\left(4,3\right),\left(6,5\right)$

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 $\left(0,-1\right),\left(2,0\right),\left(4,3\right),\left(6,5\right)$ . To get the least square line construct a table as below:

$x$	$y$	$xy$	$x^2$
$0$	$-1$	$0$	$0$
$2$	$0$	$0$	$4$
$4$	$3$	$12$	$16$
$6$	$5$	$30$	$36$
${\displaystyle \sum _{i=1}^{n}x=12}$	${\displaystyle \sum _{i=1}^{n}y=7}$	${\displaystyle \sum _{i=1}^n{x}_1{y}_1=42}$	${\displaystyle \sum _{i=1}^n{x}^2=56}$

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:

  $7=a\left(12\right)+4b$..........................(4)

  $42=a\left(56\right)+b\left(12\right)$..........................(5)

Now multiply equation (4) by $3$ , then subtract the equation (4) by equation (5) the equation is written as:

  $\begin{array}{l}21=36a+12b\\ -\left(42\right)=-56a-12b\\ -21=-20a\\ a=1.05\end{array}$

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

  $\begin{array}{l}7=\left(12\right)\times 1.5+4b\\ 7=12.6+4b\\ -5.6=4b\\ b=-1.4\end{array}$

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

  $y=1.05x-1.4$

The regression equation by graphing utility as follows,

  

The above graph shows the least square regression equation $y=1.05x-1.4$ .