Chapter 1.EA, Problem 1EA

To determine

To find:

An equation for the least squares line for given data.

Answer to Problem 1EA

Solution:

The equation for the least squares line is $y=0.132x+66.43$.

Explanation of Solution

Given:

The data is given by,

Year of Birth	Life Expectancy (years)
$1970$	$74.7$
$1975$	$76.6$
$1980$	$77.4$
$1985$	$78.2$
$1990$	$78.8$
$1995$	$78.9$
$2000$	$79.3$
$2005$	$79.9$
$2010$	$81.0$

Table $\left(1\right)$

Approach:

The general equation for least square line is as follows.

$y=mx+b$.

Here,

$m=\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{n\left(\sum x^2\right)-{\left(\sum x\right)}^2}$.

And

$b=\frac{\left(\sum y\right)-m\left(\sum x\right)}{n}$.

Calculation:

Consider $x$ represents the year of birth and $y$ represents life expectancy.

The total number of data is $9$.

So, $n=9$.

Construct the new table for find the value for $m$ and $b$.

$x$	$y$	$xy$	$x^2$
$70$	$74.7$	$5229$	$4900$
$75$	$76.6$	$5745$	$5625$
$80$	$77.4$	$6192$	$6400$
$85$	$78.2$	$6647$	$7275$
$90$	$78.8$	$7092$	$8100$
$95$	$78.9$	$7495.5$	$9025$
$100$	$79.3$	$7930$	$10000$
$105$	$79.9$	$8389.5$	$11025$
$110$	$81.0$	$8910$	$12100$
Totals $=810$	$704.8$	$63630$	$74400$

Table $\left(2\right)$

Then,

$\begin{array}{rll}m& =& \frac{9\left(63630\right)-\left(810\right)\left(704.8\right)}{9\left(74400\right)-{\left(810\right)}^2}\\ & =& \frac{572670-570888}{669600-656100}\\ & =& \frac{1782}{13500}\\ & =& 0.132\end{array}$

Now calculate for $b$.

$\begin{array}{rll}b& =& \frac{704.8-\left(0.132\right)\left(810\right)}{9}\\ & =& \frac{597.88}{9}\\ & =& 66.43\end{array}$

Therefore, the equation for the least squares line is

$y=0.132x+66.43$.