Chapter 1, Problem 1EA

The following table gives the life expectancy at birth of females born in the United States in various years from 1970 to 2010.

Source: National Center for Health Statistics.

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	80.1
2010	81.0

Find an equation for the least squares line for these data, using year of birth as the independent variable.

To determine

The equation of the least squares line for data provided in the table.

Answer to Problem 1EA

Solution: $y=\underset{\_}{0.134x+66.3}$

Explanation of Solution

Given: Consider the data provided in the table.

Year of birth	Life expectancy
1970	74.7
1975	76.6
1980	77.4
1985	78.2
1990	78.8
1995	78.9
2000	79.3
2005	80.1
2010	81.0

Explanation:

Let the data points be $\left(x_1,y_1\right),\left(x_2,y_2\right),\dots ,\left(x_n,y_n\right)$ and the least squares line $y=mx+b$ have the slope $m$ and $y$-intercept $b$.

Here, the value of slope is calculated by

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

and the intercept is calculated by

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

Now, compute the least squares line for the provided data.

Let $x$ be the years of data from $1900$. So, $1970$ is written as $70$ and $2010$ is written as $110$.

The required data is shown in the table:

$x$	$y$	$xy$	$x^2$	$y^2$
70	74.7	5229	4900	5580.09
75	76.6	5745	5625	5867.56
80	77.4	6192	6400	5990.76
85	78.2	6647	7225	6115.24
90	78.8	7092	8100	6209.44
95	78.9	7495.5	9025	6225.21
100	79.3	7930	10,000	6288.49
105	80.1	8410.5	11,025	6416.01
110	81	8910	12,100	6561
${\displaystyle \sum x}=810$	${\displaystyle \sum y}=705$	${\displaystyle \sum xy}=63,651$	${\displaystyle \sum x^2}=74,400$	${\displaystyle \sum y^2}=55,253.8$

Now, substitute these values to determine the value of slope and intercept.

Compute the slope $m$:

$\begin{array}{c}m=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{n\left({\displaystyle \sum x^2}\right)-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{9\left(63,651\right)-\left(810\right)\left(705\right)}{9\left(74,400\right)-{\left(810\right)}^2}\\ =\frac{572,859-571,050}{669,600-656,100}\\ =\frac{1809}{13,500}\\ \approx 0.134\end{array}$

Compute the intercept $b$:

$\begin{array}{c}b=\frac{\left({\displaystyle \sum y}\right)-m\left({\displaystyle \sum x}\right)}{n}\\ =\frac{\left(705\right)-\left(0.134\right)\left(810\right)}{9}\\ =\frac{705-108.54}{9}\\ =\frac{596.46}{9}\\ \approx 66.3\end{array}$

Now, the least squares line is

$\begin{array}{c}y=mx+b\\ =0.134x+66.3\end{array}$

Conclusion: The equation of the least squares line is $y=0.134x+66.3$.