Chapter 1.1, Problem 67AYU

Poverty Threshold Poverty thresholds are determined by the U.S. Census Bureau. A poverty threshold represents the minimum annual household income for a family not to be considered poor. In $2009$, the poverty threshold for a family of four with two children under the age of $18$ years was $\$21,756$. In $2017$, the poverty threshold for a family of four with two children under the age of $18$ years was $\$24,858$. Assuming that poverty thresholds increase in a straight-line fashion use the midpoint formula to estimate the poverty threshold for a family of four with two children under the age of $18$ in $2013$. How does your result compare to the actual poverty threshold in $2013$ of $\$23,624$?

To determine

To calculate: The poverty threshold for a family of four with two children under the age of $18$ in $2013$ using the midpoint formula. Compare the result to the actual poverty threshold in $2013$ of $\$23,624$

Answer to Problem 67AYU

Solution:

The estimated value of the poverty threshold for a family of four with two children under the age of $18$ in $2013$ is $\$23307$

The actual poverty threshold is $\$317$ off by calculated value

Explanation of Solution

Given Information:

In $2009$, the poverty threshold for a family of four with two children under the age of $18$ was $\$21,756$

In $2017$, the poverty threshold for a family of four with two children under the age of $18$ was $\$24,858$

Formula used:

The midpoint formula: The coordinate of midpoint of two points $P_1=\left(x_1,y_1\right)$ and $P_2=\left(x_2,y_2\right)$ is $\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Calculation:

From the information, let $P_1=(2009,21756)$ and $P_2=(2017,24858)$

By the midpoint formula, the coordinate of the midpoint of two points $P_1=(2009,21756)$ and $P_2=(2017,24858)$ is

$\left(x,y\right)=\left(\frac{2009+2017}{2},\frac{21756+24858}{2}\right)$

$\left(x,y\right)=\left(\frac{4026}{2},\frac{46614}{2}\right)$

$\left(x,y\right)=\left(2013,23307\right)$.

The estimated value of the poverty threshold for a family of four with two children under the age of $18$ in $2013$ is $23307$

The actual poverty threshold in $2013$ is $\$23624$

Therefore, the error in estimation is $=|\mathrm{Actual}\mathrm{value}-estimatedvalue|$

$=|23624-23307|$

$=317$

Therefore, the estimated value of the poverty threshold for a family of four with two children under the age of $18$ in $2013$ is $\$23307$

The actual poverty threshold is $\$317$ off by calculated value