Question

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 \(\$ 23,624 ?\)

Short answer

Using the midpoint formula, the estimated threshold for 2013 is \(\$ 23,307\), which is \(\$ 317\) less than the actual threshold of \(\$ 23,624\).

Step-by-step solution

- Identify known values

The problem provides the poverty thresholds for the years 2009 and 2017. These values are as follows: - 2009: \( \$ 21,756 \) - 2017: \( \$ 24,858 \)

- Calculate the number of years between given points

Determine the number of years between 2009 and 2017. \( 2017 - 2009 = 8 \) years.

- Determine the total increase in poverty threshold

Calculate the difference in poverty thresholds between 2009 and 2017. \( 24,858 - 21,756 = 3,102 \)

- Calculate the annual increase

Determine the annual increase by dividing the total increase by the number of years. \( \frac{3,102}{8} = 387.75 \) per year.

- Calculate the number of years from 2009 to 2013

Determine the number of years between 2009 and 2013. \( 2013 - 2009 = 4 \) years.

- Calculate the poverty threshold for 2013

Estimate the poverty threshold for 2013 by adding the total increase over these 4 years to the 2009 threshold. \( 21,756 + (4 \times 387.75) = 21,756 + 1,551 = 23,307 \)

- Compare with the actual value

Compare the estimated threshold for 2013 to the actual threshold. Estimated: \( 23,307 \) Actual: \( 23,624 \) Difference: \( 23,624 - 23,307 = 317 \)

Key concepts

Poverty Thresholds

Poverty thresholds are set by the U.S. Census Bureau. They represent the minimum household income needed for a family not to be considered poor. These thresholds vary depending on family size and composition. For example, in 2009, a family of four with two children under 18 years old had a poverty threshold of \( \$ 21,756 \). By 2017, this threshold increased to \( \$ 24,858 \). Understanding poverty thresholds helps us assess the economic well-being of families over time.

Linear Interpolation

Linear interpolation is a method to estimate unknown values that fall within two known values. This method assumes changes occur at a constant rate. In our exercise, we use linear interpolation to estimate the poverty threshold for 2013. Given the thresholds for 2009 and 2017, we assume the threshold increases uniformly. So, the midpoint formula can calculate the poverty threshold for any year between 2009 and 2017. For 2013, this involves finding the annual increase and applying it over the years to 2013.

Annual Increase Calculation

To estimate future values using the midpoint formula, calculating the annual increase is essential. Here’s how:

• First, find the difference between the known values (\[ 24,858 - 21,756 = 3,102 \]).
• Next, divide this difference by the number of years between the known values (\[ \frac{3,102}{8} = 387.75 \] dollars per year).

By applying this annual increase, we estimate values for any intermediate year.

Estimation

Estimations help us predict values when exact data points aren’t available. Using our calculated annual increase (\[ 387.75 \] dollars per year), we predict the 2013 poverty threshold. Start with the 2009 threshold (\[ 21,756 \]) and add the total increase from 2009 to 2013 (\[ 4 \times 387.75 = 1,551 \]). This gives an estimated threshold of \[ 21,756 + 1,551 = 23,307 \]. Comparing this to the actual 2013 threshold (\[ 23,624 \]), we see a small difference, demonstrating the accuracy of our estimation method.