Question

the table lists how poverty level income cutoffs (in dollars) for a family of four have changed over time. use the midpoint formula to approximate the poverty level cutoff in 1987 to the nearest dollar. the poverty level cutoff in 1987 was 11925 dollars (round to the nearest dollar as needed).
year\tincome (in dollars)
1970\t3133
1980\t8330
1990\t13719
2000\t17169
2004\t19936

Explanation:

Step1: Identify the two - point data

We use the data for 1980 ($x_1 = 1980$, $y_1=8330$) and 1990 ($x_2 = 1990$, $y_2 = 13719$) since 1987 is between 1980 and 1990.

Step2: Write the mid - point formula

The mid - point formula for a linear interpolation between two points $(x_1,y_1)$ and $(x_2,y_2)$ for a value $x$ is $y=y_1+\frac{x - x_1}{x_2 - x_1}(y_2 - y_1)$. Here $x = 1987$, $x_1 = 1980$, $x_2 = 1990$, $y_1=8330$, $y_2 = 13719$.

Step3: Calculate the difference in $x$ values

$x_2 - x_1=1990 - 1980 = 10$.

Step4: Calculate the difference in $y$ values

$y_2 - y_1=13719 - 8330 = 5389$.

Step5: Calculate $x - x_1$

$x - x_1=1987 - 1980 = 7$.

Step6: Calculate the interpolated value

$y=8330+\frac{7}{10}\times5389=8330 + 3772.3=12102.3\approx12102$.

Answer:

12102