Question

1. the table below shows how a persons age is related to their medium income.

estimate the median income for someone 32 years old.

Explanation:

Step1: Plot the points

Plot the points \((26,16.8)\), \((27,19.1)\), \((28,23.3)\), \((29,25.8)\), \((30,33.9)\) on a graph.

Step2: Draw a line of best - fit

Visually estimate a line that best represents the trend of the data points.

Step3: Extend the line

Extend the line of best - fit to \(x = 32\) (where \(x\) represents age).

Step4: Read the \(y\) - value

Read the \(y\) - value (where \(y\) represents median income) corresponding to \(x = 32\) on the extended line of best - fit.

Another way (using linear regression approximation conceptually):
Let \(x\) be age and \(y\) be median income.
We can assume a linear relationship \(y=mx + b\). Using two points \((x_1,y_1)=(26,16.8)\) and \((x_2,y_2)=(30,33.9)\)
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{33.9-16.8}{30 - 26}=\frac{17.1}{4}=4.275\)
Using the point - slope form \(y - y_1=m(x - x_1)\), with \((x_1,y_1)=(26,16.8)\)
\(y-16.8 = 4.275(x - 26)\)
\(y=4.275x-4.275\times26 + 16.8\)
\(y=4.275x-111.15+16.8\)
\(y=4.275x - 94.35\)
When \(x = 32\)
\(y=4.275\times32-94.35\)
\(y = 136.8-94.35\)
\(y = 42.45\)

Answer:

An estimate of the median income for a 32 - year - old is approximately \(\$42450\) (using the linear regression approach. If using a graphical line of best - fit, the value may vary slightly depending on the visual estimation, but a reasonable range is around \(\$40000-\$45000\))