Find a linear model that models the​ data, where x is the number of years after 2000 and y is the total disposable income in billions of dollars.

Question

Disposable income is the amount left after taxes have been paid and is one measure of health of the economy. The table gives the total disposable income for a​ country, in billions of​ dollars, for selected years from 2010 and projected to 2040.

Answer 1

To create a linear model for the total disposable income data based on the information given, we first need to extract the data from the years specified. Generally, we convert the year into a variable \( x \) based on how many years after 2000 they are (i.e., for the year 2010, \( x = 10 \), for 2011, \( x = 11 \), etc.).

Next, let's assume you have a table of data for total disposable income. It should look something like this:

| Year | Total Disposable Income (in billions) | |------|---------------------------------------| | 2010 | Y0 | | 2011 | Y1 | | 2012 | Y2 | | 2013 | Y3 | | ... | ... | | 2020 | Y10 | | ... | ... | | 2040 | Y40 |

1. Convert years to x values:

   • 2010 = 10
   • 2011 = 11
   • ...
   • 2040 = 40

2. List the income values (y values):

   • For each year listed, you will have a corresponding total disposable income (in billions), which you will label as \( Y0, Y1, Y2 \), etc.

3. Perform linear regression:

   • You can use the least squares method or software/calculators capable of linear regression to find the best-fit line \( y = mx + b \), where:
     • \( m \) is the slope of the line,
     • \( b \) is the y-intercept.

4. Interpret the model:

   • Once you have \( m \) and \( b \), you will be able to predict the total disposable income for any year after 2000 by plugging \( x \) into the linear equation.

If you provide the actual disposable income numbers for the specified years, I can assist you in performing the calculations and creating the linear model effectively!