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.
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.

Answer 1

To find a linear model that describes the relationship between years after 2000 (represented by \(x\)) and total disposable income in billions of dollars (represented by \(y\)), we start by defining our data set based on your information. I will assume a generic dataset of total disposable income for the selected years you might have. Once we have that data, we can proceed with fitting a linear model.

Let's follow these steps:

1. **Define the Data Points**: First, you need to represent your years and the corresponding disposable incomes in a tabular form. For example, if you have data like:
```
Year | Disposable Income (in billions)
2010 | 1000
2015 | 1200
2020 | 1400
2025 | 1600
2030 | 1800
2035 | 2000
2040 | 2200
```

Assuming x is the number of years after 2000, your data would be represented as follows:
```
x (Years after 2000) | y (Disposable Income)
-----------------------|----------------------
10 | 1000
15 | 1200
20 | 1400
25 | 1600
30 | 1800
35 | 2000
40 | 2200
```

2. **Formulate the Linear Model**: The linear equation can be expressed as:
\[
y = mx + b
\]
where \(m\) is the slope of the line and \(b\) is the y-intercept.

3. **Calculate the Slope and Intercept**:
You can calculate the slope \(m\) using the formula:
\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]
and the intercept \(b\) with:
\[
b = \frac{\sum y - m(\sum x)}{N}
\]

Where:
- \(N\) is the number of data points,
- \(\sum xy\) is the sum of the products of each \(x\) and \(y\),
- \(\sum x\) is the sum of all \(x\) values,
- \(\sum y\) is the sum of all \(y\) values,
- \(\sum x^2\) is the sum of the squares of \(x\) values.

4. **Example Calculation**: If you use the previous data points, let's assume you calculate \(N\), \(\sum xy\), \(\sum x\), \(\sum y\), and \(\sum x^2\):

For example:
- \(\sum x = 10 + 15 + 20 + 25 + 30 + 35 + 40 = 175\)
- \(\sum y = 1000 + 1200 + 1400 + 1600 + 1800 + 2000 + 2200 = 11200\)
- You would perform similarly for \(\sum xy\) and \(\sum x^2\).

5. **Create the Final Model**: After computing \(m\) and \(b\), substitute them back into the linear equation \(y = mx + b\) to obtain the model.

Please provide specific data if you want to complete calculations or the specific years and their corresponding disposable income figures, and I'll help you derive the linear model!