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.
Complete parts ​(a) through ​(c) below.
Year
Income​ ($ billions)
2010
10.712
2015
11.394
2020
12.324
2025
14.725
2030
15.006
2035
17.806
2040
19.996

b. 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.
yequals
  
enter your response herexplus
  
enter your response here
​(Type integers or decimals rounded to three decimal places as​ needed.)

Answer 1

To find a linear model for the disposable income data, we can use the given points from the table. First, we need to express the years in terms of \( x \), where \( x \) is the number of years after 2000 (i.e., 2000 corresponds to \( x = 0 \), 2010 corresponds to \( x = 10 \), etc.).

Here is the data converted to \( (x, y) \) format:

- \( (10, 10.712) \)
- \( (15, 11.394) \)
- \( (20, 12.324) \)
- \( (25, 14.725) \)
- \( (30, 15.006) \)
- \( (35, 17.806) \)
- \( (40, 19.996) \)

Next, we can calculate the slope (\( m \)) and the y-intercept (\( b \)) of the line using the formula for linear regression.

### Step 1: Calculate the slope (\( m \))

The slope \( m \) can be calculated using the formula:

\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]

### Step 2: Calculate the y-intercept (\( b \))

The y-intercept \( b \) can be calculated using the formula:

\[
b = \frac{\sum y - m(\sum x)}{N}
\]

### Step 3: Compute the necessary sums

Let's compute the sums needed for the calculations.

- \( N = 7 \) (number of points)
- \( \sum x = 10 + 15 + 20 + 25 + 30 + 35 + 40 = 175 \)
- \( \sum y = 10.712 + 11.394 + 12.324 + 14.725 + 15.006 + 17.806 + 19.996 = 101.963 \)
- \( \sum xy = (10 \cdot 10.712) + (15 \cdot 11.394) + (20 \cdot 12.324) + (25 \cdot 14.725) + (30 \cdot 15.006) + (35 \cdot 17.806) + (40 \cdot 19.996) \)

Let's calculate \( \sum xy \):

- \( 10 \cdot 10.712 = 107.12 \)
- \( 15 \cdot 11.394 = 170.91 \)
- \( 20 \cdot 12.324 = 246.48 \)
- \( 25 \cdot 14.725 = 368.13 \)
- \( 30 \cdot 15.006 = 450.18 \)
- \( 35 \cdot 17.806 = 623.21 \)
- \( 40 \cdot 19.996 = 799.84 \)

Adding these up:
\[
\sum xy = 107.12 + 170.91 + 246.48 + 368.13 + 450.18 + 623.21 + 799.84 = 2765.07
\]

Now calculate \( \sum x^2 \):
\[
\sum x^2 = 10^2 + 15^2 + 20^2 + 25^2 + 30^2 + 35^2 + 40^2 = 100 + 225 + 400 + 625 + 900 + 1225 + 1600 = 4175
\]

### Step 4: Plug values into formulas

Now substitute these values into the slope formula:

\[
m = \frac{7(2765.07) - (175)(101.963)}{7(4175) - (175)^2}
\]

Calculating the numerator:
\[
7(2765.07) = 19355.49
\]
\[
(175)(101.963) = 17844.025
\]
So,
\[
\text{Numerator} = 19355.49 - 17844.025 = 1511.465
\]

Calculating the denominator:
\[
7(4175) = 29225
\]
\[
(175)^2 = 30625
\]
So,
\[
\text{Denominator} = 29225 - 30625 = -1400
\]

Thus the slope:
\[
m = \frac{1511.465}{-1400} \approx -1.078
\]

### Step 5: Calculate \( b \)

Now calculate \( b \):

\[
b = \frac{101.963 - (-1.078)(175)}{7}
\]
Calculating:
\[
(-1.078)(175) = -188.65
\]
\[
b = \frac{101.963 + 188.65}{7} = \frac{290.613}{7} \approx 41.516
\]

### Conclusion

Putting it all together, the linear model is:

\[
y \approx -1.078x + 41.516
\]

To round your results to three decimal places:

\[
\text{Linear model: } y = -1.078x + 41.516
\]

Feel free to adjust if needed!