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.

Question

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.
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​(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!