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.)
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.)
Answers
GPT-4o mini
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!
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!