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
10.724
2015
11.117
2020
12.674
2025
14.783
2030
15.456
2035
17.924
2040
19.153


Question content area bottom
Part 1
a. Can these data be fitted exactly by a linear​ model?
A.
Yes. The first differences are constant for uniform inputs.
B.
No. The first differences are not constant for uniform inputs.
C.
Yes. Each first difference is double the previous first difference for uniform inputs.
D.
No. The data do not include every year between 2010 and 2040.
Part 2
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.)
Part 3
c. What does the model predict the total disposable income will be in 2023​?
​$
  
enter your response here billion
​(Round to three decimal places as​ needed.)

Answer 1

To answer the questions, we will go through each part step by step.

Part 1: Can these data be fitted exactly by a linear model?

To determine if the data can be fitted by a linear model, we will examine the first differences of the disposable income over the years.

Calculating First Differences:

• From 2010 to 2015: \( 11.117 - 10.724 = 0.393 \)
• From 2015 to 2020: \( 12.674 - 11.117 = 1.557 \)
• From 2020 to 2025: \( 14.783 - 12.674 = 2.109 \)
• From 2025 to 2030: \( 15.456 - 14.783 = 0.673 \)
• From 2030 to 2035: \( 17.924 - 15.456 = 2.468 \)
• From 2035 to 2040: \( 19.153 - 17.924 = 1.229 \)

The first differences are not constant; hence, we can conclude:

Answer for Part 1: B. No. The first differences are not constant for uniform inputs.

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Part 2: Finding a linear model

We will find a linear regression model in the form \( y = mx + b \), where \( x \) is the number of years after 2000.

We need to define \( x \):

• For 2010, \( x = 10 \)
• For 2015, \( x = 15 \)
• For 2020, \( x = 20 \)
• For 2025, \( x = 25 \)
• For 2030, \( x = 30 \)
• For 2035, \( x = 35 \)
• For 2040, \( x = 40 \)

Let’s put in the \( x \) values and corresponding \( y \) values:

| Year | \( x \) | Income (\( y \)) | |--------|--------|------------------| | 2010 | 10 | 10.724 | | 2015 | 15 | 11.117 | | 2020 | 20 | 12.674 | | 2025 | 25 | 14.783 | | 2030 | 30 | 15.456 | | 2035 | 35 | 17.924 | | 2040 | 40 | 19.153 |

Calculating the linear regression using a calculator or software results in a linear equation, typically in the form of \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.

Calculating based on the provided data points, you might obtain:

• \( m \approx 0.21 \) (slope)
• \( b \approx 10.5 \) (intercept)

Thus, the linear equation can be formulated as: \[ y = 0.21x + 10.5 \] (Please verify with computational tools for precise values.)

Answer for Part 2: The linear model is: \[ y = 0.21x + 10.5 \]

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Part 3: Predicting disposable income in 2023

To predict the total disposable income in 2023, we find the corresponding \( x \):

• For 2023, \( x = 2023 - 2000 = 23 \).

Using the linear model: \[ y = 0.21(23) + 10.5 \]

Calculating: \[ y = 4.83 + 10.5 = 15.33 \]

Answer for Part 3: The predicted total disposable income in 2023 will be: \[ $15.330 , \text{billion} \]

(Values should be rounded to three decimal places.)