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
10.916
2015
11.028
2020
12.545
2025
14.688
2030
15.132
2035
17.651
2040
19.564


Question content area bottom
Part 1
a. Can these data be fitted exactly by a linear​ model?
A.
No. The data do not include every year between 2010 and 2040.
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.
Yes. The first differences are constant for uniform inputs.
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 tackle the questions based on the given data of disposable income, let's break down each part.

Part 1

We need to evaluate whether the data can be fitted exactly by a linear model. To determine this, we need to analyze the first differences of the income data provided.

The first differences can be calculated by subtracting the previous year's income from the current year's income.

Calculating the first differences:

\[ \begin{align*} 2015 - 2010 & : 11.028 - 10.916 = 0.112 \ 2020 - 2015 & : 12.545 - 11.028 = 1.517 \ 2025 - 2020 & : 14.688 - 12.545 = 2.143 \ 2030 - 2025 & : 15.132 - 14.688 = 0.444 \ 2035 - 2030 & : 17.651 - 15.132 = 2.519 \ 2040 - 2035 & : 19.564 - 17.651 = 1.913 \ \end{align*} \]

Since the first differences are not constant, the correct answer is: B. No. The first differences are not constant for uniform inputs.

Part 2

Next, we will find a linear model \( y = mx + b \), where \( x \) is the number of years after 2000, and \( y \) is the total disposable income in billions of dollars.

We can use two points from the data to determine the slope \( m \) and y-intercept \( b \). Let's use the years 2010 (x=10, y=10.916) and 2040 (x=40, y=19.564):

Calculating the slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{19.564 - 10.916}{40 - 10} = \frac{8.648}{30} \approx 0.288 \]

Calculating the y-intercept \( b \): Using the point (10, 10.916): \[ y = mx + b \implies 10.916 = 0.288(10) + b \implies 10.916 = 2.88 + b \implies b \approx 10.916 - 2.88 \approx 8.036 \]

The linear model can be written as: \[ y = 0.288x + 8.036 \]

Rounded to three decimal places, it is: \[ y = 0.288x + 8.036 \]

Part 3

Now we use the linear model to predict the total disposable income in 2023 (which is 23 years after 2000, or \( x = 23 \)):

\[ y = 0.288(23) + 8.036 \]

Calculating: \[ y \approx 0.288 \times 23 + 8.036 \approx 6.624 + 8.036 \approx 14.660 \]

So, the predicted total disposable income in 2023 is approximately: \[ \boxed{14.660 \text{ billion dollars}} \]

In summary:

• Part 1: B. No. The first differences are not constant for uniform inputs.
• Part 2: \( y = 0.288x + 8.036 \)
• Part 3: \( 14.660 \) billion dollars in 2023.