Problem 60
Question
ECONOMICS: Per Capita Personal Income In the short run, per capita personal income (PCPI) in the United States grows approximately linearly. In 2001 PCPI was \(30.4\), and in 2009 it had grown to \(39.2\) (both in thousands of dollars). a. Use the two given (year, PCPI) data points \((1,30.4)\) and \((9,39.2)\) to find the linear relationship \(y=m x+b\) between \(x=\) years since 2000 and \(y=\mathrm{PCPI}\). b. Use your linear relationship to predict PCPI in 2020 .
Step-by-Step Solution
Verified Answer
The linear equation is \(y = 1.1x + 29.3\), predicting a PCPI of 51.3 in 2020.
1Step 1: Identify the Two Points
We are given two data points: \((1,30.4)\) for the year 2001 and \((9,39.2)\) for the year 2009. These are coordinates \((x_1, y_1)\) and \((x_2, y_2)\) on a graph where \(x\) is the number of years since 2000 and \(y\) is the PCPI in thousands.
2Step 2: Calculate the Slope
The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}.\] Substitute the given values: \[m = \frac{39.2 - 30.4}{9 - 1} = \frac{8.8}{8} = 1.1.\] So, the slope \(m\) is 1.1.
3Step 3: Write the Linear Equation
The linear equation has the form \(y = mx + b\). We have the slope \(m = 1.1\), and one of the points \((x_1, y_1) = (1, 30.4)\) can be used to find \(b\). Substitute into the equation: \[30.4 = 1.1(1) + b.\] Solving for \(b\), we find: \[b = 30.4 - 1.1 = 29.3.\] So, the equation is \(y = 1.1x + 29.3.\)
4Step 4: Predict PCPI in 2020
To predict PCPI in 2020, substitute \(x = 20\) (since 2020 is 20 years after 2000) into the linear equation: \[y = 1.1(20) + 29.3.\] Calculate the result: \[y = 22 + 29.3 = 51.3.\] Thus, the predicted PCPI for 2020 is 51.3 thousand dollars.
Key Concepts
Slope CalculationEquation of a LineData Prediction
Slope Calculation
In linear regression, the slope calculation helps us understand how one variable changes in relation to another. When given two points, we can find the slope of the line that passes through these points. The formula to calculate the slope \(m\) when given points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]This formula gives the rate at which \(y\) changes with respect to \(x\). For instance, if the slope \(m\) is positive, it indicates that as \(x\) increases, \(y\) also increases, and vice-versa for a negative slope.
In our exercise, we used two data points: \((1, 30.4)\) and \((9, 39.2)\). Substituting these into the formula, we calculated the slope as \(1.1\). This value tells us that for every additional year after 2000, the per capita personal income increased by \(1.1\) thousand dollars.
In our exercise, we used two data points: \((1, 30.4)\) and \((9, 39.2)\). Substituting these into the formula, we calculated the slope as \(1.1\). This value tells us that for every additional year after 2000, the per capita personal income increased by \(1.1\) thousand dollars.
Equation of a Line
After calculating the slope, the next step in linear regression is to form the equation of a line. This equation helps describe the relationship between the two variables. The general form of the equation is:\[y = mx + b,\]where \(m\) represents the slope and \(b\) is the y-intercept. The y-intercept \(b\) can be found by substituting one of the data points into the equation and solving for \(b\).
In our example, we used the point \((1, 30.4)\) and the calculated slope \(m = 1.1\) to find:\[30.4 = 1.1(1) + b.\]Solving for \(b\) gives us \(b = 29.3\). Thus, the full equation for our line becomes:\[y = 1.1x + 29.3.\]This equation allows you to predict the per capita personal income for any year by substituting \(x\) with the number of years since 2000.
In our example, we used the point \((1, 30.4)\) and the calculated slope \(m = 1.1\) to find:\[30.4 = 1.1(1) + b.\]Solving for \(b\) gives us \(b = 29.3\). Thus, the full equation for our line becomes:\[y = 1.1x + 29.3.\]This equation allows you to predict the per capita personal income for any year by substituting \(x\) with the number of years since 2000.
Data Prediction
Once the equation of a line is established, it becomes a powerful tool for prediction. By inputting values for \(x\) into the equation, we can predict the likely value of \(y\). This process is known as data prediction.
In the exercise, we wanted to forecast the per capita personal income for the year 2020. Given that 2020 is 20 years after 2000, we set \(x = 20\) and substitute into our linear equation:\[y = 1.1(20) + 29.3.\]The calculation results in:\[y = 22 + 29.3 = 51.3.\]Hence, using our linear model, we predict that the per capita personal income in 2020 would be \(51.3\) thousand dollars. This method can be used to predict future values as long as the data trend remains consistent.
In the exercise, we wanted to forecast the per capita personal income for the year 2020. Given that 2020 is 20 years after 2000, we set \(x = 20\) and substitute into our linear equation:\[y = 1.1(20) + 29.3.\]The calculation results in:\[y = 22 + 29.3 = 51.3.\]Hence, using our linear model, we predict that the per capita personal income in 2020 would be \(51.3\) thousand dollars. This method can be used to predict future values as long as the data trend remains consistent.
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