Question

The numbers of working-age civilians (in millions) in the United States that were not involved in the labor force from 1995 to 2005 are given by the following ordered pairs. \((1995,66.3) \quad(1996,66.6) \quad(1997,66.8)\) \((1998,67.5) \quad(1999,68.4)\) \((2000,70.0)\) \(\begin{array}{lll}(2001,71.4) & (2002,72.7) & (2003,74.7)\end{array}\) \((2004,76.0) \quad(2005,76.8)\) A linear model that approximates the data is \(y=1.16 t+59.1,5 \leq t \leq 15\), where \(y\) is the number of civilians (in millions) and \(t=5\) represents \(1995 .\) Plot the actual data with the model. How closely does the model represent the data? (Source: U.S. Bureau of Labor Statistics)

Short answer

To determine how closely the linear model represents the actual data, one must plot both on the same graph and compare. The visual comparison will provide an approximation of how well the model fits the data. A detailed analysis might require a calculation of residuals or other statistical measures that were not asked for in this exercise.

Step-by-step solution

- Understanding Components of the Linear Model

The linear model provided is given by the equation \(y = 1.16t+59.1\), where \(y\) represents the number of civilians in millions and \(t\) represents the time in years. In this equation, the year 1995 is represented by \(t = 5\), which means the \(t\) value is calculated by subtracting 1990 from the actual year.

- Plotting the Actual Data

First, convert all the years into their corresponding 't' values by subtracting 1990 from the each year. Once that is done, plot these ordered pairs on a graph where the x-axis represents the value of \(t\) (time) and the y-axis represents the value of \(y\) (number of civilians in millions).

- Plotting the Linear Model

Next, plot the linear model \(y = 1.16t+59.1\) on the same graph. As the value of \(t\) ranges from \(5 \leq t \leq 15\), plot this equation within this range.

- Comparing the Linear Model with the Actual Data

After both plots are on the same graph, visually analyze how closely the line representing the linear model fits the actual data points. This can be done by checking the proximity of the points to the line and the distribution of the points around the line.

Key concepts

Plotting Data

Plotting data is a foundational skill in statistics and algebra that helps us visualize the relationship between different variables. In our case, we're interested in the number of working-age civilians in the United States not involved in the labor force over a period from 1995 to 2005. Each ordered pair, such as (1995, 66.3), represents a year and the corresponding number of civilians (in millions) not in the labor force.
To plot these pairs, we first convert the years to a more manageable number that indicates time passed since 1990. This is where 't' comes in, with each 't' representing the year subtracted by 1990, simplifying our graph and ensuring a tidy x-axis. With the years converted, we place points on a graph with the x-axis labeled as 't' and the y-axis labeled with the number of civilians. Correct plotting provides a clear visual representation that will be essential for comparing with our linear model and understanding the trend.

Linear Regression

Linear regression is a powerful tool used to create a model that best fits a set of data. In the context of our exercise, the equation \(y = 1.16t + 59.1\) represents the linear regression line — this line is the result of calculating the trend that best represents the distribution of our data points. The number 1.16 is the slope of the line, showing the average change in the number of non-working civilians for each year. The 59.1 is the y-intercept, indicating where the line would cross the y-axis when \(t = 0\).In constructing this line, the linear regression process has taken into consideration all of the data, working to minimize the distance between the plotted points and the line itself, which results in a model that provides the approximate number of civilians not involved in the labor force for any given year within our data range. It's an essential technique that allows us to predict values and analyze trends over time.

Modeling with Algebra

Modeling with algebra involves utilizing algebraic equations to represent real-world situations. The linear model \(y = 1.16t + 59.1\) is a prime example of this, translating the behavior of a sociological phenomenon into mathematical terms. The linear model simplifies the complex factors affecting workforce involvement into an equation that's easy to work with and understand. With algebraic modeling, we are not only engaging in an exercise of mathematical representation but also gaining the means to infer, predict, and propose hypotheses about future scenarios. It equips us with a mathematical language to articulate patterns we observe in data, thereby transforming abstract numbers into meaningful narratives. Whether it be forecasting economic trends or understanding population dynamics, algebraic models are invaluable tools for both the classroom and professional research.