Question

The table lists the average monthly Social Security benefits \(B\) (in dollars) for retired workers aged 62 and over from 1998 through 2005 . A model for the data is \(B=\frac{582.6+38.38 t}{1+0.025 t-0.0009 t^{2}}, \quad 8 \leq t \leq 15\) where \(t=8\) corresponds to 1998 . $$ \begin{array}{|l|l|l|l|l|l|l|l|l|} \hline t & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline B & 780 & 804 & 844 & 874 & 895 & 922 & 955 & 1002 \\ \hline \end{array} $$ (a) Use a graphing utility to create a scatter plot of the data and graph the model in the same viewing window. How well does the model fit the data? (b) Use the model to predict the average monthly benefit in \(2008 .\) (c) Should this model be used to predict the average monthly Social Security benefits in future years? Why or why not?

Short answer

The steps do not provide actual numerical answers because they involve the use of a graphing utility and critical thinking. For step 2, you would get a numerical value when you substitute \(t = 18\) into the model equation. For step 3, the answer would be a detailed explanation based on your critical analysis of the model and its fit to the data.

Step-by-step solution

Create a scatter plot of the data and graph the model

Normally, you would use a graphing calculator or graphing software to input the model equation and the actual data points. Since the exercise is theoretical, it is understood that you should be able to represent actual data as points in a scatter plot. For the model data, it should appear as a curve on the plot. After creating both graphs, you can determine the model's validity by visually comparing the scatter plot and the model.

Predict the average monthly benefit for 2008

To do this, you would use the given model and substitute the approtiate 't' value corresponding to 2008. In this case, \(t = 18\) (2008 is 18 years after 1990). So, you will substitute \(t = 18\) into the model \(B=\frac{582.6+38.38 t}{1+0.025 t-0.0009 t^{2}}\) and calculate the value of 'B'.

Evaluate the model's future predictive ability

Here, you must consider whether this model is reasonable based on previous results and understanding of mathematical models. This step typically does not involve calculations; instead, you would consider factors such as the fit of the model to the data, the trend of the data, and the nature of the underlying phenomenon being modeled. It's also important to remember that all models have limitations and might not fully capture all influences on the variable being predicted.

Key concepts

Scatter Plot

A scatter plot is a type of graph used to display data points on a Cartesian coordinate system. It allows us to see the relationship between two variables by plotting them against each other. In the context of algebraic modeling, scatter plots are essential to visually assess how well a mathematical model fits real-world data.

When creating a scatter plot for the Social Security benefits data, each year, denoted as variable 't', corresponds to a point on the x-axis, and each average benefit, 'B', is a point on the y-axis. By plotting these points, we can observe the trend in benefits over the years from 1998 to 2005.

Once the scatter plot is prepared, we can add the model's graph on the same plot. This helps in comparing the data points with the model's prediction curve. If the model fits well, the curve will closely follow the pattern of the scattered data points, indicating that the model accurately predicts the changes in Social Security benefits over that time period.

Predictive Analysis

Predictive analysis involves using historical data to make informed predictions about future events. In mathematical modeling, it is an invaluable tool for forecasting future outcomes based on established patterns.

To predict Social Security benefits for the year 2008 using the given model, you would set the model's equation with 't' as 18 (because 2008 is 18 years after 1990). By substituting this into the mathematical formula:\[ B = \frac{582.6 + 38.38 t}{1 + 0.025 t - 0.0009 t^2} \]and calculating, you can find the corresponding benefit amount. This prediction helps policymakers and planners manage expectations and make informed decisions about Social Security allocations in the future.

However, it's important to remember that predictive analysis relies on the assumption that past patterns will continue. If the underlying conditions change dramatically, the predictive capability of a model might diminish.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent real-world systems. It helps in understanding, analyzing, and predicting phenomena by translating complex situations into simpler, mathematical terms.

In the Social Security benefits scenario, a rational function is used to model the benefit values over time. The equation:\[ B = \frac{582.6 + 38.38 t}{1 + 0.025 t - 0.0009 t^2} \]represents how the benefits change with time, 't'. This model captures trends like increase or decrease in benefits over the specified period and allows for both interpolation and extrapolation.

When contemplating the use of such a model for future predictions, it's crucial to evaluate the suitability of the model. Some aspects to consider are the accuracy of the model fit, its complexity relative to error margins, and external factors affecting Social Security benefits. All mathematical models have inherent limitations and may require adjustments as new data becomes available or as underlying circumstances evolve.