Question

Domestic Demand The domestic demands \(D\) (in thousands of barrels) for refined oil products in the United States from 1995 to 2005 are shown in the table. (Source: U.S. Energy Information Administration)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Demand } \\ \hline 1995 & 6,469,625 \\ \hline 1996 & 6,701,094 \\ \hline 1997 & 6,796,300 \\ \hline 1998 & 6,904,705 \\ \hline 1999 & 7,124,435 \\ \hline 2000 & 7,210,566 \\ \hline \end{array} $$$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Demand } \\ \hline 2001 & 7,171,885 \\ \hline 2002 & 7,212,765 \\ \hline 2003 & 7,312,410 \\ \hline 2004 & 7,587,546 \\ \hline 2005 & 7,539,440 \\ \hline \end{array} $$(a) Use a spreadsheet software program to create a scatter plot of the data. Let \(t\) represent the year, with \(t=5\) corresponding to 1995 . (b) Use the regression feature of a spreadsheet software program to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use the regression feature of a spreadsheet software program to find a logarithmic model \((y=a+b \ln x)\) for the data. (d) Use a spreadsheet software program to graph the exponential model in base \(e\) and the logarithmic model with the scatter plot. (e) Use both models to predict domestic demands in 2008 , 2009, and \(2010 .\) Do both models give reasonable predictions? Explain.

Short answer

The short answer can't be determined without a spreadsheet program. However, to provide one, this information would follow the procedure highlighted in the steps: creating a scatter plot, running regression analyses to define an exponential and a logarithmic model, graphically representing the models, and predicting future data points. The models' reasonableness would be judged based on how closely they follow the observed trend in the data.

Step-by-step solution

Scatter plot

Firstly, enter the provided data into two columns of a spreadsheet, where the first column represents the Year and the second column represents the corresponding Demand. Using these columns, generate a scatter plot. This provides a visual representation of the data.

Exponential regression

Using the regression feature of the spreadsheet, compute an exponential regression model for the data, which results in an equation of the form \(D = a \cdot b^t\). Use the expression \(b=e^{ln(b)}\) to rewrite this model in base e, yielding \(D=a \cdot e^{kt}\).

Logarithmic regression

Reperform the regression, but now choose a logarithmic model. This results in a model of the form \(D = a + b \cdot ln(t)\).

Graphing Models

Using the spreadsheet program, generate graphs for both models (the exponential model and the logarithmic model) on top of the scatter plot to show their fit to the data.

Prediction and Analysis

Substitute values of \(t\) corresponding to the years 2008, 2009 and 2010 into both models, to predict demand. The reasonableness of the models can be determined by analysing how closely the predicted values align with known trends and/or values.

Key concepts

Scatter Plot

A scatter plot is a type of data visualization that shows the relationship between two sets of data. In algebra, we often use scatter plots to visualize the relationship between two variables. For instance, in our exercise, the scatter plot would display the years (independent variable) along the horizontal axis and the domestic demand for oil (dependent variable) along the vertical axis.

Each point on the plot would represent the demand for a specific year, allowing us to see trends over time at a glance. Creating a scatter plot is usually the first step in data analysis as it provides a foundation for determining the appropriate model to fit the data.

Exponential Regression

Exponential regression fits an exponential model to data which is believed to increase (or decrease) at a rate proportional to its current value. This type of regression is helpful when dealing with growth or decay processes, like population growth or radioactive decay. The general form of an exponential model is \( D = a \times b^t \), where \(a\) is the initial amount, \(b\) is the growth factor, and \(t\) is time.

In our exercise, by using the provided data and exponential regression, we find a model that best fits the growth pattern of domestic oil demand. Rewriting the model using the natural base \(e\) provides a simplified version that can be used for further analysis and predictions.

Logarithmic Regression

Logarithmic regression is used when data growth slows over time, leveling off as it progresses. In contrast to exponential growth, logarithmic models are characterized by a rapid increase initially that decreases and approaches a straight line over time. The general equation for a logarithmic trend is \( D = a + b \cdot \ln(t) \), with \(a\) and \(b\) being coefficients fitted to the data.

When applying this to our problem, logarithmic regression can provide insight into the oil demand's rate of change, possibly correlating to factors that cause the demand increase to slow down, such as the saturation of a market or the introduction of alternative energy sources.

Predictive Modeling in Mathematics

Predictive modeling in mathematics involves using statistical techniques to make predictions about future events based on historical data. By analyzing past trends and patterns, we can develop a model that, when given new input data, forecasts likely outcomes. This is an integral part of many fields such as finance, meteorology, and engineering.

In the context of our oil demand example, we use previous years' data to build a model that predicts future demand. This is crucial for decision-makers as it helps in planning for production, setting prices, and understanding market dynamics. The exponential and logarithmic regression models built from past data exemplify predictive modeling.

Data Analysis in Algebra

Data analysis in algebra involves collecting, graphical plotting, summarizing, and interpreting data to discern patterns or trends. Algebraic tools such as equations and functions are central to the analysis as they provide a framework for modeling and understanding the relationships in the data. Key steps include visualization with scatter plots, fitting data with regression models, and using these models for prediction.

The use of spreadsheets and software greatly enhances this process, allowing students and professionals to quickly run regression analyses and perform accurate predictions. Consequently, one can make data-driven decisions or hypothesize about future trends, as we've seen with the domestic demand for oil in our exercise.