Problem 3 - College Algebra and Calculus: An Applied Approach

Question

The yearly revenues (in billions of dollars) of UPS from 1997 to 2005 are given by the following ordered pairs. \(\begin{array}{lll}(1997,22.5) & (1998,24.8) & (1999,27.1) \\ (2000,29.8) & (2001,30.6) & (2002,31.3) \\\ (2003,33.5) & (2004,36.6) & (2005,42.6)\end{array}\) Use a graphing utility to create a scatter plot of the data. Let \(x=7\) represent 1997 . Then use the regression feature of the graphing utility to find a best-fitting line for the data. Graph the model and the data together. How closely does the model represent the data? (Source: United Parcel Service)

Step-by-Step Solution

Answer

The exact representation of the data by the model cannot be given without the actual graph. However, depending on how closely the data points align along the line of best fit and follow the same direction, one could say the model accurately or inadequately represents the data.

Step 1: Prepare the data

Data are given as years from 1997 and the corresponding revenue for UPS in those years. The data is already provided as points in this case, so it is important to list them out clearly. Let \(x = 7\) represent the year 1997, and so on. Convert the years into this format.

Step 2: Create a scatter plot

Using a graphing utility, input the data points and create a scatter plot. Each point in the scatter plot represents the revenue for a certain year. Thus, the scatter plot should be showing an upward trend as the revenue increases each year.

Step 3: Find the best-fitting line

Once the scatter plot is created, use the regression feature of the graphing utility to find a best-fitting line for the data. This line will be a straight line that best fits the scatter plot points. It may not pass all points but it represents the overall trend of the scatter data.

Step 4: Graph the model and the data

Graph the best-fitting line along with the scatter plot on the same graph. The best-fitting line will serve as the model for the data.

Step 5: Analyze the model

Finally, look at how closely the model (the best-fitting line) represents the data. If the points lie close to this line and follow the same direction of this line, then the model is said to fit the data well. Otherwise, if the data points are scattered in such a way that the direction of the points does not match that of the line, the model doesn't represent the data accurately.

Key Concepts

Best-Fitting Line

When creating graphs from data, the best-fitting line, also known as the line of best fit or the trend line, plays a crucial role. It is essentially a straight line that passes through the 'cloud' of scatter plot points in such a way that minimizes the distance of all the points from the line.

Think of it as playing a game of balance; the line tries to stay close to as many points as possible. Mathematically, the best-fitting line is often found using the least squares method, which calculates the line that minimizes the sum of the squares of the vertical distances of the points from the line itself. In the context of the UPS revenue problem:

• Each year is converted to a number starting with 7 for 1997.
• Each year's revenue becomes a point on the scatter plot with its respective year number.
• A graphing utility then mathematically determines the straight line that best represents the upward revenue trend over time.

The accuracy of this line with respect to the actual data points can tell us much about the consistency and predictability of UPS's revenue growth.

Regression Analysis

The process of finding the best-fitting line through a scatter plot is known as regression analysis. It's a form of predictive modelling technique which investigates the relationship between a dependent (target) and independent (predictor) variables. For example, in our UPS exercise, the dependent variable is the revenue, and the independent variable is the year.

Using regression analysis:

• We can quantify the rate at which revenue has increased over the years.
• Determine if the increase is consistent.
• And even make future predictions assuming the trend continues similarly.

Linear regression, which is what we're discussing here, results in a best-fitting line also known as the regression line. This can be represented by the equation of the line, which is often in the form of \( y = mx + b \), where \m\ is the slope of the line, and \b\ is the y-intercept. The regression line gives us a mathematical model that we can use to predict UPS's future revenues given the data from 1997 to 2005.

Graphing Utility

A graphing utility is an indispensable tool in statistical analysis, especially when working with large datasets or complex calculations, such as finding the best-fitting line through regression analysis.

In the case of our UPS data:

• Firstly, the graphing utility provides a visual representation of the scatter plot, which is the graphical distribution of all the data points.
• Then, it offers a regression feature that automatically calculates the best-fitting line by considering all the data points and applying the least squares method or another algorithm.
• The graphing utility might include other useful features, such as plotting multiple sets of data, zooming in/out of specific sections of the graph, or even calculating other types of regression lines, like polynomial or exponential, if needed.

By driving the computational side, the graphing utility frees students and researchers to focus on the analysis and interpretation of the data, which are critical skills in both educational and professional settings.