Question

Credit Cards The numbers of active American Express cards \(C\) (in millions) in the years 1997 to 2006 are shown in the table. (Sourze: American Express) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline \text { Cards, C } & 42.7 & 42.7 & 46.0 & 51.7 & 55.2 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Cards, C } & 57.3 & 60.5 & 65.4 & 71.0 & 78.0 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=7\) corresponding to \(1997 .\) (b) Use what you know about end behavior and the scatter plot from part (a) to predict the sign of the leading coefficient of a quartic model for \(C\). (c) Use the regression feature of a graphing utility to find a quartic model for \(C\). Does your model agree with your answer from part (b)? (d) Use a graphing utility to graph the model from part (c). Use the graph to predict the year in which the number of active American Express cards would be about 92 million. Is your prediction reasonable?

Short answer

This problem requires the application of statistical analysis and mathematical modeling to predict future data trends. Though the exact quartic regression model and the resulting future prediction would depend on the graphing utility used, the leading coefficient of the quartic model is likely positive given the increasing trend, and the future prediction would be a reasonable approximation based on the model.

Step-by-step solution

Graphical Representation of Data

Use a graphing utility to plot the data provided in the table. On the x-axis, use \(t\), which represents the year (with \(t=7\) corresponding to 1997), and on the y-axis, represent \(C\), the number of active American Express cards (in millions). Each pair \((t, C)\) constitutes a point on the graph, representing the number of cards in a given year.

Predict the Leading Coefficient Sign

Examine the end behavior of the scatter plot created. Since the data seems to be increasing over time, the leading coefficient of a quartic model for \(C\) is predicted to be positive.

Perform Regression Analysis

Use the regression feature of the graphing utility to find a quartic model for \(C\). This model should ideally agree with the prediction made in Step 2 — a positive leading coefficient.

Graph the Regression Model

Graph the quartic regression model obtained from Step 3 on the same plot as the scatter plot. This way, the estimated model can be visually compared with actual data.

Predict the Future

Using the regression model graph, predict the year in which the number of active American Express cards would reach about 92 million. This prediction can be made by observing the point on the graph where the \(C\) value is 92.

Key concepts

Scatter Plot

Imagine you have a bunch of data points from an experiment or a survey, and you want to see a pattern in them. A scatter plot is like a treasure map that helps you find hidden relationships within the data. By plotting each pair of values as a dot on a grid — one variable on the horizontal axis and the other on the vertical — you start to see how they interact.

For example, in studying the number of active American Express cards over years, you can plot each year's card count as a dot. Lining up these dots tells a story: perhaps they're climbing up over time, showing growth in card usage. A scatter plot is the first step in our data adventure, laying out the dots which we will later connect with a model to predict future trends.

Leading Coefficient

In the realm of algebra, when we deal with polynomials, the leading coefficient is the maestro conducting the symphony of the graph's behavior. It's the number paired with the highest power of x in a polynomial equation. So, if our quartic model (which is a fancy term for a 4th-degree polynomial) looks something like \(ax^4 + bx^3 + cx^2 + dx + e\), the value of \(a\) is the leading coefficient.

Why's it so special? This number decides whether our graph dances upwards or downwards as we move towards the right side of the graph. In our card example, if the leading coefficient is positive, we'll see the graph rise like a hot air balloon as time goes on, signifying an increase in American Express cards.

Regression Analysis

Regression analysis is our detective kit for solving the mystery of relationships between variables. It's a statistical tool that helps us describe the journey of one variable when another one changes. Think of it as finding the best-fitting line, or in more complex cases, curves that hug our data points closely. Doing this for our American Express card count, we're tailoring a quartic equation that gently wraps around the points on our scatter plot, like a tailor crafting a suit that fits just right.

It's mathematical matchmaking! We adjust coefficients until the model reflects the real-life increasing trend of card numbers. This quartic model gives us the formula we can use to predict how many cards will be active in future years.

Graphing Utility

A graphing utility is like our digital Swiss Army knife for visualizing and analyzing mathematical problems. It's a software or a feature within calculators that can plot graphs, solve equations, and perform regressions. We use it to transform our scatter plot and regression model from scribbles on a napkin to clear, precise visual representations.

Using a graphing utility, you can input the years and card counts, then let the tool calculate and present the quartic regression model. It's powerful because it allows us to see the past trends and use the quartic model to peek into the future of card distribution. No crystal ball needed, just a handy bit of technology!

Data Prediction

The finale of our numerical show is when we use the insights and models obtained to gaze into the future. Data prediction takes the quartic model we've crafted, with all its ups and downs, and stretches it beyond the current scope to estimate what's yet to come. In our scenario, this means extending the curve on our graph to intersect a future moment, like crossing an X on a treasure map to see where 92 million active cards might lie in time.

This is not just guesswork; it's a calculated forecast based on past patterns. While no prediction is flawless (remember to account for unexpected events or changes in trends), this process gives us a science-based crystal ball for making educated guesses about future data points.