Question

Super Bowl Ad Revenue The table shows Super Bowl TV ad revenues \(R\) (in millions of dollars) for several years from 1987 to 2006. (Source: TNS Media Intelligence)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Revenue } \\ \hline 1987 & 31.5 \\ \hline 1992 & 48.2 \\ \hline 1997 & 72.2 \\ \hline 2002 & 134.2 \\ \hline 2006 & 162.5 \\ \hline \end{array} $$(a) Use a spreadsheet software program to create a scatter plot of the data. Let \(t\) represent the year, with \(t=7\) corresponding to 1987 . (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 a spreadsheet software program to graph the exponential model in base \(e\). (d) Use the exponential model in base \(e\) to predict the Super Bowl ad revenues in 2009 and in 2010 .

Short answer

The steps will lead to a scatter plot of the data, an exponential model for the data, a graph of this model, and predictions of revenues for specific years. Note: Precise numerical results cannot be given without actually applying these steps using the actual data on a spreadsheet software.

Step-by-step solution

Creating the Scatter Plot

Use a spreadsheet software such as Microsoft Excel or Google Sheets to input the data given in the exercise. Plot the data on a scatter plot with year(t) on the x-axis and revenue(R) on the y-axis. Ensure that t=7 is synonymous with the year 1987.

Finding an Exponential Model

With the scatter plot in place, utilize the regression feature of the spreadsheet software to find an exponential model that best fits the data. Given the rising nature of the revenues with time, an exponential model is a suitable representation of this relationship. After finding the model \(R = ab^t\), use the Inverse Property \(b = e^{ln b}\) to rewrite the model as \(R = ae^{(ln b)t}\). This is the exponential model in base \(e\).

Graphing the Exponential Model

Next is to graph this new exponential model \(R = ae^{(ln b)t}\) using the same spreadsheet software.

Predicting Future Revenues

Finally, use the exponential model in base \(e\) to predict the Super Bowl ad revenues for the year 2009 (for which t=29) and 2010 (for which t=30). To do this, substitute these t-values into the model and solve for R to get the respective predictions.

Key concepts

Scatter Plot

A scatter plot is a type of data visualization that shows the relationship between two quantitative variables. In the context of Super Bowl ad revenues, a scatter plot helps visualize how revenues have changed over the years. To create one, you can use spreadsheet software like Google Sheets or Microsoft Excel. Here's how you can do it:

* Input your data into the spreadsheet, with years in one column and corresponding revenues in another.
* Select the data and insert a scatter plot graph.
* On the x-axis, plot the years, adjusting them based on a variable such as "t" where, for example, "t=7" represents the year 1987, and increase "t" by 5 for subsequent values.

This visual distribution of data points helps identify patterns or trends, such as exponential growth, that might otherwise go unnoticed in a table format.

Regression Analysis

Regression analysis is a statistical method that allows you to examine the relationship between two or more variables. In the case of determining an exponential growth model for Super Bowl ad revenues, it involves finding a mathematical equation that best fits the scatter plot data. This is crucial because it provides a predictive formula that can be used for future forecasting.

* With your scatter plot ready, use the built-in regression feature in your spreadsheet software to perform the analysis.
* Choose an exponential regression model, as it typically fits data showing rapid growth over fixed intervals.

The result of this analysis is an equation often in the form of \( R = ab^t \), which can be further refined to \( R = ae^{(\ln b)t} \). This final form reflects the exponential model in base \( e \). Utilizing regression analysis thus provides a powerful tool for making sense of historical data and employing it for projections.

Spreadsheet Software

Spreadsheet software like Microsoft Excel or Google Sheets is extremely beneficial for data analysis due to its powerful graphing and computational features. For the exercise on Super Bowl ad revenues, it serves as the main tool to perform tasks such as plotting data on a scatter plot and conducting regression analysis.

* Begin by entering your data into the grid format of the spreadsheet.
* Use functions to manipulate and analyze your data for better insights; particularly, use the graphing tools to visualize the data and ascertain patterns.
* Most importantly, utilize the statistical functions such as regression analysis to derive predictive models from your data.

Understanding how to leverage these tools within spreadsheet software enhances your ability to organize, analyze, and predict data trends efficiently and accurately.

Revenue Prediction

Revenue prediction utilizes mathematical models derived from regression analysis to forecast future trends based on historical data. For Super Bowl ad revenues, once you've established your exponential growth model, predicting future revenue becomes a straightforward process.

* Use the derived exponential model in base \( e \), for example, \( R = ae^{(\ln b)t} \), where you substitute future "t" values to estimate the revenues.
* Calculate for years like 2009 and 2010 by replacing "t" with the respective numerical values corresponding to these years (e.g., "t=29" for 2009).

These predictions give key insights into business growth, helping stakeholders make informed decisions. Accurate revenue predictions are crucial for setting sales targets, budgeting, and strategic planning.