Question

Restaurants The total sales \(y\) (in billions of dollars) for fast-food and full-service restaurants for the years 1999 to 2005 are shown in the table. (Source: National Restaurant Association $$ \begin{array}{|c|c|c|} \hline \text { Year } & \text { Fast-food } & \text { Full-service } \\ \hline 1999 & 103.0 & 125.4 \\ \hline 2000 & 107.1 & 133.8 \\ \hline 2001 & 111.6 & 139.9 \\ \hline 2002 & 115.1 & 141.9 \\ \hline 2003 & 120.5 & 148.3 \\ \hline 2004 & 129.4 & 157.0 \\ \hline 2005 & 135.6 & 164.9 \\ \hline \end{array} $$ (a) Use a spreadsheet software program to create a scatter plot of the data for fast-food sales and use the regression feature to find a linear model. Let \(x\) represent the year, with \(x=9\) corresponding to 1999. Repeat the procedure for the data for full-service sales. (b) Assuming that the amounts for the given 7 years are representative of future years, will fast-food sales ever equal full-service sales?

Short answer

The exact answer will depend on the specific parameter values obtained from the regression analysis. The method involves creating linear regression models for both fast-food and full-service sales and then solving these simultaneously to predict the year when the sales will be equal.

Step-by-step solution

- Creating the Fast-Food Model

Use a spreadsheet software like Excel to input the data for fast-food sales. The Year column should be converted to start from 9, which represents 1999. Then, create a scatter plot of the data. Use the regression feature in the software to generate a linear regression model. The model will be in the form \(y = mx + c\), where \(y\) represents the sales, \(x\) is the year, \(m\) is the gradient and \(c\) is the y-intercept.

- Creating the Full-Service Model

Perform the same process as in Step 1, but this time for full-service sales. This will result in another linear regression model of the form \(y = mx + c\).

- Predicting Sales Equality

Assuming that the models obtained in Steps 1 and 2 hold for future years, our task is to find out if sales for fast-food will ever equal full-service sales. This requires setting the two equations equal to each other and solving for \(x\). If \(x\) is greater than 9 and within a reasonable future range, then one can say that fast-food sales will eventually equal full-service sales. If \(x\) is not reasonable (indicating a very far future), then the assumption that fast-food and full-service restaurant sales will be equal in the future might not make sense. The definite answer will, however, hugely depend on the specific model parameters obtained from the regression analysis.

Key concepts

Scatter Plot

A scatter plot is an excellent visualization tool used to display the relationship between two variables. When you create a scatter plot, you place one variable on the x-axis and the other on the y-axis. This type of plot is especially useful in identifying trends and patterns.
In the context of restaurant sales data, you can place the years on the x-axis and sales amounts on the y-axis. Each point on the scatter plot represents sales data for a particular year. This visualization can immediately show you if there is a trend, like an increase or decrease in sales over time.

• Helps identify trends and correlations
• Aids in visual comparison between datasets
• Provides a foundation for further analysis, such as regression

By visually appealing and easily interpreted, scatter plots serve as a straightforward first step in data analysis.

Sales Data Analysis

Sales data analysis involves examining sales figures to understand and forecast future performance. In the restaurant industry, this can inform business decisions and strategy formation.
The sales data from fast-food and full-service restaurants from 1999 to 2005 provides valuable insight. Analyzing this data helps identify trends in sales growth or decline. It allows businesses to explore factors that contribute to these trends, such as economic changes or consumer behavior shifts.

• Identifies key patterns or anomalies
• Helps understand market trends
• Lays groundwork for future strategic planning

Using analysis tools like trend lines in a spreadsheet offers a deeper understanding of historical data, potentially guiding future sales strategies.

Forecasting

Forecasting uses historical data to predict future trends. In sales, this asks whether historical sales behavior will continue on the same path.
Applying forecasting with our restaurant sales data entails utilizing our linear regression models generated from the sales data. By analyzing the line of best fit, which emerges from the scatter plot, you can estimate future sales. This is often done using a linear model, which predicts sales depending on the passage of time.
Forecasting is crucial as it helps:

• Plan inventory and staffing
• Set realistic business goals
• Identify potential sales challenges

Although forecasts are never 100% certain, they are informed and improved through careful analysis.

Educational Spreadsheet Usage

Educational spreadsheet tools like Microsoft Excel or Google Sheets are invaluable for data analysis tasks such as creating scatter plots or applying regression analysis.
These programs offer built-in features to facilitate the analysis of complex datasets. For example, you can quickly compute a linear regression model—a fundamental tool for forecasting—using their "trend lines" or "regression analysis" functions.
Working with spreadsheets for educational purposes helps students:

• Develop analytical and critical thinking skills
• Familiarize with data manipulation tools
• Strengthen problem-solving abilities

By learning to use these tools effectively, students can handle real-world data analysis issues in various domains beyond sales.