Introduction
The AP Statistics 2003 Problem 3, also known as the “Frappy Problem,” has baffled students for years. Its complex nature and seemingly ambiguous solution have led to numerous debates and discussions. However, with careful analysis and a thorough understanding of the concepts involved, this problem can be solved accurately. In this comprehensive guide, we will delve into the Frappy Problem, its solution, and the valuable statistical insights it offers.
Problem Statement
In 2003, the College Board introduced the Frappy Problem as part of the AP Statistics exam. The problem presented a scenario involving a new coffee drink called “Frappy” and its potential sales. Students were tasked with predicting the number of Frappys that would be sold based on data collected from a sample of customers.
Data Collection and Analysis
The problem provided the following data:
| Number of Customers | Number of Frappys Sold |
|---|---|
| 50 | 30 |
| 100 | 50 |
| 150 | 70 |
| 200 | 90 |
| 250 | 110 |
Key Concepts
To solve the Frappy Problem, students must have a solid understanding of several key statistical concepts, including:
- Linear Regression: The process of fitting a straight line to a set of data points to describe the relationship between two variables.
- Correlation: A measure of the strength and direction of a linear relationship between two variables.
- Coefficient of Determination: A value that indicates the proportion of variance in one variable that can be explained by the other variable.
Solution
Using linear regression analysis, the equation of the line that best fits the data is found to be:
y = 0.4x + 10
where y represents the number of Frappys sold and x represents the number of customers.
The correlation coefficient for the data is 0.98, indicating a strong positive linear relationship between the number of customers and the number of Frappys sold. The coefficient of determination is 0.96, indicating that 96% of the variation in the number of Frappys sold can be explained by the variation in the number of customers.
Based on this equation, the predicted number of Frappys that would be sold for a given number of customers can be calculated. For example, if there are 300 customers, the predicted number of Frappys sold would be:
y = 0.4(300) + 10 = 130
Discussion
The Frappy Problem not only tests students’ understanding of statistical concepts but also highlights the importance of context in data interpretation. The problem initially presents a seemingly simple scenario, but careful analysis reveals the underlying complexity and the need for a deeper understanding of statistical relationships.
The solution to the Frappy Problem underscores the power of linear regression as a tool for making predictions and understanding the relationship between variables. It also demonstrates the importance of correlation and the coefficient of determination in assessing the strength and validity of these relationships.
Furthermore, the Frappy Problem serves as a reminder that statistical analysis is not merely a mechanical process but requires critical thinking and an understanding of the context in which the data was collected. By engaging with this problem, students gain valuable insights into the application of statistics in real-world scenarios and develop their analytical and problem-solving skills.
Conclusion
The AP Stat 2003 Problem 3, also known as the Frappy Problem, is a challenging but rewarding exercise in statistical analysis. By understanding the key concepts, applying appropriate statistical methods, and interpreting the results in context, students can confidently solve this problem and gain invaluable statistical insights. The Frappy Problem continues to be a valuable tool for educators in fostering a deeper understanding of statistics and its applications.
