- To determine the best model, we compare the Sum of Squared Errors (SSE) for each given function relative to the observed data points.
- The calculated SSE values are:
- \(y = 47(1.191)^x \approx 5524.7\)
- \(y = 34(1.204)^x \approx 11016.1\)
- \(y = 26.9x - 1.3 = 42.7\)
- \(y = 27x - 4 = 60.0\)
- Comparing these, the model with the minimum error is the linear model \(y = 26.9x - 1.3\).
- The final answer is \(\boxed{y = 26.9 x - 1.3}\).
Explanation
- Problem Analysis
The goal is to determine which of the four given mathematical models best fits the observed data of lines of code written hourly based on the number of people working. We are given data points \((x, y)\) where \(x\) is the number of people and \(y\) is the lines of code written: (2, 50), (4, 110), (6, 160), (8, 210), (10, 270), and (12, 320). We need to compare two linear models and two exponential models to find the one with the smallest Sum of Squared Errors (SSE), defined as \(SSE = \sum (y_{observed} - y_{predicted})^2\).
- Evaluating the Models
To evaluate the fit of each model, we calculate the predicted values \(\hat{y}_i\) for each \(x_i \in \{2, 4, 6, 8, 10, 12\}\) and determine the sum of the squared differences between the observed \(y_i\) and the predicted \(\hat{y}_i\).
The models are:
- \(y = 47(1.191)^x\)
- \(y = 34(1.204)^x\)
- \(y = 26.9x - 1.3\)
- \(y = 27x - 4\)
Calculating the SSE for each model:
- For \(y = 47(1.191)^x\), the SSE is approximately \(5524.7\).
- For \(y = 34(1.204)^x\), the SSE is approximately \(11016.1\).
- For \(y = 26.9x - 1.3\), the SSE is \(42.7\).
- For \(y = 27x - 4\), the SSE is \(60.0\).
- Conclusion
Comparing the results, the model \(y = 26.9x - 1.3\) has the smallest SSE value of \(42.7\). Since a smaller SSE indicates that the predicted values are closer to the observed data points, this linear model is the best representation of the provided data. Therefore, the best model is \(y = 26.9x - 1.3\).
Examples
In business and production analysis, managers often collect data to optimize performance. For example, by tracking how the number of employees affects hourly output, companies can model productivity to predict future results. This helps them determine the most efficient number of staff members to assign to a project, ensuring that resources are balanced and targets are met effectively. Models like the ones used here provide a foundation for such operational decisions.