To determine which model best represents the data given in the table, we need to evaluate how well each model fits the data. We do this by calculating the sum of squared errors (SSE) for each model. The SSE measures the differences between the observed values (lines of code written hourly) and the values predicted by each model. The model with the smallest SSE will be our best fit.
Let's analyze the given data and models:
Data from Table:
- When 2 people are working, 50 lines of code are written hourly.
- When 4 people are working, 110 lines of code are written hourly.
- When 6 people are working, 160 lines of code are written hourly.
- When 8 people are working, 210 lines of code are written hourly.
- When 10 people are working, 270 lines of code are written hourly.
- When 12 people are working, 320 lines of code are written hourly.
Models to compare:
1. \( y = 47(1.191)^x \)
2. \( y = 34(1.204)^x \)
3. \( y = 26.9 x - 1.3 \)
4. \( y = 27 x - 4 \)
Step-by-Step SSE Calculations:
To find the SSE for each model, we follow these steps:
1. Calculate the predicted values using the model.
2. Find the difference between the predicted values and the observed values.
3. Square these differences.
4. Sum all squared differences.
Sum of Squared Errors (SSE):
For all models, the calculated SSE values are:
- Model 1: \(y = 47(1.191)^x\) has \(\text{SSE} = 5524.7386102429155\)
- Model 2: \(y = 34(1.204)^x\) has \(\text{SSE} = 11016.094163149735\)
- Model 3: \(y = 26.9 x - 1.3\) has \(\text{SSE} = 42.69999999999972\)
- Model 4: \(y = 27 x - 4\) has \(\text{SSE} = 60\)
Determining the Best Model:
The smallest SSE indicates the best fit to the data. Comparing the SSE values, we observe that Model 3, \( y = 26.9 x - 1.3 \), has the smallest SSE value of 42.69999999999972.
Conclusion:
The model that best represents the data is:
[tex]\[ y = 26.9 x - 1.3 \][/tex]
