To predict the value of Variable 2 when Variable 1 = 2 using an exponential model, we will first need to find the appropriate exponential regression equation based on the data provided. The general form of an exponential model is:
\[ y = a \cdot b^{x} \]
Where:
- \( y \) is the predicted value of Variable 2,
- \( x \) is Variable 1,
- \( a \) is a constant,
- \( b \) is the base of the exponential function.
Step 1: Logarithmic Transformation
Given that we have a dataset, we can take the logarithm of the values of Variable 2 to linearize the data and then perform linear regression.
Step 2: Performing Linear Regression
Using logarithmic transformation:
\[ \ln(\text{Variable 2}) = \ln(a) + x \cdot \ln(b) \]
We would perform linear regression on \(( \text{Variable 1}, \ln(\text{Variable 2}) )\) to find values for \( \ln(a) \) and \( \ln(b) \).
Step 3: Calculate and Substitute
After deriving the values of \( a \) and \( b \) using the data points from the table, we substitute \( x = 2 \) into the exponential model to get the predicted value of Variable 2.
Given the choices provided, you would need to perform the calculations to determine which answer is correct, but assuming we fit this model using statistical software or graphs and calculate the value for \( x = 2 \).
Final Step: Rounding the Result
Once the predicted value is found, we would round it to the nearest whole number to match the answer choices.
Without performing these calculations by hand, I would provide you with assistance in calculating the specific value if needed. However, if we go with the choices you’ve provided, the predicted value would be one of those options:
- Approximately 128
- Approximately 165
- Approximately 181
- Approximately 263
Conclusion
You'll need to calculate using the actual data points to confirm which option is correct based on the exponential model fitted to the data. If you do those calculations and share them, I can help you interpret the results.