To answer the question regarding the selection of the best model (linear or exponential) based on residuals, let’s analyze the situation:

Key Points from the Question:

1. $k$: Sum of the absolute values of residuals for the linear model $f(x)$.
2. $p$: Sum of the absolute values of residuals for the exponential model $g(x)$.
3. Residuals: Represent the differences between the actual values $y$ and the predicted values from the model.

The model with lower residuals (smaller sum of absolute residuals) is considered the better fit.

Analysis of the Options:

• If $k > p$: The exponential model $g(x)$ has smaller residuals, making it a better fit.
  • This corresponds to option B: "Because $k > p$, the exponential model is preferred."
• If $k < p$: The linear model $f(x)$ has smaller residuals, making it a better fit.
  • This corresponds to option C: "Because $k < p$, the linear model is preferred."

Correct Answer:

The correct selection depends on the relative sizes of $k$ and $p$. Since the model with smaller residuals is preferred, the options are interpreted as:

• $k > p$: Exponential model preferred → Option B
• $k < p$: Linear model preferred → Option C

If we assume that the residuals $k$ for the linear model are greater than $p$ for the exponential model, then:

Final Answer: B. Because $k > p$, the exponential model is preferred.

Let me know if you'd like a more detailed explanation of residuals, regression models, or additional examples!

Here are some related questions for practice:

1. What are residuals, and how are they used to evaluate model performance?
2. How do linear and exponential regressions differ in fitting data?
3. How can residual plots help in identifying the best-fitting model?
4. What role does the sum of absolute residuals play in model evaluation?
5. How would you calculate the residuals for a linear and an exponential regression model?

Tip: Always compare residual values (or errors) when deciding between two models. Smaller residuals indicate a better-fitting model.