Let's break down the problem:

The regression equation is $\hat{y} = 0.496 \cdot 1.3^x$, which represents an exponential regression model.

Analyzing the Scatterplot:

• The scatterplot shows an increasing trend in the points, which matches the expected pattern of an exponential model.

Analyzing the Residual Plot:

• In the residual plot, residuals (the differences between observed and predicted values) appear to follow a somewhat patterned curve instead of being randomly distributed around zero.

Now, let's review the options:

• A. "The residual plot changes direction, so a quadratic model would fit the data better."

  • False: The residual plot shows a curve, which may suggest a better fit with a non-linear model, but it doesn't mean quadratic is necessarily better.

• B. "The points in the scatterplot follow a line, so a linear model would fit the data better."

  • False: The scatterplot shows an exponential growth, not a straight line. A linear model would likely not fit well here.

• C. "The model is not a good fit because there is a pattern in the residual plot."

  • True: The residual plot shows a clear pattern (a curve), which indicates that the exponential model might not be the best fit. Ideally, residuals should be randomly scattered around zero.

• D. "The model is a good fit because the patterns in the plots are both curves."

  • False: Even though both plots show curves, the pattern in the residuals suggests that the model is not capturing all the trends in the data.

Correct answer: C

Would you like further details on any of these points?

Here are 5 related questions to expand on this topic:

1. How can you identify if an exponential model is a good fit based on a residual plot?
2. What does it mean when a residual plot shows a pattern?
3. How would a quadratic model be more appropriate for some data sets?
4. What are the key differences between linear and exponential regression models?
5. How do we interpret the parameters of an exponential equation like $\hat{y} = 0.496 \cdot 1.3^x$?

Tip: A good residual plot should display no clear patterns, meaning the model has captured all the information in the data.