To match the regression equation $\hat{y} = -0.667x + 52.6$ with the appropriate graph, we need to analyze the properties of the equation and determine which graph best represents it. The slope is $-0.667$ and the y-intercept is $52.6$.

Given these parameters:

• The y-intercept is $52.6$, meaning the line will cross the y-axis at $52.6$.
• The slope of $-0.667$ indicates the line is decreasing, with a relatively moderate decline.

Let's evaluate the given options:

Graph A

• The y-values range from 0 to 60.
• The x-values range from 0 to 30.
• The graph seems to depict a linear relationship with a negative slope and the y-intercept might be around 52.6, which matches our equation.

Graph B

• The y-values range from 0 to 250.
• The x-values range from 0 to 150.
• This range is much larger than needed for the given y-intercept of 52.6.

Graph C

• The y-values range from 0 to 60.
• The x-values range from 0 to 20.
• This graph also shows a linear relationship with a negative slope and the y-intercept appears to be around 52.6.

Graph D

• The y-values range from 0 to 8000.
• The x-values range from 0 to 12.
• The range is significantly larger and does not fit the equation's y-intercept and slope.

Based on these analyses, both Graph A and Graph C seem to fit the requirements of the given regression equation. However, let's consider the moderate decline of the slope $-0.667$. Graph A has a larger range for x-values (0 to 30) compared to Graph C (0 to 20), which might be a better fit for the moderate slope.

Thus, the best match for the given regression equation is:

A.

Do you want details or have any questions?

Here are 8 related questions to expand on the information:

1. What does the slope $-0.667$ tell us about the relationship between $x$ and $\hat{y}$?
2. How would the graph look if the slope was positive instead of negative?
3. What are the implications of the y-intercept being $52.6$?
4. How would you find the x-intercept of the given regression line?
5. What does the correlation coefficient tell you in the context of this regression?
6. How do you interpret the residuals of this regression model?
7. What is the importance of plotting the regression line on a scatter plot?
8. How can you use this regression equation to make predictions for new values of $x$?

Tip: When interpreting the slope and y-intercept of a regression equation, always consider the context of the data and what these values represent in real-world terms.