Final answer:
To analyze the relationship between state ranking and area, the ranking is the independent variable, and the area is the dependent variable. A scatter plot and regression analysis, including calculating the least-squares line and correlation coefficient, help determine the strength of the relationship. The model's estimates for specific states can be compared to actual values for validation.
Step-by-step explanation:
To analyze the relationship between the ranking of a state and the area of the state, we must identify the independent variable and the dependent variable. The independent variable is the factor that is manipulated or varied to see its effect on the dependent variable. In this case, the ranking of a state would be the independent variable, and the area of the state would be the dependent variable. Using a scatter plot, we can visually assess the relationship between these two variables. The scatter plot might show a trend where states with a higher or lower ranking have larger or smaller areas, respectively.
Applying regression analysis, we calculate the least-squares line to find the best linear fit for the data points on the scatter plot. The equation for the least-squares line is typically written in the form ŷ = a + bx. The correlation coefficient, often represented by r, quantifies the strength and direction of the relationship between the variables. A correlation coefficient close to 1 or -1 indicates a strong linear relationship, while a coefficient close to 0 suggests a weak or no linear relationship.
Finally, we use the least-squares line to estimate the areas of specific states, such as Alabama and Colorado. The accuracy of these estimates can then be compared to the actual areas to evaluate the model's performance.