A scatter plot will show if there's a trend suggesting a relationship between a state's ranking and its area. The least-squares line equation predicts area based on ranking, while the correlation coefficient indicates the relationship's strength and significance.
The independent variable in this scenario would typically be the ranking of a state, as it does not depend on other variables, while the dependent variable is often the area of the state, which may vary and be assessed in relation to the ranking. When creating a scatter plot to visualize this data, one may expect a spread of points.
The scatter plot could potentially show clusters, trends, or no obvious patterns at all, depending on the data. The relationship between the variables can be deduced from the scatter plot. If the points suggest a trend, either increasing or decreasing, a relationship may exist; if the points are scattered without direction, the relationship could be weak or nonexistent.
To quantify the relationship, the least-squares line can be calculated, which provides a way to predict the area (dependent variable) based on a state's ranking (independent variable). The equation for the line will take the form î = a + bx, where î is the predicted area, a is the y-intercept, b is the slope of the line, and x is the state ranking.
The correlation coefficient, an important indicator of the strength and direction of a linear relationship, would be calculated next. If this value is close to 1 or -1, it indicates a strong linear correlation. The significance of this coefficient can be determined using a statistical test or by comparing it to a critical value from a correlation table.
Answer : The least-squares line equation will provide a model for predicting state areas based on rankings, while the correlation coefficient will reveal the strength and significance of the linear relationship between these variables.