Final answer:
The independent variable is the year the state entered the Union, and the number of letters in the state's name is the dependent variable. A scatter plot and least-squares line are used to analyze the linear relationship, and the significance is determined through the correlation coefficient.
Step-by-step explanation:
In analyzing the relationship between two variables, the independent variable is typically the one that is presumed to be the cause, while the dependent variable is the effect. For instance, when examining the number of letters in a state name in relation to the year the state entered the Union, the year would be the independent variable and the number of letters would be the dependent variable.
To visualize the relationship between these variables, a scatter plot can be drawn. A scatter plot helps to determine whether there appears to be a correlation or trend between the two variables. Upon drawing this scatter plot, if it shows a general upward or downward trend, it suggests there is a relationship. Without the actual data, we cannot visually confirm the presence of a relationship.
The next step would be to calculate the least-squares line, which provides a linear model of the form ý = a + bx. Here, 'a' represents the y-intercept, and 'b' represents the slope of the line. The correlation coefficient can then be calculated to quantify the strength of the relationship between the variables. A correlation coefficient that is closer to 1 or -1 implies a strong relationship, while a coefficient closer to 0 implies a weak relationship.
If the correlation is significant and the line fits the data well, this line can be used to predict the number of letters in the name of a state based on the year it entered the Union. This model, however, may have limitations if the data does not fit a linear pattern well, or if predictions are made for years far outside the range of the data set.