Final answer:
To infer if the number of letters in a state's name is associated with the year it joined the Union, the independent variable is the year of entry and the dependent variable is the number of letters. A scatter plot and least-squares line help analyze the data, while the correlation coefficient assesses the relationship's significance.
Step-by-step explanation:
When examining if the number of letters in a state name depends on the year the state entered the Union, a clear approach is required:
The independent variable would logically be the year the state entered the Union, as it is not affected by other variables. In contrast, the number of letters would be the dependent variable as it might change based on the entry year.
To visualize the data, scatter plot should be drawn with the independent variable on the x-axis and the dependent variable on the y-axis.
An inspection of the scatter plot might give a preliminary idea if there's a discernible relationship between the two variables.
The least-squares line is calculated to fit the data points best, and it is typically expressed in the equation form ý = a + bx.
The correlation coefficient, r, helps determine the strength and direction of the relationship between the variables.
Using the least-squares line equation, the estimated state name letter count for any given year can be forecasted.
Whether a linear model is the best fit for the data can be assessed by looking at the scatter plot and considering the correlation coefficient validity.
Finally, the utility of the least-squares line for making predictions in the current year can be evaluated, keeping in mind potential limitations of extrapolating far beyond the original data set.