Question

Calculus project: Look up the population of your state (the first year it was measured will depend on which state you have, 1790 was the first year in general). While you should note the population in 2020, do not use it in your data when finding parameters; we will use that data point to test the model. You may find it helpful to rename your first year as: a. Year 0 b. Year 1 c. Year A d. Year X

Answer 1

When examining if the number of letters in a state's name relates to the year it entered the Union, we plot the year (independent variable) against the number of letters (dependent variable). A scatter plot and the calculation of the least-squares line can determine the presence of any correlation, which is further clarified by the correlation coefficient. The line can predict the number of letters for hypothetical recent entries if the model remains relevant.

Step-by-step explanation:

To determine if the number of letters in a state name depends on the year the state entered the Union, we would designate the year the state entered the Union as the independent variable and the number of letters in the state name as the dependent variable. Using this data, a scatter plot can be drawn to visualize any potential correlation between the two variables.

Upon visual inspection of the scatter plot, it may or may not be clear if there is a relationship. If there appears to be some form of correlation, we may proceed to calculate the least-squares line, which is a line of best fit that minimizes the sum of the squares of the vertical distances of the points from the line. The equation for this line takes the form ý = a + bx.

The correlation coefficient, denoted as r, is a statistical measure that illustrates how strongly variables are related. The closer the value of r is to 1 or -1, the stronger the correlation. After calculating r, if it is near 0, it suggests that there is no significant relationship between the variables.

For predictions, we use the least-squares line to estimate the number of letters in state names for states that entered the Union in 1900 and 1940. Furthermore, whether a line is the best fit or not can be assessed by evaluating the scatter plot and residual patterns.

Lastly, using the least-squares line to predict the number of letters for a state entering the Union in the current year may not be viable if the data does not support such extrapolation, and it's important to assess whether the trend modelled historically applies to the present.