Chapter 9.2, Problem 69E

The population center of the $48$ contiguous states of the United States is the point where a flat, rigid map of the contiguous states would balance if the location of each person was represented on the map by a weight of equal measure. In $1790$, the population center was $23$ miles east of Baltimore, Maryland. By $1990$, the center had shifted about $800$ miles west and $100$ miles south to a point in southeast Missouri. To study this shifting population, the U.S. Census Bureau divides the states in to four regions as shown in the figure. Problems 69 and 70 deal with population shifts among these regions.

Population shifts. Table 3 gives the percentage of the U.S. population living in the south region during the indicated years.

The following transition matrix $P$ is proposed as a model for the data, where $S$ represents the population that lives in the south region:

$\begin{array}{l}\text{ Next decade}\\ \begin{array}{cc}S& S^{\prime }\end{array}\\ \begin{array}{c}\text{Current}\\ \text{decade}\end{array}\begin{array}{c}S\\ S^{\prime }\end{array}\left[\begin{array}{cc}.61& .39\\ .21& .79\end{array}\right]=P\end{array}$

(A) Let $S_0=\left[\begin{array}{cc}.309& .691\end{array}\right]$ and find $S_1,S_2,S_3$, and $S_4$. Compute the matrices exactly and then round entries to three decimal places.

(B) Construct a new table comparing the results from part (A) with the data in Table 3.

(C) According to this transition matrix, what percentage of the population will live in the south region in the long run?