Chapter 9, Problem 43RE

Smoking. Table 2 gives the percentage of U.S. adults who were smokers in the given year.

The following transition matrix $P$ is proposed as a model for the data, where $S$ represents the population of U.S. adult smokers.

$\begin{array}{l}\text{ Five years later}\\ \begin{array}{cc}S& S^{\prime }\end{array}\\ \begin{array}{c}\text{Current}\\ \text{year}\end{array}\begin{array}{c}S\\ S^{\prime }\end{array}\left[\begin{array}{cc}.74& .26\\ .03& .97\end{array}\right]=P\end{array}$

(A) Let $S_0=\left[\begin{array}{cc}.301& .699\end{array}\right]$ and find $S_1,S_2$, and $S_3$. 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 2.

(C) According to this transition matrix, what percentage of the adult U.S. population will be smokers in the long run?