Chapter 8, Problem 2PS

a.

To determine

Find the percent form of the given matrices.

Answer to Problem 2PS

The Percentage matrix for $2000$ is

  $\left[\begin{array}{ccc}4.64& 11.79& 2.62\\ 5.92& 14.03& 2.93\\ 9.08& 22.11& 4.42\\ 6.05& 13.94& 2.46\end{array}\right]$

and the Percentage matrix for $2010$ is

  $\left[\begin{array}{ccc}3.99& 11.39& 2.53\\ 5.22& 13.53& 2.92\\ 9.00& 23.28& 4.82\\ 5.81& 14.73& 2.77\end{array}\right]$

Explanation of Solution

Given:

The number of population (in thousands) matrix of $2000$ is

  $\begin{array}{l}\begin{array}{ccc}0-17& 18-64& 65+\end{array}\\ \begin{array}{c}\text{Northeast}\\ \text{Midwest}\\ \text{South}\\ \text{West}\end{array}\left[\begin{array}{ccc}13,048& 33,174& 7,372\\ 16,648& 39,486& 8,259\\ 25,567& 62,232& 12,438\\ 17,031& 39,244& 6,922\end{array}\right]\end{array}$

and the number of population (in thousands) matrix of $2010$ is

  $\begin{array}{l}\begin{array}{ccc}0-17& 18-64& 65+\end{array}\\ \begin{array}{c}\text{Northeast}\\ \text{Midwest}\\ \text{South}\\ \text{West}\end{array}\left[\begin{array}{ccc}12,333& 35,179& 7,805\\ 16,128& 41,777& 9,022\\ 27,789& 71,873& 14,894\\ 17,931& 45,467& 8,547\end{array}\right]\end{array}$

The total population in $2000$ was approximately $281,422,000$ and the total population in $2010$ was approximately $308,746,000$ .

Calculation:

The percentage matrix for $2000$ is

  $\begin{array}{l}=\frac{100}{281,422}\left[\begin{array}{ccc}13,048& 33,174& 7,372\\ 16,648& 39,486& 8,259\\ 25,567& 62,232& 12,438\\ 17,031& 39,244& 6,922\end{array}\right]\\ =\left[\begin{array}{ccc}4.64& 11.79& 2.62\\ 5.92& 14.03& 2.93\\ 9.08& 22.11& 4.42\\ 6.05& 13.94& 2.46\end{array}\right]\end{array}$

The percentage matrix for $2010$ is

  $\begin{array}{l}=\frac{100}{308,746}\left[\begin{array}{ccc}12,333& 35,179& 7,805\\ 16,128& 41,777& 9,022\\ 27,789& 71,873& 14,894\\ 17,931& 45,467& 8,547\end{array}\right]\\ =\left[\begin{array}{ccc}3.99& 11.39& 2.53\\ 5.22& 13.53& 2.92\\ 9.00& 23.28& 4.82\\ 5.81& 14.73& 2.77\end{array}\right]\end{array}$

b.

To determine

Find the matrix for change in the percent of the population.

Answer to Problem 2PS

The matrix for change in the percent of the population is

  $\left[\begin{array}{ccc}-0.65& -0.40& -0.09\\ -0.70& -0.50& -0.01\\ -0.08& 1.17& 0.40\\ 0.24& 0.79& 0.31\end{array}\right]$

Explanation of Solution

Given:

The number of population (in thousands) matrix of $2000$ is

  $\begin{array}{l}\begin{array}{ccc}0-17& 18-64& 65+\end{array}\\ \begin{array}{c}\text{Northeast}\\ \text{Midwest}\\ \text{South}\\ \text{West}\end{array}\left[\begin{array}{ccc}13,048& 33,174& 7,372\\ 16,648& 39,486& 8,259\\ 25,567& 62,232& 12,438\\ 17,031& 39,244& 6,922\end{array}\right]\end{array}$

and the number of population (in thousands) matrix of $2010$ is

  $\begin{array}{l}\begin{array}{ccc}0-17& 18-64& 65+\end{array}\\ \begin{array}{c}\text{Northeast}\\ \text{Midwest}\\ \text{South}\\ \text{West}\end{array}\left[\begin{array}{ccc}12,333& 35,179& 7,805\\ 16,128& 41,777& 9,022\\ 27,789& 71,873& 14,894\\ 17,931& 45,467& 8,547\end{array}\right]\end{array}$

The total population in $2000$ was approximately $281,422,000$ and the total population in $2010$ was approximately $308,746,000$ .

Calculation:

The percentage matrix for $2000$ is

  $\begin{array}{l}=\frac{100}{281,422}\left[\begin{array}{ccc}13,048& 33,174& 7,372\\ 16,648& 39,486& 8,259\\ 25,567& 62,232& 12,438\\ 17,031& 39,244& 6,922\end{array}\right]\\ =\left[\begin{array}{ccc}4.64& 11.79& 2.62\\ 5.92& 14.03& 2.93\\ 9.08& 22.11& 4.42\\ 6.05& 13.94& 2.46\end{array}\right]\end{array}$

The percentage matrix for $2010$ is

  $\begin{array}{l}=\frac{100}{308,746}\left[\begin{array}{ccc}12,333& 35,179& 7,805\\ 16,128& 41,777& 9,022\\ 27,789& 71,873& 14,894\\ 17,931& 45,467& 8,547\end{array}\right]\\ =\left[\begin{array}{ccc}3.99& 11.39& 2.53\\ 5.22& 13.53& 2.92\\ 9.00& 23.28& 4.82\\ 5.81& 14.73& 2.77\end{array}\right]\end{array}$

The percent change is

  $\begin{array}{l}\left[\begin{array}{ccc}3.99& 11.39& 2.53\\ 5.22& 13.53& 2.92\\ 9.00& 23.28& 4.82\\ 5.81& 14.73& 2.77\end{array}\right]-\left[\begin{array}{ccc}4.64& 11.79& 2.62\\ 5.92& 14.03& 2.93\\ 9.08& 22.11& 4.42\\ 6.05& 13.94& 2.46\end{array}\right]\\ =\left[\begin{array}{ccc}-0.65& -0.40& -0.09\\ -0.70& -0.50& -0.01\\ -0.08& 1.17& 0.40\\ 0.24& 0.79& 0.31\end{array}\right]\end{array}$

Hence the matrix for change in the percent of the population is

  $\left[\begin{array}{ccc}-0.65& -0.40& -0.09\\ -0.70& -0.50& -0.01\\ -0.08& 1.17& 0.40\\ 0.24& 0.79& 0.31\end{array}\right]$

c.

To determine

Find the region(s) and age group(s) have percent that decreased from $2000$ to $2010$ .

Answer to Problem 2PS

The regions, Northeast and Midwest have percent decreased for all age groups and the age group $0-17$ has percent decreased in all regions.

Explanation of Solution

Given:

The number of population (in thousands) matrix of $2000$ is

  $\begin{array}{l}\begin{array}{ccc}0-17& 18-64& 65+\end{array}\\ \begin{array}{c}\text{Northeast}\\ \text{Midwest}\\ \text{South}\\ \text{West}\end{array}\left[\begin{array}{ccc}13,048& 33,174& 7,372\\ 16,648& 39,486& 8,259\\ 25,567& 62,232& 12,438\\ 17,031& 39,244& 6,922\end{array}\right]\end{array}$

and the number of population (in thousands) matrix of $2010$ is

  $\begin{array}{l}\begin{array}{ccc}0-17& 18-64& 65+\end{array}\\ \begin{array}{c}\text{Northeast}\\ \text{Midwest}\\ \text{South}\\ \text{West}\end{array}\left[\begin{array}{ccc}12,333& 35,179& 7,805\\ 16,128& 41,777& 9,022\\ 27,789& 71,873& 14,894\\ 17,931& 45,467& 8,547\end{array}\right]\end{array}$

The total population in $2000$ was approximately $281,422,000$ and the total population in $2010$ was approximately $308,746,000$ .

Calculation:

The percentage matrix for $2000$ is

  $\begin{array}{l}=\frac{100}{281,422}\left[\begin{array}{ccc}13,048& 33,174& 7,372\\ 16,648& 39,486& 8,259\\ 25,567& 62,232& 12,438\\ 17,031& 39,244& 6,922\end{array}\right]\\ =\left[\begin{array}{ccc}4.64& 11.79& 2.62\\ 5.92& 14.03& 2.93\\ 9.08& 22.11& 4.42\\ 6.05& 13.94& 2.46\end{array}\right]\end{array}$

The percentage matrix for $2010$ is

  $\begin{array}{l}=\frac{100}{308,746}\left[\begin{array}{ccc}12,333& 35,179& 7,805\\ 16,128& 41,777& 9,022\\ 27,789& 71,873& 14,894\\ 17,931& 45,467& 8,547\end{array}\right]\\ =\left[\begin{array}{ccc}3.99& 11.39& 2.53\\ 5.22& 13.53& 2.92\\ 9.00& 23.28& 4.82\\ 5.81& 14.73& 2.77\end{array}\right]\end{array}$

The percent change is

  $\begin{array}{l}\left[\begin{array}{ccc}3.99& 11.39& 2.53\\ 5.22& 13.53& 2.92\\ 9.00& 23.28& 4.82\\ 5.81& 14.73& 2.77\end{array}\right]-\left[\begin{array}{ccc}4.64& 11.79& 2.62\\ 5.92& 14.03& 2.93\\ 9.08& 22.11& 4.42\\ 6.05& 13.94& 2.46\end{array}\right]\\ =\left[\begin{array}{ccc}-0.65& -0.40& -0.09\\ -0.70& -0.50& -0.01\\ -0.08& 1.17& 0.40\\ 0.24& 0.79& 0.31\end{array}\right]\end{array}$

Hence the regions, Northeast and Midwest have percent decreased for all age groups and the age group $0-17$ has percent decreased in all regions.