Chapter 5.4, Problem 120AYU

(a)

To determine

The diversity index of the United States in $2010$, of Shannon’s diversity index, is given by the formula $H=-\left(p_1\log p_1+p_2\log p_2+\cdot \cdot \cdot +p_n\log p_n\right)$, where $p_1$ is the proportion of the population that is species $1$, $p_2$ is the proportion of the population that is species $2$, and so on. The population is the people in the United States and the species is race. According to the U.S. Census Bureau, the distribution of the race in the United State in $2010$ was as follows:

$\begin{array}{|cc|}\hline \text{Race}& \text{Proportion}\\ \text{White}& 0.724\\ \text{Black or African American}& 0.126\\ \text{American Indian and Alaska Native}& 0.009\\ \text{Asian}& 0.048\\ \text{Native Hawaiian and Other Pacific Islander }& 0.002\\ \text{Some Other Race}& 0.062\\ \text{Two or More Races}& 0.029\\ \hline\end{array}$

(b)

To determine

The largest value of the diversity index $H_{\max }$, where $H_{\max }=\log \left(S\right)$ and $S$ is the number of categories of race .

(c)

To determine

The evenness ratio $E_H$ where $E_H=\frac{H}{H_{\max }},0\le E_H\le 1$, if $E_H=1$, there is complete evenness.

(d)

To determine

The distribution of the race for the United States in $2000$ from the Census Bureau and find the Shannon’s diversity index also decide the United State becoming more diverse.