Chapter 5.4, Problem 118AYU

(a)

To determine

To calculate: The diversity index of the United States in $2015$ , 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 States in $2015$ was as follows: $\begin{array}{cc}\text{Race}& \text{Proportion}\\ \text{White}& 0.617\\ \text{Black or African American}& 0.124\\ \text{American Indian and Alaska Native}& 0.007\\ \text{Asian}& 0.053\\ \text{Native Hawaiian and Other Pacific Islander }& 0.002\\ \text{Hispanic}& 0.177\\ \text{Two or More Races}& 0.020\end{array}$

Answer to Problem 118AYU

Solution:

The diversity index of the United States in $2015$ is $0.49699$ .

Explanation of Solution

Given information:

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 $2015$ was as follows: $\begin{array}{cc}\text{Race}& \text{Proportion}\\ \text{White}& 0.617\\ \text{Black or African American}& 0.124\\ \text{American Indian and Alaska Native}& 0.007\\ \text{Asian}& 0.053\\ \text{Native Hawaiian and Other Pacific Islander }& 0.002\\ \text{Hispanic}& 0.177\\ \text{Two or More Races}& 0.020\end{array}$

Formula used:

The 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.

Calculation:

Here, the Shannon’s Diversity Index is given by the formula $H=-\left(p_1\log p_1+p_2\log p_2+\cdot \cdot \cdot +p_n\log p_n\right)$ , and there are 7 species of race, so the value of $n$ is $7$ .

The diversity index = $-\left(0.617\log \left(0.617\right)+0.124\log \left(0.124\right)+0.007\log \left(0.007\right)+0.053\log \left(0.053\right)+0.002\log \left(0.002\right)+0.177\log \left(0.177\right)+0.020\log \left(0.020\right)\right)$

  $=-\left(-0.12939-0.11242-0.01508-0.06761-0.00540-0.13311-0.03398\right)$

  $=-\left(-0.49699\right)$

  $=0.49699$

Therefore, the diversity index of the United States in $2015$ is $0.49699$

(b)

To determine

To calculate: 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.

Answer to Problem 118AYU

Solution:

The largest value of the diversity index $H_{\max }$ is $0.8451$

Explanation of Solution

Given information:

According to the U.S. Census Bureau, the distribution of the race in the United State in $2015$ was as follows: $\begin{array}{cc}\text{Race}& \text{Proportion}\\ \text{White}& 0.617\\ \text{Black or African American}& 0.124\\ \text{American Indian and Alaska Native}& 0.007\\ \text{Asian}& 0.053\\ \text{Native Hawaiian and Other Pacific Islander }& 0.002\\ \text{Hispanic}& 0.177\\ \text{Two or More Races}& 0.020\end{array}$

Formula used:

The largest value of the diversity index $H_{\max }=\log \left(S\right)$

Calculation:

The largest value of the diversity index, $H_{\max }=\log \left(S\right)$

Here, $S$ is the number of categories of race, so there are $7$ different categories of race.

  $H_{\max }=\log \left(7\right)$

  $=0.84509804$

Therefore, the largest value of the diversity index $H_{\max }$ is $0.84509804\approx 0.8451$ .

(c)

To determine

To calculate: The evenness ratio $E_H$ for the United States, where $E_H=\frac{H}{H_{\max }},0\le E_H\le 1$ , if $E_H=1$ there is complete evenness.

Answer to Problem 118AYU

Solution:

The evenness ratio is $E_H=0.5881$

Explanation of Solution

Given information:

The Shannon’s Diversity Index is given by the formula $H=-\left(p_1\log p_1+p_2\log p_2+\cdot \cdot \cdot +p_n\log p_n\right)$ and $H_{\max }=\log \left(S\right)$ , where $S$ is the number of categories of race.

Formula used:

The evenness ratio is given by the formula $E_H=\frac{H}{H_{\max }},0\le E_H\le 1$

Calculation:

From the part (a) and (b), $H=0.49699$ and $H_{\max }=0.8451$

  $E_H=\frac{H}{H_{\max }},0\le E_H\le 1$

  $=\frac{0.49699}{0.8451}$

  $=0.5881$

Therefore, the evenness ratio is $E_H=0.5881$

(d)

To determine

The distribution of the race for the United States in $2010$ from the Census Bureau and find the Shannon’s diversity index also decide the United States becoming more diverse.

Answer to Problem 118AYU

Solution:

The diversity index of the United States in $2010$ is $0.4215$ and the United States is becoming more diverse because the diversity index of the United States in $2015$ is more than the diversity index of the United States in $2010$ .

Explanation of Solution

Given information:

The distribution of race in the United States in $2015$ is,

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

Here, the Shannon’s Diversity Index is given by the formula: $H=-\left(p_1\log p_1+p_2\log p_2+\cdot \cdot \cdot +p_n\log p_n\right)$ , and there are 7 species of race, so the value of $n$ is $7$

The diversity index =

  $-\left(0.724\log \left(0.724\right)+0.126\log \left(0.126\right)+0.009\log \left(0.009\right)+0.048\log \left(0.048\right)+0.002\log \left(0.002\right)+0.062\log \left(0.062\right)+0.029\log \left(0.029\right)\right)$

  $=-\left(-0.1015492781-0.1133533113-0.01841181742-0.06330042061-0.005397940009-0.07487171525-0.04459045806\right)$

  $=-\left(-0.4214749407\right)$

  $=0.4214749407$

Therefore, the diversity index of the United States in $2010$ is $0.4214749407\approx 0.4215$

From part (a),

The diversity index of the United States in $2015$ is $0.49156$

The diversity index of the United States in $2015$ is more than the diversity index of the United States in $2010$ , so the United States is becoming more diverse.