Q. 24 - Elementary Statistics

Question

24. Exotic Plants. In the article "Effects of Human Population, Area, and Time on Non-native Plant and Fish Diversity in the United States" (Biological Consevation, Vol, 100, No. 2, pp. 243-252), M. McKinney investigated the relationship of various factors on the number of exotic plants in each state. On the WeissStats site, you will find the data on population (in millions), area (in thousands of square miles), and number of exotic plants for each state. Use the technology of your choice to determine the linear correlation coefficient between each of the following:
a. population and area
b. population and number of exotic plants
c. area and number of exotic plants
d. Interpret and explain the results you got in parts (a)-(c).

Step-by-Step Solution

Answer

(a) The correlation coefficient between the population and area is $r\approx 0.1078$.

(b) The correlation coefficient between the population and number of exotic plants is $r\approx 0.7209$.

(c) The correlation coefficient between the area and number of exotic plants is $r\approx -0.3092$.

(d) The population and area have a weak positive linear link, population and number of exotic plants have a moderate relationship, and area and number of exotic plants have a weak negative linear relationship.

Part (a) Step 1: Given information

To determine the the linear correlation coefficient between the population and area.

Part (a) Step 2:Explanation

Calculate the correlation coefficient between the population and area  using the given value:

$\sum x_i{y}_i=23075.59$

$\sum x_i^2=3568.86$

$\sum y_i^2=613510.8$

Using the formula, determine the correlation coefficient as follows:

$r=\frac{\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n}{\sqrt{\left[\sum x_i^2-{\left(\sum x_i\right)}^2/n\right]\left[\sum y_i^2-{\left(\sum y_i\right)}^2/n\right]}}$

$=\frac{23075.59-(284.8)\left(3549\right)/50}{\sqrt{\left[3568.86-(284.8{)}^2/50\right]\left[613510.8-(3549{)}^2/50\right]}}$

$r\approx 0.1078$

As a result, the correlation coefficient between the population and area is $r\approx 0.1078$.

Part (b) Step 1: Given information

To determine the correlation coefficient between the population and number of exotic plants.

Part (b) Step 2: Explanation

Calculate the correlation coefficient between the population and number of exotic plants using the given value:
$\sum x_i{y}_i=213236.3$
$\sum x_i^2=3568.86$
$\sum y_i^2=19052833$
Using the formula, determine the correlation coefficient as follows:
$r=\frac{\sum x_{i_i}-\left(\sum x_i\right)\left(\sum y_i\right)/n}{\sqrt{\left[\sum x_i^2-{\left(\sum x_i\right)}^2/n\right]\left[\sum y_i^2-{\left(\sum y_i\right)}^2/n\right]}}$$=\frac{213236.3-(284.8)\left(28919\right)/50}{\sqrt{\left[3568.86-(284.8{)}^2/50\right]\left[19052833-(28919{)}^2/50\right]}}$
$r\approx 0.7209$

As a result, the correlation coefficient between the population and number of exotic plants is $r\approx 0.7209$.

Part (c) Step 1: Given information

To determine the correlation coefficient between the area and number of exotic plants.

Part (c) Step 2: Explanation

Calculate the correlation coefficient between the area and number of exotic plants using the given value:
$\sum x_i{y}_i=1769050$
$\sum x_i^2=613510.8$
$\sum y_i^2=19052833$
Using the formula, determine the correlation coefficient as follows:
$r=\frac{\sum x_{i{y}_i-}\left(\sum x_i\right)\left(\sum y_i\right)/n}{\sqrt{\left[\sum x_i^2-{\left(\sum x_i\right)}^2/n\right]\left[\sum y_i^2-{\left(\sum y_i\right)}^2/n\right]}}$
$=\frac{1769050-(284.8)\left(28919\right)/50}{\sqrt{\left[613510.8-(3549{)}^2/50\right]\left[19052833-(28919{)}^2/50\right]}}$
$r\approx -0.3092$

As a result, the correlation coefficient between the area and number of exotic plants is $r\approx -0.3092$.

Part (d) Step 1: Given information

To interpret and explain the results that got in parts (a)-(c).

Part (d) Step 2: Explanation

Part (a)-(c) results should be interpreted.
The linear relationship will be positive if the correlation is positive.
The linear relationship will be negative if it is negative.
The linear relationship is poor if correlation varies between $0<\left|r\right|<0.5$.
The linear relationship is moderate if it is between $0.5<\left|r\right|<0.8$.
The linear association is strong if it is between $0.8<\left|r\right|<1$.
Population and area have a weak positive linear link, population and number of exotic plants have a moderate relationship, and area and number of exotic plants have a weak negative linear relationship.