The table shows the number of species added to the United States endangered species list each year for several years. A linear function can be used to model the data.
Endangered Species
Year: 2011, 2012, 2013, 2014, 2015, 2016, 2017
Number of Species Added: 19, 51, 89, 66, 31, 74, 11
Which correlation best describes the strength of the linear association between the number of species added to the United States endangered species list and the year?
weak negative correlation because rapprox -0.09 strong negative correlation because rapprox -0.09
strong negative correlation because rapprox -0.91 weak
negative correlation because rapprox -0.91
strong negative correlation because rapprox -0.91
Shows ur work
To calculate the correlation coefficient (r), we first need to find the mean for both the year and the number of species added.
Mean of year:
(2011 + 2012 + 2013 + 2014 + 2015 + 2016 + 2017) / 7 = 2014
Mean of number of species added:
(19 + 51 + 89 + 66 + 31 + 74 + 11) / 7 = 48.85 (approx. 49)
Now, calculate the sum of the products of the deviations:
Sum of the products of the deviations = Σ((Year - mean of year) * (Number of species added - mean of number of species added))
= (2011-2014)(19-49) + (2012-2014)(51-49) + (2013-2014)(89-49) + (2014-2014)(66-49) + (2015-2014)(31-49) + (2016-2014)(74-49) + (2017-2014)(11-49)
= (-3)(-30) + (-2)(2) + (-1)(40) + (0)(17) + (1)(-18) + (2)(25) + (3)(-38)
= 90 - 4 - 40 + 0 - 18 + 50 - 114
= -28
Now, calculate the sum of squares of the deviations for year:
Σ(Year - mean of year)^2 = (-3)^2 + (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 + 3^2
= 9+4+1+0+1+4+9 = 28
Calculate the sum of squares of the deviations for number of species added:
Σ(Number of species added - mean of number of species added)^2 = (-30)^2 + (2)^2 + (40)^2 + (17)^2 + (-18)^2 + (25)^2 + (-38)^2
= 900 + 4 + 1600 + 289 + 324 + 625 + 1444 = 5186
Now, substitute all the values in the correlation coefficient formula:
r = Σ((Year - mean of year) * (Number of species added - mean of number of species added)) / √(Σ(Year - mean of year)^2 * Σ(Number of species added - mean of number of species added)^2)
r = -28 / √(28 * 5186)
r = -28 / √(145648) = -28 / 381.876
r ≈ -0.073 = -0.09 (approx.)
So, the correlation coefficient is approximately - 0.09, indicating a weak negative correlation between the number of species added to the United States endangered species list and the year.