Question

name: stephanie benhinc date: 9/18/23 period: 4 exercise 10: spearmans rank correlation practice spearmans rank is a statistical analysis that can be used to determine whether a correlation (relationship) exists between two variables. for example, between the distribution - abundance of a species and some other factor. when determining spearmans rank, the researcher will state null hypotheses, which always states that there is no correlation between the variables. once you have determined the r_s value, you are able to accept or reject the null hypothesis. r_s=1 - \frac{6\sum d^{2}}{n(n^{2}-1)} where: \\(r_s\\) = calculated that ranges between - 1 and 1; \\(\sum\\) = sum of (total); n = number of paired items in the sample; d = difference in rank between each pair of measurements. green crabs and eelgrass density green crabs (carcinus maenas) are one of the most invasive species in the oceans. they arrived in u.s. waters (both atlantic and pacific coasts) from europe. these crabs eat large quantities of juvenile king crab and juvenile salmon, making them a threat to alaskas fishery. green crabs destroy native seagrasses by uprooting them while foraging for prey and disturbing sediment, leading to habitat degradation. an investigation was carried out to determine if a relationship exists between the density of green crabs and the density of eelgrass. 1. state the null hypothesis, specific to this investigation. 2. complete the table below by using the spearmans ranking process in order to determine the r_s value for this data - set. eelgrass per m^{2} rank green crabs per m^{2} rank d d^{2} 132 0 1.5 164 0 1.5 94 1 5.5 84 1 5.5 51 3 4 31 5 3 15 7 15 21 7 \\(\sum d^{2}=\\) in this data - set, n is equal to 8

Explanation:

Step1: State null hypothesis

The null hypothesis ($H_0$) for a correlation - based investigation is that there is no relationship between the two variables. So, for the investigation of the relationship between green crab density and eelgrass density, the null hypothesis is: There is no correlation between the density of green crabs and the density of eelgrass.

Step2: Calculate $D$ and $D^{2}$

$D$ is the difference in rank between the two variables (eelgrass density rank and green - crab density rank). For the first row, eelgrass density per $m^{2}=132$ has rank 0 and green - crab density per $m^{2}=0$ has rank 1.5, so $D = 0 - 1.5=- 1.5$ and $D^{2}=(-1.5)^{2}=2.25$.
For the second row, eelgrass density per $m^{2}=164$ has rank 0 and green - crab density per $m^{2}=0$ has rank 1.5, so $D = 0 - 1.5=-1.5$ and $D^{2}=(-1.5)^{2}=2.25$.
For the third row, eelgrass density per $m^{2}=94$ has rank 1 and green - crab density per $m^{2}=1$ has rank 5.5, so $D = 1 - 5.5=-4.5$ and $D^{2}=(-4.5)^{2}=20.25$.
For the fourth row, eelgrass density per $m^{2}=84$ has rank 1 and green - crab density per $m^{2}=1$ has rank 5.5, so $D = 1 - 5.5=-4.5$ and $D^{2}=(-4.5)^{2}=20.25$.
For the fifth row, eelgrass density per $m^{2}=51$ has rank 3 and green - crab density per $m^{2}=3$ has rank 4, so $D = 3 - 4=-1$ and $D^{2}=(-1)^{2}=1$.
For the sixth row, eelgrass density per $m^{2}=31$ has rank 5 and green - crab density per $m^{2}=5$ has rank 3, so $D = 5 - 3 = 2$ and $D^{2}=2^{2}=4$.
For the seventh row, eelgrass density per $m^{2}=15$ has rank 7 and green - crab density per $m^{2}=7$ has rank 1.5, so $D = 7 - 1.5 = 5.5$ and $D^{2}=(5.5)^{2}=30.25$.
For the eighth row, eelgrass density per $m^{2}=21$ has rank 7 and green - crab density per $m^{2}=7$ has rank 1.5, so $D = 7 - 1.5 = 5.5$ and $D^{2}=(5.5)^{2}=30.25$.
Then $\sum D^{2}=2.25 + 2.25+20.25+20.25 + 1+4+30.25+30.25=100.5$.

Step3: Calculate $r_s$

The formula for the Spearman's rank - correlation coefficient is $r_s = 1-\frac{6\sum D^{2}}{n(n^{2}-1)}$, where $n = 8$.
Substitute $\sum D^{2}=100.5$ and $n = 8$ into the formula:
\[

$$\begin{align*} r_s&=1-\frac{6\times100.5}{8\times(8^{2}-1)}\\ &=1-\frac{603}{8\times63}\\ &=1-\frac{603}{504}\\ &=1 - 1.1964\\ &\approx - 0.196 \end{align*}$$

\]

Answer:

1. $H_0$: There is no correlation between the density of green crabs and the density of eelgrass.
2. $\sum D^{2}=100.5$, $r_s\approx - 0.196$