Question

name: stephanie ben - hur date: alaskas 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. when determining spearmans rank, the researcher will state a null hypothesis, which always states that there is no relationship between the variables. once you have determined the r_s value, you are able to accept or reject the null hypothesis. an r_s value is calculated that ranges between - 1 and 1 with: r_s=1 - \frac{6\sum d^{2}}{n(n^{2}-1)} \sum = sum (total) n = number of pairs of items in the sample d = difference in rank between each pair of measurements elgrass 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 green 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}
164	0
94	1	5.5
84	1	5.5
51	3
31	5
15	7	15
21	7

in this data set, n is equal to 8

Explanation:

Step1: State the null hypothesis

The null hypothesis ($H_0$) for a correlation study is that there is no relationship between the two variables. In this case, the two variables are the density of green - crabs and the density of eelgrass. So, $H_0$: There is no correlation between the density of green crabs and the density of eelgrass.

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

1. For the first row:

• Eelgrass per $m^{2}=132$, rank = 0; Green crabs per $m^{2}=7.5$, rank = 1.5. Then $D=0 - 1.5=- 1.5$, and $D^{2}=(-1.5)^{2}=2.25$.

1. For the second row:

• Eelgrass per $m^{2}=164$, rank = 0; Green crabs per $m^{2}=0$, rank = 7.5. Then $D=0 - 7.5=-7.5$, and $D^{2}=(-7.5)^{2}=56.25$.

1. For the third row:

• Eelgrass per $m^{2}=94$, rank = 1; Green crabs per $m^{2}=5.5$, rank = 2.5. Then $D=1 - 2.5=-1.5$, and $D^{2}=(-1.5)^{2}=2.25$.

1. For the fourth row:

• Eelgrass per $m^{2}=84$, rank = 1; Green crabs per $m^{2}=5.5$, rank = 2.5. Then $D=1 - 2.5=-1.5$, and $D^{2}=(-1.5)^{2}=2.25$.

1. For the fifth row:

• Eelgrass per $m^{2}=51$, rank = 3; Green crabs per $m^{2}=4$, rank = 4. Then $D=3 - 4=-1$, and $D^{2}=(-1)^{2}=1$.

1. For the sixth row:

• Eelgrass per $m^{2}=31$, rank = 5; Green crabs per $m^{2}=3$, rank = 5. Then $D=5 - 5 = 0$, and $D^{2}=0^{2}=0$.

1. For the seventh row:

• Eelgrass per $m^{2}=15$, rank = 7; Green crabs per $m^{2}=15$, rank = 1. Then $D=7 - 1 = 6$, and $D^{2}=6^{2}=36$.

1. For the eighth row:

• Eelgrass per $m^{2}=21$, rank = 7; Green crabs per $m^{2}=7$, rank = 3. Then $D=7 - 3 = 4$, and $D^{2}=4^{2}=16$.

1. Calculate $\sum D^{2}$:

• $\sum D^{2}=2.25 + 56.25+2.25 + 2.25+1+0+36+16=116$.

Answer:

1. $H_0$: There is no correlation between the density of green crabs and the density of eelgrass.
2. $\sum D^{2}=116$