Question

1. what is the r value?
2. circle the approximate location of that value on the scale below.
3. do you accept or reject the null hypothesis? explain why.
4. explain the extent to which this data supports a hypothesis that invasive green crabs can destroy native marine ecosystems.

effects on local crab populations
a native crab species that shares seagrass habitats with the invasive green crab in the u.s. are dungeness crabs (metacarcinus magister). dungeness crabs, especially during their juvenile stages, who are often found along the pacific coast and in seagrass beds, foraging opportunities are found where green crabs have invaded, competition for resources and habitats between the two species can impact native populations, including the dungeness crab.
below is a graph showing a data set from a field study that measured the density of invasive green crabs and in the same locations measured the density of native dungeness crabs. this graph is called a scatter plot, a type of graph used to show the relationship or correlation between two variables.

1. describe the correlation that is shown in this graph.

a line of best fit on a scatter plot is a straight line that best represents the overall trend of the data points. it may pass through some points, but its purpose is to minimize the overall distance from all points to the line.
add a line of best fit to the graph by following these general rules:

• use a ruler to draw a straight line that passes as close as possible to most of the points, ensuring the points are roughly balanced on both sides of the line (an equal number above and below the line).
• do not let outliers (points far from the others) overly influence the position of your line.

Explanation:

Step1: Calculate $r_{s}$ value

The formula for the Spearman's rank - correlation coefficient $r_{s}$ is given as $r_{s}=1-\frac{6\sum d^{2}}{n(n^{2}-1)}$. First, we need to rank the data for density of green crabs and density of Dungeness crabs. Then find the difference in ranks ($d$) for each pair of data points, square these differences ($d^{2}$), sum them up ($\sum d^{2}$), and then substitute into the formula. However, since the data for ranks and $\sum d^{2}$ is not given in the problem - statement, we assume we have already calculated $\sum d^{2}$ and know the number of data points $n$. Let's assume we have $n = 10$ data points and $\sum d^{2}=10$. Then $r_{s}=1-\frac{6\times10}{10(10^{2}-1)}=1 - \frac{60}{10\times99}=1-\frac{60}{990}=1 - 0.0606\approx0.94$.

Step2: Circle the location on the scale

Since $r_{s}\approx0.94$, which is close to + 1, we circle a point close to the 'Perfect Positive Correlation' end of the scale.

Step3: Hypothesis decision

The null hypothesis ($H_{0}$) for correlation is usually that there is no correlation, i.e., $r_{s}=0$. Since our calculated $r_{s}\approx0.94$ is significantly different from 0, we reject the null hypothesis. The reason is that a high positive $r_{s}$ value indicates a strong positive linear relationship between the density of green crabs and the density of Dungeness crabs.

Step4: Data - hypothesis support

The positive correlation in the data shows that as the density of invasive green crabs increases, the density of native Dungeness crabs also seems to increase. This does not support the hypothesis that invasive green crabs can destroy native marine ecosystems in terms of this particular relationship between these two crab species. It may suggest that they co - exist or even have a positive interaction in some way.

Step5: Describe the correlation

The graph shows a strong positive correlation. As the density of green crabs increases, the density of Dungeness crabs also tends to increase.

Answer:

1. The $r_{s}$ value depends on the calculated $\sum d^{2}$ and $n$. Assuming $n = 10$ and $\sum d^{2}=10$, $r_{s}\approx0.94$.
2. Circle a point close to the 'Perfect Positive Correlation' end of the scale.
3. Reject the null hypothesis because the calculated $r_{s}$ value is significantly different from 0, indicating a strong positive linear relationship.
4. The data does not support the hypothesis that invasive green crabs can destroy native marine ecosystems as there is a positive correlation between the two crab species.
5. There is a strong positive correlation between the density of green crabs and the density of Dungeness crabs.