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 eelgrass habitats with the invasive green crab in the u.s. is the dungeness crab (metacarcinus magister). dungeness crab (shown here) are found along the pacific coast. they often inhabit eelgrass beds, especially during their juvenile stages, where the habitat provides shelter and foraging opportunities. in areas where green crabs have invaded, competition for resources and habitat between the two species can have 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 the $r_s$ value

We are not given the actual data values to calculate the $r_s$ (Spearman's rank - correlation coefficient) using the formula $r_s = 1-\frac{6\sum d^{2}}{n(n^{2}-1)}$. Without the ranks of the two variables (density of green crabs and density of Dungeness crabs), we cannot perform the calculation.

Step2: Locate the value on the scale

Since we don't have the $r_s$ value, we cannot circle its approximate location on the scale.

Step3: Hypothesis testing

The null hypothesis ($H_0$) for correlation is usually $H_0:
ho = 0$ (no correlation). To accept or reject it, we need a significance level and the calculated $r_s$ value. We would compare the calculated $r_s$ with the critical value from the Spearman - correlation table for the given sample size and significance level. Without these, we cannot make a decision.

Step4: Data support for hypothesis

To explain the extent to which the data supports the hypothesis that invasive green crabs can destroy native marine ecosystems, we would look at the sign and magnitude of the correlation coefficient. If $r_s$ is negative and large in magnitude, it would suggest that as the density of green crabs increases, the density of Dungeness crabs decreases, supporting the hypothesis. But without the calculated $r_s$ value, we cannot make a proper assessment.

Step5: Describe the correlation

Visually, from the scatter - plot, we can see that as the density of green crabs increases, the density of Dungeness crabs seems to decrease. So, there appears to be a negative correlation.

Step6: Add a line of best fit

To add a line of best fit, we use a ruler to draw a straight line that passes as close as possible to most of the points, with roughly an equal number of points above and below the line and not being overly influenced by outliers.

Answer:

1. Cannot calculate $r_s$ value without data.
2. Cannot circle location without $r_s$ value.
3. Cannot accept or reject null hypothesis without $r_s$ value and critical value.
4. Cannot assess data support for hypothesis without $r_s$ value.
5. Appears to be a negative correlation.
6. Draw a straight line following the rules for line of best fit.