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 that area is the dungeness crab (metacarcinus magister). dungeness crabs (shown in figure 1) are found along the pacific coast and often inhabit eelgrass beds, especially during their juvenile stages. the habitat provided by eelgrass beds offers foraging opportunities for both species and often includes shelter and 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 $r$ value

The formula for the correlation coefficient $r$ is given as $r = \frac{\sum xy}{n(\sigma_x\sigma_y)}$. However, since the formula in the problem is $r = \frac{\sum xy}{n(\overline{x}^2 - 1)}$ (assuming it's a mis - written form and we need to calculate based on given data points from the scatter - plot which are not fully provided in a way to calculate directly. But if we had data points $(x_i,y_i)$ we would calculate $\sum xy=\sum_{i = 1}^{n}x_iy_i$, $n$ is the number of data points, $\overline{x}$ is the mean of $x$ values etc.). Without actual data values for calculation, we can't get a numerical value for $r$. But for the sake of general steps, if we had data:

1. Calculate the means of $x$ and $y$ values: $\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$ and $\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$.
2. Calculate the deviations $x_i-\overline{x}$ and $y_i - \overline{y}$ for each data point.
3. Calculate $\sum xy=\sum_{i = 1}^{n}(x_i-\overline{x})(y_i - \overline{y})$, $\sigma_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\overline{x})^2}{n}}$ and $\sigma_y=\sqrt{\frac{\sum_{i = 1}^{n}(y_i-\overline{y})^2}{n}}$.
4. Then $r=\frac{\sum xy}{n(\sigma_x\sigma_y)}$.

Step2: Locate $r$ on the scale

Once we have the $r$ value, if $r=- 1$, it is a perfect negative correlation, if $r = 0$ it is no correlation and if $r = 1$ it is a perfect positive correlation. We would circle the appropriate place on the scale.

Step3: Hypothesis testing

The null hypothesis $H_0$ is usually that there is no correlation ($r = 0$). If the calculated $r$ value is far from 0 (either positive or negative), we reject the null hypothesis. We compare the calculated $r$ value with a critical $r$ value (which depends on the sample size $n$ and the significance level $\alpha$). If $|r|>r_{critical}$, we reject $H_0$.

Step4: Analyze data support for hypothesis

If there is a strong negative correlation between the density of invasive green crabs and the density of native Dungeness crabs (a large negative $r$ value), it supports the hypothesis that invasive green crabs can destroy native marine ecosystems as it shows an inverse relationship between the two populations.

Step5: Describe correlation

By looking at the scatter - plot, if the points seem to form a pattern sloping downwards from left to right, it is a negative correlation. If they form a pattern sloping upwards from left to right, it is a positive correlation. If the points are randomly scattered, it is no correlation.

Step6: Draw line of best fit

Using a ruler, we draw a straight line that passes as close as possible to most of the points, balancing the points above and below the line and not being overly influenced by outliers.

Answer:

Since no data values are given to calculate the $r$ value, we can't provide a specific numerical answer for the $r$ value, location on the scale, hypothesis decision, data - support analysis, correlation description or the exact line of best fit. But the steps to solve each part are as described above.