When points are plotted in a linear regression model, trendlines or best-fit lines are used to make inferences and predictions about the data. There are several common trendline types: logarithmic, polynomial, exponential, power, and linear. Contrary to popular belief, a linear trendline is not always the best fit for every data set. In other words, we need to test the trendline to figure out whether or not it possesses strong associations of linearity between points. We can test this by graphing the plot’s residuals. 

What is meant by “residuals”? The residual of a point on a graph is calculated by subtracting the predicted y-value from its actual value. It is written using the following equation:

In this equation:

The actual values are represented by the points plotted on the graph, while the predicted values are represented by the trend line. The difference between each actual value and its predicted counterpart is the point's residual.

The question provided a table of the x- and y-values for the scatterplot. It also provided the equation of the linear trendline. Given this information, we can calculate the predicted y-values and the residuals of the scatterplot.

Let’s start by calculating the predicted y-values using the equation of the trendline and the x-values.

Lets start with the first x-value:

Now, calculate each predicted value for every x-coordinate in the scatter plot. Afterwards, calculate the residual for each point. For example,

Calculate the residuals for every point in the graph.

Now, we have calculated the predicted y-values and the residuals; therefore, we can create a graph of the residuals in the series. The graph will contain the residual values on the y-axis and the original x-values on the x-axis. 

Now, we can fit a trendline to the data. Notice that in this case the trendline is nearly horizontal. This indicates that there is a random spread in the residual data, which indicates that there is a linear correlation between points. The correct answer is "The graph is linear because the plot of the residuals possesses a random distribution." Now, we can determine a scatter plot's linearity using a graph of the plot's residuals.

When points are plotted in a linear regression model, trendlines or best-fit lines are used to make inferences and predictions about the data. There are several common trendline types: logarithmic, polynomial, exponential, power, and linear. Contrary to popular belief, a linear trendline is not always the best fit for every data set. In other words, we need to test the trendline to figure out whether or not it possesses strong associations of linearity between points. We can test this by graphing the plot’s residuals. 

What is meant by “residuals”? The residual of a point on a graph is calculated by subtracting the predicted y-value from its actual value. It is written using the following equation:

In this equation:

The actual values are represented by the points plotted on the graph, while the predicted values are represented by the trend line. The difference between each actual value and its predicted counterpart is the point's residual.

The question provided a table of the x- and y-values for the scatterplot. It also provided the equation of the linear trendline. Given this information, we can calculate the predicted y-values and the residuals of the scatterplot.

Let’s start by calculating the predicted y-values using the equation of the trendline and the x-values.

Lets start with the first x-value:

Now, calculate each predicted value for every x-coordinate in the scatter plot. Afterwards, calculate the residual for each point. For example,

Calculate the residuals for every point in the graph.

Now, we have calculated the predicted y-values and the residuals; therefore, we can create a graph of the residuals in the series. The graph will contain the residual values on the y-axis and the original x-values on the x-axis. 

Now, we can fit a trendline to the data. Notice that in this case the trendline is nearly horizontal. This indicates that there is a random spread in the residual data, which indicates that there is a linear correlation between points. The correct answer is "The graph is linear because the plot of the residuals possesses a random distribution." Now, we can determine a scatter plot's linearity using a graph of the plot's residuals.