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.

In order to solve this problem, we need to define what is meant by the phrase "least squares regression equation." Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

In this equation, the variables are represented in the following manner:

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

We can see that the slope of the least squares regression equation is the following:

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

We can see that the y-intercept of the least squares regression equation is the following:

Now, we can write the least squares regression equation:

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

Let's input this information into a least squares regression equation in the slope-intercept format:

Substitute in our calculated values and solve.

In order to solve this problem, we need to define what is meant by the phrase 'least squares regression equation.' Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

In this equation, the variables are represented in the following manner:

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

We can see that the slope of the least squares regression equation is the following:

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

We can see that the y-intercept of the least squares regression equation is the following:

Now, we can write the least squares regression equation:

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

Let's input this information into a least squares regression equation in the slope intercept format:

Substitute in our calculated values and solve.

In order to solve this problem, we need to define what is meant by the phrase 'least squares regression equation.' Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

In this equation, the variables are represented in the following manner:

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

We can see that the slope of the least squares regression equation is the following:

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

We can see that the y-intercept of the least squares regression equation is the following:

Now, we can write the least squares regression equation:

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

y-intercept= 
slope= 
Let's input this information into a least squares regression equation in the slope intercept format:

Substitute in our calculated values and solve.

In order to solve this problem, we need to define what is meant by the phrase 'least squares regression equation.' Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

In this equation, the variables are represented in the following manner:

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

We can see that the slope of the least squares regression equation is the following:

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

We can see that the y-intercept of the least squares regression equation is the following:

Now, we can write the least squares regression equation:

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

y-intercept= 
slope= 
Let's input this information into a least squares regression equation in the slope intercept format:

Substitute in our calculated values and solve.

In order to solve this problem, we need to define what is meant by the phrase 'least squares regression equation.' Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

In this equation, the variables are represented in the following manner:

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

We can see that the slope of the least squares regression equation is the following:

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

We can see that the y-intercept of the least squares regression equation is the following:

Now, we can write the least squares regression equation:

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

y-intercept= 
slope= 
Let's input this information into a least squares regression equation in the slope intercept format:

Substitute in our calculated values and solve.

In order to solve this problem, we need to define what is meant by the phrase 'least squares regression equation.' Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

In this equation, the variables are represented in the following manner:

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

We can see that the slope of the least squares regression equation is the following:

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

We can see that the y-intercept of the least squares regression equation is the following:

Now, we can write the least squares regression equation:

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

y-intercept= 
slope= 
Let's input this information into a least squares regression equation in the slope intercept format:

Substitute in our calculated values and solve.

In order to solve this problem, we need to define what is meant by the phrase 'least squares regression equation.' Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

In this equation, the variables are represented in the following manner:

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

We can see that the slope of the least squares regression equation is the following:

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

We can see that the y-intercept of the least squares regression equation is the following:

Now, we can write the least squares regression equation:

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

y-intercept= 
slope= 
Let's input this information into a least squares regression equation in the slope intercept format:

Substitute in our calculated values and solve.

In order to solve this problem, we need to define what is meant by the phrase 'least squares regression equation.' Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

In this equation, the variables are represented in the following manner:

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

We can see that the slope of the least squares regression equation is the following:

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

We can see that the y-intercept of the least squares regression equation is the following:

Now, we can write the least squares regression equation:

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

y-intercept= 
slope= 
Let's input this information into a least squares regression equation in the slope intercept format:

Substitute in our calculated values and solve.