Simple linear regression is a great way to make observations and interpret data. In this lesson, you will learn to find the regression line of a set of data using a ruler and a graphing calculator.

## Simple Linear Regression

Hannah is a scientist studying the time management and study skills of college students.

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She conducts an experiment at a local college with 50 students. She asks each student to track their time spent on social media, time spent studying, time spent sleeping and time spent working over the course of a semester. She also asks the students to record their final GPA for the semester.In this lesson, you will be learning about the simple linear regression and how to find a regression line using a graphing calculator. A regression line is a straight line that attempts to predict the relationship between two points, also known as a trend line or line of best fit.

Simple linear regression is a prediction when a variable (y) is dependent on a second variable (x) based on the regression equation of a given set of data. We’ll see how Hannah uses simple linear regression to help interpret her data.

## Scatterplots and Regression

Once all 50 students turn in their data, Hannah creates a scatterplot, which is a graph of ordered pairs showing a relationship between two sets of data. For her first scatterplot, Hannah uses two variables: time spent on social networking and amount of sleep.

To simplify her information, we are going to look at the average time per week each student spent sleeping and on social media. Take a look at the scatterplot:

Since we are using two variables, we can call this bivariate data. Bivariate data is two sets of variables that can change and are compared to find relationships. Bivariate data is most often displayed visually using a scatterplot.You may notice that this data has several points that create a sort of pattern.

Many of the points increase in the x value as they decrease in the y value. This is a relationship between the two sets of data known as a correlation. A correlation is the relationship between two sets of variables used to describe or predict information.A regression line is one way of predicting this information and finding a correlation in the data. There are two ways you can find the regression line of a set of data.

The first way is to find the regression line by using a ruler, and the second way is to use a graphing calculator. Let’s talk briefly about how to find a regression line by hand before we use a calculator.

## Finding the Regression Line by hand

To find a regression line by hand, follow these steps:

1. Draw a line that is closest to as many points as possible.
2. Choose two points, and calculate slope.
3. Write the equation of the line.

Let’s use the scatterplot above to practice finding the regression line using a ruler.First, use the ruler to find a place that is closest to as many points as possible. Sometimes you can find two points to use.Second, pick two points you think would be on the regression line. You can use points that are on the line or you can make up new points. Use these points and plug them into the following equation:

If you are unfamiliar with using this equation, check out our algebra lessons!Third, now you have the slope of your line.

You can create an equation based on this information. Find the y-intercept by extending the line all the way to the y-axis.Use the slope-intercept equation to create the equation for your line like this:

• y = mx + by = -1x + 66

## Using a Graphing Calculator

Now that you know how to find a regression line by hand, let’s talk about how to find a regression line using a calculator. Every calculator is a little bit different.

However, you should be able to get by with just about any graphing calculator using these steps:

1. Set the calculator to Statistics mode.
2. Enter the data (1st Set on L1, 2nd Set on L2).
3. Adjust settings for a scatterplot, and then graph the points.
4. Set the calculator for regression line.

We will use the data that Hannah collected about the amount of sleep and the amount of work the students did during the semester:

Data table for example
This data is a weekly average for each student. To save time, I’m only using 20 students, rather than the original 50. In this example, I am using a TI-83 graphing calculator.First, set your calculator to statistics by pressing the Stat button.

This will take you to a screen with the options of Edit, Calc and Tests. Select the Edit option by pressing enter. The L1 is the x-coordinates, and the L2 is for the y-coordinates. Enter each of the coordinates using the number pad and hitting ‘Enter’ when you are done entering each coordinate. After hitting the enter button, the calculator will take you to the next line for the second coordinate. You can also use the arrow buttons to move between L1 and L2.

Next, adjust the settings on your calculator to display a scatterplot. To do this, hit the 2nd button then hit Statplot. You should see this screen:

Hit Enter to go into the next screen, which looks like this:

Settings
Make sure your settings match mine by moving the cursor around with the arrow buttons and selecting each item with the Enter button. Selected items will have a black background and light text.You will also want to make sure you can see all of your points. I’ve adjusted the values in each field using the arrow keys, the enter button and the number pad.

Push the Graph key to see the points plotted on the graph. Now you are ready to find the equation of your regression line.First, press the STAT button again. This time, use the arrow keys to move to the CALC option at the top of the screen.

Using the arrow keys, move your cursor down to item number four labeled LinReg(ax + b). Press the enter button.The calculator will show you the same LinReg(ax + b) at the top of the screen.

Hit enter a second time to calculate the regression line.

• a = 1.3
• b = 40.6

Therefore, the equation for the regression line is y = 1.

3x + 40.6. My graph would look like this:

Final graph for example

## Practice Problems

Now Hannah wants to compare the amount of time a student spends studying to the amount of time the student spends sleeping. Can you find the regression line and its equation from this set of data? Pause the video here to work on this problem.

Sleep Time (Hours) Study Time (Hours)
70 6
64 4
60 7
57 8
56 6
54 10
52 12
50 10
49 11
47 9
47 8
46 4
50 6
45 7
42 10
40 8
38 9
35 10
49 5
49 4

The equation for the regression line on this graph is y = -0.0989x + 12.643. If you rounded numbers here, that’s okay for this problem.

Don’t forget to press ‘enter’ when you see the LinReg(ax + b) on your calculator. It will not show you the values for a and b if you don’t press enter, telling the calculator to find these variables.Okay, try this. Hannah now wants to compare the time a student spends studying to his or her GPA. Can you find the regression line and its equation from this set of data? Feel free to pause the video here while you work.

Study Time (Hours) GPA (Semester)
6 2.

6

4 2.2
7 2.9
8 3.

4

3 3.5
10 4
12 4
10 3.8
11 3.6
9 4
8 3.7
4 4
6 2.

4

7 3.5
10 3.2
8 3.3
9 3.9
10 3.

7

5 2.6
4 2.3

The equation for the regression line on this graph is y = .

1683x + 2.0343. If you rounded numbers here, that’s okay for this problem. If you did not get the correct answer here, feel free to go back in the video and follow the steps again with me.

## Lesson Summary

In summary, a regression line, also known as a trend line or line of best fit, is a straight line that attempts to predict the relationship between two points. Simple linear regression is a prediction when a variable (y) is dependent on a second variable (x) based on the regression equation of a given set of data.Every calculator is a little bit different. However, you should be able to get by with just about any graphing calculator using these steps:

1. Set the calculator to Statistics mode.
2. Enter the data (1st Set on L1, 2nd Set on L2).
3. Adjust settings for a scatterplot, and graph the points.
4. Set the calculator for regression line.

Finally, if you get any errors or the information doesn’t look correct, double check the points you entered first, and then review the settings on the calculator.

## Learning Outcomes

Watch this video lesson, then see how well you can:

• Provide definitions for ‘regression line’ and ‘simple linear regression’
• Create a scatterplot
• Present a step-by-step method for using a graphing calculator
• Find the regression line of a set of data using a ruler and a graphing calculator

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