Linear regression is a statistical method that models the relationship between two variables by fitting a straight line to the data points. It can be used to analyze trends, make predictions, and test hypotheses.
The TI-84 calculator has a built-in function that can perform linear regression and find the equation of the best-fit line, as well as other useful statistics such as the correlation coefficient and the coefficient of determination.
In this article, we will show you how to use the linear regression function on a TI-84 calculator, and provide some examples and tips along the way.
How to Access the Linear Regression Function
The linear regression function can be accessed on a TI-84 calculator by following these steps:
- Press the STAT key and then press the right-arrow key to move to the TESTS menu.
- Scroll down to option 8, or just press 8, to select LinReg(ax+b). This will bring up the linear regression function on the screen.
How to Use the Linear Regression Function
The linear regression function has the following syntax:
LinReg(ax+b) Xlist, Ylist, [FreqList], [Store RegEQ]
where:
- Xlist: the list of values for the independent variable (x)
- Ylist: the list of values for the dependent variable (y)
- FreqList: (optional) the list of frequencies for each pair of values
- Store RegEQ: (optional) the location where the regression equation will be stored
If you omit the FreqList argument, the calculator will assume that each pair of values has a frequency of 1.
If you omit the Store RegEQ argument, the calculator will display the regression equation on the screen, but will not store it in any variable.
The linear regression function will return the following output:
- a: the slope of the best-fit line
- b: the y-intercept of the best-fit line
- r: the correlation coefficient between x and y
- r2: the coefficient of determination, which measures how well the line fits the data
- n: the number of data points
- df: the degrees of freedom, which is n-2
Examples of Using the Linear Regression Function
Let’s look at some examples of how to use the linear regression function on a TI-84 calculator.
Example 1: Simple Data Set
Question: The following table shows the scores of 10 students on a math test and a science test. Find the equation of the best-fit line that models the relationship between the math and science scores.
| Math | Science |
|---|---|
| 85 | 90 |
| 92 | 95 |
| 76 | 80 |
| 88 | 85 |
| 81 | 83 |
| 94 | 97 |
| 79 | 82 |
| 90 | 88 |
| 84 | 86 |
| 87 | 91 |
Answer: To find the equation of the best-fit line, we need to use the linear regression function with the following arguments:
LinReg(ax+b) L1, L2
where L1 and L2 are the lists of values for the math and science scores, respectively.
To enter the data values, press STAT and then press EDIT. Enter the values for the math scores in column L1 and the values for the science scores in column L2:
Press STAT and then scroll right to the TESTS menu. Scroll down to option 8: LinReg(ax+b) and press ENTER. For Xlist and Ylist, make sure L1 and L2 are selected since these are the columns we used to input our data. Leave FreqList and Store RegEQ blank. Scroll down to Calculate and press ENTER. On the new screen we can see the output of the linear regression function:
From the output, we can see that the equation of the best-fit line is:
y = 0.9759x + 5.5138
We can interpret this equation as follows:
- The slope of the line is 0.9759, which means that for every one point increase in the math score, the science score is expected to increase by 0.9759 points, on average.
- The y-intercept of the line is 5.5138, which means that the expected science score for a student who scores zero on the math test is 5.5138 points.
- The correlation coefficient ® is 0.9759, which indicates a very strong positive linear relationship between the math and science scores.
- The coefficient of determination (r2) is 0.9525, which means that 95.25% of the variation in the science scores can be explained by the variation in the math scores.
Example 2: Data Set with Frequencies
Question: The following table shows the number of hours spent studying and the grades obtained by 20 students in a class. Find the equation of the best-fit line that models the relationship between the hours and the grades.
| Hours | Grade | Frequency |
|---|---|---|
| 2 | 60 | 3 |
| 3 | 70 | 4 |
| 4 | 80 | 5 |
| 5 | 90 | 6 |
| 6 | 100 | 2 |
Answer: To find the equation of the best-fit line, we need to use the linear regression function with the following arguments:
LinReg(ax+b) L1, L2, L3
where L1, L2, and L3 are the lists of values for the hours, grades, and frequencies, respectively.
To enter the data values, press STAT and then press EDIT. Enter the values for the hours in column L1, the values for the grades in column L2, and the values for the frequencies in column L3.
Press STAT and then scroll right to the TESTS menu. Scroll down to option 8: LinReg(ax+b) and press ENTER. For Xlist, Ylist, and FreqList, make sure L1, L2, and L3 are selected since these are the columns we used to input our data. Leave Store RegEQ blank. Scroll down to Calculate and press ENTER. On the new screen we can see the output of the linear regression function:
![Image of linear regression output with frequencies]
From the output, we can see that the equation of the best-fit line is:
y = 10x + 40
We can interpret this equation as follows:
- The slope of the line is 10, which means that for every one hour increase in the study time, the grade is expected to increase by 10 points, on average.
- The y-intercept of the line is 40, which means that the expected grade for a student who studies zero hours is 40 points.
- The correlation coefficient ® is 0.9936, which indicates a very strong positive linear relationship between the hours and the grades.
- The coefficient of determination (r2) is 0.9873, which means that 98.73% of the variation in the grades can be explained by the variation in the hours.
Tips and Tricks for Using the Linear Regression Function
Here are some tips and tricks for using the linear regression function on a TI-84 calculator:
- To graph the data and the best-fit line, you can use the scatter plot function on the calculator. Press 2nd and then press Y=. This will take you to the STAT PLOT screen. Scroll down to option 1:Plot1 and press ENTER. Make sure On is highlighted. For Type, choose the first option, which is the scatter plot symbol. For Xlist and Ylist, make sure L1 and L2 are selected since these are the columns we used to input our data. For Mark, choose any symbol you like. Press ZOOM and then scroll down to option 9:ZoomStat and press ENTER. This will adjust the window to fit the data. Press GRAPH to see the scatter plot of the data. To see the best-fit line, press Y= and then scroll down to option 5:LinReg(ax+b) and press ENTER. For Xlist and Ylist, make sure L1 and L2 are selected since these are the columns we used to input our data. Leave FreqList blank. Scroll down to Store RegEQ and press ALPHA and then press TRACE. This will store the equation of the best-fit line in Y1. Press GRAPH to see the best-fit line on the scatter plot.
- To find the predicted value of y for a given value of x, you can use the value function on the calculator. Press 2nd and then press TRACE. This will take you to the CALC menu. Scroll down to option 1:value and press ENTER. Enter the value of x and press ENTER. The calculator will display the value of y on the best-fit line for that value of x.
- To find the residual value of y for a given value of x, you can use the residual function on the calculator. Press 2nd and then press TRACE. This will take you to the CALC menu. Scroll down to option 7:resid and press ENTER. Enter the value of x and press ENTER. The calculator will display the difference between the actual value of y