Quadratic regression is a statistical method that models the relationship between two variables by fitting a parabola 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 quadratic regression and find the equation of the best-fit parabola, 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 quadratic regression function on a TI-84 calculator, and provide some examples and tips along the way.
How to Access the Quadratic Regression Function
The quadratic 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 CALC menu.
- Scroll down to option 5, or just press 5, to select QuadReg. This will bring up the quadratic regression function on the screen.
How to Use the Quadratic Regression Function
The quadratic regression function has the following syntax:
QuadReg 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 quadratic regression function will return the following output:
- a: the coefficient of the x2 term of the best-fit parabola
- b: the coefficient of the x term of the best-fit parabola
- c: the constant term of the best-fit parabola
- r: the correlation coefficient between x and y
- r2: the coefficient of determination, which measures how well the parabola fits the data
- n: the number of data points
- df: the degrees of freedom, which is n-3
Examples of Using the Quadratic Regression Function
Let’s look at some examples of how to use the quadratic regression function on a TI-84 calculator.
Example 1: Simple Data Set
Question: The following table shows the distance traveled by a car and the corresponding fuel consumption. Find the equation of the best-fit parabola that models the relationship between the distance and the fuel consumption.
| Distance (km) | Fuel (L) |
|---|---|
| 10 | 1.2 |
| 20 | 2.1 |
| 30 | 3.4 |
| 40 | 5.1 |
| 50 | 7.2 |
| 60 | 9.8 |
| 70 | 13 |
| 80 | 16.7 |
Answer: To find the equation of the best-fit parabola, we need to use the quadratic regression function with the following arguments:
QuadReg L1, L2
where L1 and L2 are the lists of values for the distance and the fuel consumption, respectively.
To enter the data values, press STAT and then press EDIT. Enter the values for the distance in column L1 and the values for the fuel consumption in column L2:
Press STAT and then scroll right to the CALC menu. Scroll down to option 5: QuadReg 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 quadratic regression function:
From the output, we can see that the equation of the best-fit parabola is:
y = 0.003x2 + 0.057x + 0.6
We can interpret this equation as follows:
- The coefficient of the x2 term is 0.003, which means that the fuel consumption increases at an increasing rate as the distance increases.
- The coefficient of the x term is 0.057, which means that the fuel consumption increases by 0.057 liters for every one kilometer increase in the distance, when the distance is zero.
- The constant term is 0.6, which means that the fuel consumption is 0.6 liters when the distance is zero.
- The correlation coefficient ® is 0.9999, which indicates a very strong positive quadratic relationship between the distance and the fuel consumption.
- The coefficient of determination (r2) is 0.9998, which means that 99.98% of the variation in the fuel consumption can be explained by the variation in the distance.
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 parabola 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 parabola, we need to use the quadratic regression function with the following arguments:
QuadReg 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 CALC menu. Scroll down to option 5: QuadReg 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 quadratic regression function:
From the output, we can see that the equation of the best-fit parabola is:
y = 10x2 – 100x + 240
We can interpret this equation as follows:
- The coefficient of the x2 term is 10, which means that the grade increases at an increasing rate as the hours increase.
- The coefficient of the x term is -100, which means that the grade decreases by 100 points for every one hour increase in the hours, when the hours are zero.
- The constant term is 240, which means that the grade is 240 points when the hours are zero.
- The correlation coefficient ® is 0.9936, which indicates a very strong positive quadratic relationship between the hours and the grade.
- The coefficient of determination (r2) is 0.9873, which means that 98.73% of the variation in the grade can be explained by the variation in the hours.
Tips and Tricks for Using the Quadratic Regression Function
Here are some tips and tricks for using the quadratic regression function on a TI-84 calculator:
- To graph the data and the best-fit parabola, 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 parabola, press Y= and then scroll down to option 5:QuadReg 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. Scroll down to Store RegEQ and press ALPHA and then press TRACE. This will store the equation of the best-fit parabola in Y1. Press GRAPH to see the best-fit parabola 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 parabola 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 and the predicted value of y on the best-fit parabola for that value of x.
- To check your answer, you can substitute the solution into the equation and see if it makes the equation true. You can do this by pressing the 2nd key and then the ENTER key to recall the equation, and then replacing the X with the solution using the STO> key. For example, to check the solution x = 2 for the equation y = 10x2 – 100x + 240, you can press:
2nd ENTER 2 STO> X ENTER
The calculator will display:
10(2)2-100(2)+240=0
true
This means that the solution is correct.