Quadratic Regression: Videos & Practice Problems

Quadratic regression models data with a curve using the equation $y=a{x}^2+bx+c$, where a causes curvature. Using a TI-84 calculator, input data into lists, select quadratic regression, and store the equation as y₁. The calculator outputs coefficients and the coefficient of determination (R2), indicating fit quality. An R2 near 1, like 0.977, confirms a strong fit, useful for analyzing nonlinear trends in business analytics and statistics.

Quadratic Regression Using TI-84

Quadratic regression is a powerful statistical method used to model data that follows a curved pattern rather than a straight line. Unlike linear regression, which fits data to a line defined by the equation y = bx + c, quadratic regression fits data to a parabola described by the equation y = ax² + bx + c. The key difference lies in the ax² term, which introduces curvature, allowing the model to capture U-shaped or inverted U-shaped trends in the data.

When analyzing datasets that exhibit nonlinear trends, quadratic regression provides a more accurate representation by estimating the coefficients a, b, and c that define the best-fit curve. These coefficients are typically calculated using technology such as a TI-84 graphing calculator, which simplifies the process by performing complex computations automatically.

To perform quadratic regression on a TI-84 calculator, first input your data points into the Stat Edit lists, usually L1 for the independent variable (x-values) and L2 for the dependent variable (y-values). Next, access the Stat menu, navigate to the Calc submenu, and select the quadratic regression function (often labeled as QuadReg). Specify the lists containing your data and choose to store the resulting regression equation in the calculator’s function editor (e.g., Y1) for easy graphing.

After running the regression, the calculator outputs the coefficients a, b, and c, which can be substituted back into the quadratic equation:

\[ \hat{y} = a x^{2} + b x + c \]

For example, if the calculator returns a = -2.07, b = 20.63, and c = -1.05, the quadratic regression equation becomes:

\[ \hat{y} = -2.07 x^{2} + 20.63 x - 1.05 \]

Evaluating the quality of the fit involves examining the coefficient of determination, denoted as R². This value ranges from 0 to 1 and indicates how well the regression curve explains the variability in the data. An R² value close to 1 signifies a strong fit. For instance, an R² of 0.977 suggests that the quadratic model explains approximately 97.7% of the variance, confirming that the curve closely matches the data points.

To visualize the quadratic regression, adjust the graphing window on the calculator to encompass the range of your data. Ensure that the Stat Plot feature is enabled to display the original data points alongside the regression curve. The resulting graph will show a smooth parabola passing near or through the data points, illustrating the model’s accuracy.

Quadratic regression is especially useful in real-world scenarios where relationships between variables are not linear, such as tracking sales trends over time, modeling projectile motion, or analyzing growth rates that accelerate or decelerate. By leveraging technology like graphing calculators, students and professionals can efficiently derive and interpret quadratic models, enhancing their ability to analyze complex datasets.

For the data points in the graphs below, which most likely suggests a quadratic relationship?

• A.
• B.
• C.
• D.

Answer: (C)

The data below shows the population of a small town (in 1000s) over a 9 year period. Using a graphing calculator, determine the linear & quadratic regression curves. Compare their $R^2$ values. Which model is a better fit for the data?

• A. Both Linear with $R^2=0.93$ and quadratic with $R^2=0.999$ are good fits for the data.
• B. Quadratic with $R^2=0.999$ is a better fit than linear with with $R^2=0.96$.
• C. Both Linear with $R^2=0.96$ and quadratic with $R^2=0.999$ are good fits for the data.
• D. Quadratic with $R^2=0.999$ is a better fit than linear with with $R^2=0.93$.

Answer: (D)

Here’s what students ask on this topic:

What is quadratic regression and how does it differ from linear regression?

Quadratic regression is a statistical method used to model data that follows a curved pattern rather than a straight line. Unlike linear regression, which fits data to a line using the equation $y=bx+c$, quadratic regression fits data to a parabola using the equation $y=a{x}^2+bx+c$. The key difference is the $a{x}^2$ term, which allows the curve to bend upward or downward, capturing nonlinear trends in the data. This makes quadratic regression useful when data points form a U-shaped or inverted U-shaped pattern, which linear regression cannot accurately model.

How do you perform quadratic regression using a TI-84 calculator?

To perform quadratic regression on a TI-84 calculator, first enter your data into lists: input your independent variable values into L1 and dependent variable values into L2. Next, press STAT, select CALC, and choose the quadratic regression option (usually QuadReg). Enter L1, L2 as your X and Y lists. You can store the regression equation by pressing VARS, selecting Y-VARS, then Function, and choosing Y1. This stores the equation for easy graphing. Finally, calculate the regression to get coefficients $a$, $b$, and $c$, and the coefficient of determination $R^2$, which indicates the fit quality.

What does the coefficient of determination (R²) tell us in quadratic regression?

The coefficient of determination, denoted as $R^2$, measures how well the quadratic regression model fits the data. It ranges from 0 to 1, where values closer to 1 indicate a better fit. For example, an $R^2$ of 0.977 means that approximately 97.7% of the variation in the dependent variable is explained by the quadratic model. Generally, an $R^2$ above 0.8 is considered a strong fit, suggesting the quadratic equation accurately represents the data's trend. This helps in assessing the reliability of predictions made using the regression equation.

How do you interpret the coefficients in a quadratic regression equation?

In a quadratic regression equation of the form $y=a{x}^2+bx+c$, each coefficient has a specific role. The coefficient $a$ controls the curvature of the parabola; if $a$ is positive, the curve opens upward (U-shaped), and if negative, it opens downward (inverted U). The coefficient $b$ affects the slope and direction of the curve, influencing where the vertex (peak or trough) is located. The constant $c$ represents the y-intercept, the point where the curve crosses the y-axis. Together, these coefficients define the shape and position of the best-fit curve through the data points.

When should you use quadratic regression instead of linear regression?

You should use quadratic regression when your data shows a nonlinear pattern that resembles a curve, such as a U-shape or an inverted U-shape, which linear regression cannot accurately model. If plotting your data reveals that the relationship between variables is not a straight line but bends upward or downward, quadratic regression is more appropriate. Additionally, if the coefficient of determination ($R^2$) for a quadratic model is significantly higher than that of a linear model, it indicates a better fit. This method is especially useful in business statistics and predictive analytics when modeling trends that change direction.

How do you graph a quadratic regression equation on a TI-84 calculator?

After performing quadratic regression and storing the equation as Y1 on your TI-84 calculator, turn on the Stat Plot by pressing 2nd then Y= and ensure the plot is on. Next, adjust the window settings by pressing WINDOW to set appropriate ranges for your x and y values, typically covering the minimum and maximum data points. For example, set Xmin slightly below your smallest x-value and Xmax above your largest. Similarly, set Ymin and Ymax to cover your y-values. Finally, press GRAPH to display the curve along with your data points, allowing you to visually assess the fit of the quadratic regression model.