To fill in the data with three regression equations and determine which one fits best, we'll need to use the data provided in the image. Since I can't directly see the numerical values in the tables from the image, I'll outline the process in general terms.

Step-by-Step Process

1. Extract the Data Points

   • For both questions, identify the independent variable (x) and dependent variable (y) from the tables provided.
   • Let's assume you have data points like this for each dataset:
     • $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$.

2. Determine the Types of Regression Equations

   • You have three types of regression equations to test: Linear, Quadratic, and Cubic. These equations generally look like:
     • Linear: $y = ax + b$
     • Quadratic: $y = ax^2 + bx + c$
     • Cubic: $y = ax^3 + bx^2 + cx + d$

3. Calculate the Regression Equations

   • Use a calculator or software (like Excel, Google Sheets, or a graphing calculator) to compute the regression equations for each type.

Example Regression Equations

For the given data points, let’s compute the three types of regression equations.

1. Linear Regression:

   • Find the linear regression equation of the form $y = ax + b$.
   • Use the least squares method or a calculator to compute the slope (a) and the intercept (b).

2. Quadratic Regression:

   • Fit the data to a quadratic equation of the form $y = ax^2 + bx + c$.
   • Use a calculator with quadratic regression capabilities or software tools to find coefficients $a, b, c$.

3. Cubic Regression:

   • Fit the data to a cubic equation of the form $y = ax^3 + bx^2 + cx + d$.
   • Compute the coefficients $a, b, c, d$ using a graphing calculator or a software tool.

Example of Filling in the Data

Regression Type	Resulting Equation
Linear	$y = ax + b$ (Insert values for a, b)
Quadratic	$y = ax^2 + bx + c$ (Insert values for a, b, c)
Cubic	$y = ax^3 + bx^2 + cx + d$ (Insert values for a, b, c, d)

How to Pick the Best Fit

1. Calculate the R-squared Value for Each Regression:

   • The R-squared value tells you how well the data fits the regression model. It ranges from 0 to 1, with 1 indicating a perfect fit.
   • Compute the R-squared values for each of the three regression types.

2. Compare the R-squared Values:

   • The regression type with the highest R-squared value is generally considered the best fit for the data.

3. Check Residual Plots (Optional):

   • A residual plot can help confirm that the chosen regression type is the best fit. Ideally, the residuals should be randomly scattered around the horizontal axis (y = 0).

Next Steps

Would you like to:

1. Provide specific data from the image to compute the regression equations?
2. Have a guide on using specific tools (Excel, graphing calculator) to compute these regressions?
3. Learn more about interpreting the R-squared value and residual plots?
4. Explore alternative methods for choosing the best regression model?
5. Understand the practical implications of different regression types?

Tip: Always validate your regression model with additional tests, like residual analysis, to ensure it's the best fit for your data.