The table shows the home market value based on square feet.

We need to define the model allowing the home market value prediction.

1. Plot the home market values vs. square feet. What can we say?
2. Generate the linear regression report and explain:
2.1. The residual plot
2.2. The regression report
3. We want to improve the model by adding \( x^2 \). Generate the report and explain:
3.1. The residual plots
3.2. The regression report
4. We want to improve the model again by introducing \( x \cdot y \). Generate the report and explain:
4.1. The residual plots
4.2. The regression report
5. Can we improve it again? Argue your answer.
6. If yes, propose a solution.

If there is a need to improve the model, one possible solution is to explore other non-linear regression techniques, such as polynomial regression or regression with splines. These methods can capture more complex relationships between the predictors and the outcome variable. Additionally, collecting more data or including additional relevant predictors might also help improve the model's accuracy.

1. To start, plot the home market values on the y-axis and square feet on the x-axis. By examining the scatter plot, you can see the relationship between the two variables. If the points form a roughly linear pattern, it suggests that there might be a linear relationship between home market values and square feet.
2. Next, generate a Linear Regression report. This report provides statistical information about the relationship between the variables. The report includes the equation of the regression line, which represents the best fit line through the data points. It also provides information about the coefficient of determination (R-squared value), which indicates the proportion of the variance in the dependent variable (home market values) that can be explained by the independent variable (square feet).
2.1 The residual plot is a graphical representation of the differences between the observed home market values and the predicted values from the regression line. It helps to assess the goodness of fit of the regression model. In a residual plot, the points should be randomly scattered around the horizontal line at zero. If there is a clear pattern or trend in the residuals, it suggests that the linear regression model may not be appropriate.
2.2 The regression report provides important information about the coefficients of the regression equation. It includes the intercept, which represents the predicted value of home market values when square feet is zero. It also includes the coefficient for square feet, which represents the change in home market values for a one-unit increase in square feet. The report may also include statistical tests, such as t-tests or p-values, to assess the significance of the coefficients.
3. To improve the model, you can add x^2 as an additional predictor variable. This allows for a quadratic relationship between home market values and square feet. Generate a new regression report to analyze the updated model.
3.1 The residual plots for the quadratic model will show if the model captures any non-linear patterns that were missed by the linear model. You may see a U-shape or inverted U-shape in the residuals if the quadratic term helps explain the variation in home market values.
3.2 The regression report for the quadratic model will provide the coefficients for both square feet and the squared term. The squared term coefficient represents the change in the rate of change of home market values for each additional unit of square feet. It is important to interpret both coefficients in the context of the problem.
4. To further improve the model, you can introduce the interaction term X*Y, where X is square feet and Y is the squared term. Generate a new regression report to analyze the updated model.
4.1 The residual plots for the model with the interaction term will help identify if there are any interactions or joint effects between the predictors. If the residuals show a systematic pattern or trend, it suggests that the model might not adequately capture the relationship between the variables.
4.2 The regression report for the model with the interaction term will provide coefficients for all the predictors, including the interaction term. The interpretation of the coefficients should consider the interaction effect between square feet and the squared term.
5. Whether or not we can further improve the model depends on the specific data and context. Adding more variables or higher-order terms might increase the complexity of the model without improving its predictive power. It is important to evaluate the goodness of fit, interpretability, and practicality of the model when deciding whether to make further improvements.

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