Price, Square Footage, Bedrooms, And Age Data

Price Sq Footage Bedrooms Age Years64000 1670 2

Use the data in Problem 22 and develop a regression model to predict selling price based on the square footage and number of bedrooms. Use this regression line to predict the selling price of a 2000-square-foot house with 3 bedrooms. Conduct F-test for the above model using F-table as well as p-value from the ANOVA table and interpret the results. Determine which among the two independent variables has a statistically significant relationship with the selling price. Develop a simple regression model to predict selling price based on just square footage, compare it with the multiple regression model, and evaluate which model is better and whether the number of bedrooms should be included in the model.

Paper For Above instruction

The purpose of this paper is to analyze the relationship between house prices and various attributes such as square footage and number of bedrooms, using regression analysis techniques. The data provided consist of house prices, square footage, number of bedrooms, and house age. This analysis aims to develop predictive models, perform hypothesis testing to evaluate the significance of predictors, and determine the most appropriate model for estimating house prices.

Given the data, the first step involves constructing a multiple regression model with the house price as the dependent variable and square footage and number of bedrooms as independent variables. The general form of this model is:

Price = β0 + β1(Square Footage) + β2(Bedrooms) + ε

Where β0 is the intercept, β1 and β2 are the coefficients representing the effect of each predictor, and ε is the error term.

Using statistical software or manual calculations, the regression coefficients are estimated based on the dataset. Once the model is fitted, we can substitute the values for a house with 2000 square feet and 3 bedrooms to predict its selling price, which would be computed as:

Predicted Price = β0 + β1(2000) + β2(3)

This calculation furnishes an estimate that assists both buyers and sellers in understanding price expectations based on house features.

Next, hypothesis testing involves conducting an F-test to evaluate the overall significance of the regression model. The null hypothesis states that all regression coefficients are zero (no effect), versus the alternative that at least one coefficient differs from zero. The F-statistic is calculated using the regression sum of squares (SSR) and residual sum of squares (SSE), as well as their respective degrees of freedom. This F-value is then compared to the critical value from the F-distribution table at the desired significance level, typically 0.05.

Concurrent with the F-test, the p-value derived from the ANOVA table offers a measure of significance for the overall model. A p-value below the significance threshold indicates the model explains a significant portion of the variance in house prices. If the F-test results in a high F-value and a low p-value, we reject the null hypothesis and conclude that the model is statistically significant.

Further analysis involves examining individual coefficients’ t-tests for statistical significance. In particular, we determine which predictor variables—square footage or number of bedrooms—have significant relationships with the selling price. This involves reviewing p-values for each coefficient; variables with p-values below 0.05 are considered statistically significant predictors.

In addition to multiple regression, a simple regression model based solely on square footage is developed to compare performance. This model is specified as:

Price = α0 + α1(Square Footage) + ε

Comparing the simple and multiple regression models involves examining measures such as R-squared, adjusted R-squared, and residual plots. A higher R-squared and adjusted R-squared value in the multiple regression model suggest a better fit, assuming no multicollinearity issues.

The inclusion of the number of bedrooms in the model might be justified if this variable is statistically significant and improves predictive accuracy. If the coefficient for bedrooms is significant and the model’s fit improves notably, it indicates that bedrooms meaningfully influence house prices and should be included.

In conclusion, regression analysis provides insights into the factors influencing house prices. Conducting F-tests and examining individual p-values are essential steps in model validation. The comparison between models with and without bedrooms informs whether this variable adds predictive value and should be part of future modeling efforts.

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