91Ó°ÊÓ

Many variables have an impact on determining the price of a house. A few of these are Size of the house (square feet), Lotsize, and number of Bathrooms. Information for a random sample of homes for sale in the Statesboro, Georgia, area was obtained from the Internet. Regression output modeling the Asking Price with Square Footage and number of Bathrooms gave the following result: Dependent Variable is Asking Price \(s=67013 \quad R-S q=71.1 \% \quad R-S q(a d j)=64.6 \%\) \(\begin{array}{lrrcc}\text { Predictor } & \text { Coeff } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & -152037 & 85619 & -1.78 & 0.110 \\ \text { Baths } & 9530 & 40826 & 0.23 & 0.821 \\\ \text { Sq ft } & 139.87 & 46.67 & 3.00 & 0.015\end{array}\) Analysis of Variance \(\begin{array}{lllll}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F-Ratio P-Value } \\ \text { Regression } & 2 & 99303550067 & 49651775033 & 11.06 & 0.004 \\ \text { Residual } & 9 & 40416679100 & 4490742122 & & \\ \text { Total } & 11 & 1.39720 \mathrm{E}+11 & & & & \end{array}\) a) Write the regression equation. b) How much of the variation in home asking prices is accounted for by the model? c) Explain in context what the coefficient of Square Footage means. d) The owner of a construction firm, upon seeing this model, objects because the model says that the number of bathrooms has no effect on the price of the home. He says that when he adds another bathroom, it increases the value. Is it true that the number of bathrooms is unrelated to house price? (Hint: Do you think bigger houses have more bathrooms?)

Step by step solution

01

Write the Regression Equation

The regression equation can be formed by using the coefficients provided for the intercept, square footage, and number of bathrooms. The equation is as follows:\[\text{Asking Price} = -152037 + 9530 \times \text{Baths} + 139.87 \times \text{Sq Ft}\]

02

Determine Variation Accounted for by the Model

The variation in the home asking prices accounted for by the model is provided by the R-Squared value, which is 71.1%. This means that 71.1% of the variation in asking prices is explained by the model that uses the number of bathrooms and square footage as predictors.

03

Interpret the Coefficient of Square Footage

The coefficient for square footage is 139.87. This indicates that for each additional square foot, the asking price of the house is expected to increase by approximately $139.87, assuming all other factors remain constant.

04

Analyze the Impact of Number of Bathrooms

The coefficient for bathrooms is 9530, but its high p-value (0.821) suggests that it is not statistically significant in predicting the asking price of a house in this model. However, this does not necessarily mean bathrooms have no effect on house price universally. It could indicate that other variables (e.g., size of the house) might already account for such effects, or there could be a correlation where larger houses naturally have more bathrooms, reducing its separate explanatory power.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficient Interpretation

When we talk about coefficient interpretation in a regression analysis, we are essentially explaining how much the dependent variable is expected to change with a one-unit change in an independent variable, while keeping all other variables constant.
The regression equation for our example is given as:\[\text{Asking Price} = -152037 + 9530 \times \text{Baths} + 139.87 \times \text{Sq Ft}\]
Here, each coefficient helps us understand the relationship between the predictor and the asking price.
- **Square Footage (Sq Ft):** The coefficient for square footage is 139.87, which means for every additional square foot, the asking price is expected to increase by approximately \(139.87, assuming the number of bathrooms stays the same.
- **Bathrooms (Baths):** The coefficient here is 9530, implying that each additional bathroom might increase the price by around \)9530. However, statistical significance needs to be considered to see if this effect is reliable.

R-Squared Value

R-squared ( R^2 ) is a key concept in regression analysis as it provides a measure of how well the independent variables explain the variability in the dependent variable. In our example concerning house pricing, R^2 is 71.1%. This means that 71.1% of the variance in asking prices is explained by the model, which includes square footage and number of bathrooms as predictors.
An R-squared value closer to 100% indicates a model that explains a high proportion of variability, while a lower value would suggest that the model doesn't capture as much of what's influencing the dependent variable.
However, it's crucial to understand that a high R-squared does not imply causation or that the model is the best predictor for future values. It also cannot capture some nuances, such as multicollinearity or omitted variable bias, which could affect the regression outcomes.

Statistical Significance

Statistical significance plays a vital role in determining whether the relationships identified in our regression model are likely due to chance or reflect a true relationship. This is typically assessed using the p-value.
In our regression model, the p-value for square footage is 0.015, which is less than the common threshold of 0.05, indicating that square footage significantly predicts asking price.
- **Statistical Significance Tips:**
- If the p-value is less than the chosen significance level (commonly 0.05), the result is said to be statistically significant.
- Statistically significant predictors offer confidence that the observed relationship is present in the larger population.
On the other hand, the p-value for the number of bathrooms is 0.821, which is far above 0.05, suggesting that this variable does not have a statistically significant effect on asking price within this model. In practice, this means that for this dataset, adding a bathroom does not independently affect the price, but it could interact with other factors like overall house size.

Multiple Regression Model

A multiple regression model allows us to assess the impact of more than one independent variable on a dependent variable. It expands simple linear regression by adding other predictors to better understand complex relationships and improve predictions.
In our real estate example, both the size of the house (square footage) and the number of bathrooms are included as predictors.
The resulting regression equation is:\[\text{Asking Price} = -152037 + 9530 \times \text{Baths} + 139.87 \times \text{Sq Ft}\]
Through this model, we can evaluate how each variable uniquely contributes to explaining variations in house prices.
- **Why Multiple Regression?**

• It captures the combined effect of multiple variables, offering a more comprehensive view.
• It helps to control for potential confounding variables by including them in the model.
• This method can improve prediction accuracy through the integration of diverse data points.

However, being thorough with variable inclusion is crucial, as irrelevant variables can decrease model efficiency and potentially lead to misleading conclusions.