Beteta - MAT 303 Project One

MAT 303 Project One Summary Report Diego Beteta diego.beteta@snhu.edu Southern New Hampshire University 1

1. Introduction In this project, we are analyzing a dataset of historical housing sales in Seattle to understand how different attributes of a house, such as its size, age, and location, affect its selling price. The primary goal is to create regression models that can accurately predict a house's selling price based on these attributes. This is crucial for our real estate company to set appropriate listing prices for homes. We will perform various statistical analyses, including multiple regression, to assess the combined effect of several factors and explore interactions between qualitative variables (like the presence of a backyard or renovation status) to see how they jointly influence price. Additionally, we might use quadratic regression to examine non-linear relationships, like how increases in square footage might lead to a disproportionate increase in price. The results of these analyses will help our company make informed decisions about pricing homes, ensuring competitiveness and profitability in the real estate market. 2. Data Preparation Several important variables stand out for analyzing and predicting house prices in this dataset, which consists of 2,692 rows and 23 columns. In this project, we'll focus on a subset of key variables from the dataset, each representing different aspects of a home and its surroundings, which are crucial for predicting its sale price: 1. price : This is the home's sale price, the primary variable we aim to predict. 2. bedrooms : The number of bedrooms in the home, indicating its accommodation capacity. 3. bathrooms : The number of bathrooms contributing to the home's convenience and comfort. 4. sqft_living : The size of the living area in square feet, a direct measure of the home's size. 5. sqft_above : The size of the upper level in square feet, giving an idea of the additional living space apart from the main level. 6. sqft_lot : The size of the lot on which the house sits, in square feet, indicating the amount of outdoor space. 7. age : The age of the home, which can impact its style, condition, and appeal. 8. grade : A measure of craftsmanship and the quality of materials used in the home, reflecting its overall build quality. 9. appliance_age : The average age of all appliances in the home, indicating the need for updates or replacements. 10. crime : The crime rate per 100,000 people in the area, a factor that can influence the desirability of a neighborhood. 11. backyard : Indicates whether the home has a backyard (1) or not (0), an important feature for many buyers. 12. school_rating : The average rating of schools in the area is often a significant consideration for families. 13. view : Describes whether the home backs out to a lake (2), trees (1), or a road (0), affecting its aesthetic and potentially its value. Analyzing how these variables interact and influence the home's sale price will be central to developing effective predictive models in this project. 2

The scatterplot of home prices versus the age of the homes does not display a clear or strong trend. Unlike the previous plot with the living area, the relationship between age and price appears more scattered and less predictable. Homes of various ages are spread across a wide range of prices, suggesting that age alone may not be a strong predictor of a home's price. While there are clusters of newer homes (lower age) at higher price points, older homes (higher age) are distributed across a broad spectrum of prices. This indicates that factors other than age, location, condition, or renovations play a significant role in determining the price of a home. Overall, the plot suggests a more complex relationship between a home's age and selling price, underscoring the need to consider additional variables for accurate price prediction. The correlation coefficient between the price of homes and their age is approximately -0.075. This value indicates a very weak inverse relationship, suggesting that as the age of a home increases, there is a slight tendency for its price to decrease. However, the correlation is so weak that it implies age is not a significant predictor of home prices in this dataset. Other factors beyond the age of the home play a more crucial role in determining its price, and the impact of age on price is relatively negligible. A correlation coefficient closer to -1 or 1 would signify a stronger relationship, but a value near zero, as seen here, points to a need for a more meaningful correlation. Reporting Results The general form and the prediction equation of the multiple regression model using price as the response variable and living area, upper-level area, age of the home, number of bathrooms, and view as predictor variables are as follows: 4

 General Form: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + β 5 x 5  Prediction Equation: ^ y = ^ β 0 + ^ β 1 sqf t living + ^ β 2 sqf t above + ^ β 3 age + ^ β 4 bathrooms + ^ β 5 view 1 + ^ β 6 view 2  Multiple Regression Model: ^ price = 7709 + 129.3 sqftliving + 19.51 sqft above + 1451 age + 43970 bathrooms + 167500 vie w 1 + 249000 vie  R-squared (0.6029 or 60.29%) : This value indicates that approximately 60.29% of the variability in home prices is explained by the model's predictors (living area, upper-level area, age of the home, number of bathrooms, and view). In simple terms, this measures how well the model fits the data. An R-squared of 60.29% is relatively strong, suggesting that the model captures a significant portion of the price variation. However, it also means that other factors that account for around 40% of the price variability should be included in the model.  Adjusted R-squared (0.602 or 60.2%) : The Adjusted R-squared adjusts the R-squared value based on the number of predictors in the model and the number of observations. It's a more accurate measure of model fit, especially when you have multiple predictors, as it penalizes the model for including variables that do not improve its predictive capability. An Adjusted R- squared of 60.2% is very close to the R-squared value, indicating that the model's predictors are relevant and explain the variation in home prices. These statistics suggest that the model is reasonably good at explaining the variability in home prices. However, a portion of the variability is still unexplained by the model, which could be due to factors not included in the analysis.  Living Area (Beta estimate: 129.3) : This coefficient means that for every additional square foot in a home's living area, the home's price is expected to increase by approximately $129.3, holding all other factors constant. This positive coefficient suggests that larger living areas are valued higher in the housing market. It's a straightforward interpretation, reflecting the general preference for more spacious homes.  Lake View (Beta estimate: 249,000) : This coefficient indicates that homes with a lake view (classified as 'view2') are, on average, $249,000 more expensive than homes without this feature, all else being equal. This substantial increase in price highlights the premium value placed on homes with lake views. It reflects the desirability of such a feature in the housing market, possibly due to the aesthetic appeal, tranquility, and exclusivity of lakefront properties. These beta estimates quantify the market's value on these specific home features, showing the importance of living areas and scenic views in determining home prices. 5

The Normal Q-Q plot shown here assesses whether the residuals from the multiple regression model follow a normal distribution, which is one of the key assumptions of linear regression. In a Q-Q plot, the points should fall approximately along the straight line if the residuals are normally distributed. While many of the points align with the line in this plot, some deviations exist, particularly at the ends. This pattern suggests that the residuals may not follow a perfectly normal distribution. Such deviations could be due to outliers or skewness in the data. However, it's important to note that minor deviations from normality are often acceptable, especially in large datasets, and do not necessarily invalidate the model. The overall interpretation of this plot suggests that while the assumption of normality is not perfectly met, the model may still provide useful predictions. Evaluating Significance of Model The overall significance of the model can be determined by carrying out an F-test, which essentially tests whether the model provides a better fit to the data than a model with no predictors.  Null Hypothesis (H0) : The null hypothesis for the F-test states that none of the predictors in the model affect the response variable, i.e., all the coefficients of the predictor variables are zero except the intercept.  Alternative Hypothesis (H1) : The alternative hypothesis states that at least one predictor has a non-zero effect on the response variable.  The P-value from your model's F-statistic is reported as < 2.2e-16, which is essentially zero. 7

Given such a small P-value, well below the 5% level of significance, we reject the null hypothesis. This means we have strong evidence that at least one of the predictors in the model significantly affects house prices. The F-test confirms that your model is statistically significant. This suggests that the variables you've included in the model (living area, upper-level area, age of the home, number of bathrooms, and view) collectively have a significant impact on predicting the price of a home, much more than a model that doesn't include these predictors. Individual beta tests are conducted for each predictor to determine which terms in the model are significant at a 5% level of significance. These tests assess whether each predictor has a statistically significant effect on the response variable.  Null Hypothesis (H0) : The coefficient of the predictor is zero, meaning the predictor does not affect the response variable.  Alternative Hypothesis (H1) : The coefficient of the predictor is not zero, implying the predictor does affect the response variable. The following is the evaluation of each predictor based on its P-value and the conclusion of the tests: 1. sqft_living : o P-value: < 2e-16 (essentially zero) o Reject H0. The living area is a significant predictor of the home price at a 5% level of significance. 2. sqft_above : o P-value: 0.00894 o Reject H0. The upper-level area is a significant predictor, as the P-value is less than 0.05. 3. age : o P-value: < 2e-16 (essentially zero) o Reject H0. The age of the home significantly affects its price. 4. bathrooms : o P-value: 9.13e-13 (very close to zero) o Reject H0. The number of bathrooms is a significant predictor of the home price. 5. view1 : o P-value: < 2e-16 (essentially zero) o Reject H0. Homes with a view of trees (view1) are significantly different in price compared to homes without this view. 6. view2 : o P-value: < 2e-16 (essentially zero) o Reject H0. Homes with a lake view (view2) have a significantly different price. All these predictors are significant at the 5% level. This means each of these variables (living area, upper- level area, age, number of bathrooms, and views of trees and lakes) has a statistically significant impact on the price of a home. 8

The prediction interval is wider than the confidence interval because it accounts for more sources of uncertainty. While the confidence interval provides a range for the average price of homes with specific characteristics, the prediction interval covers the variability of individual home prices around this average. The confidence interval tells us where the average price for similar homes will likely fall. In contrast, the prediction interval tells us where the price of a specific home might fall, considering additional factors unique to each home. This makes the prediction interval broader, including the uncertainty in estimating the average price (like the confidence interval) and the natural variability in individual home prices. 4. Model #2 - Complete Second Order Regression Model with Quantitative Variables Correlation Analysis The scatterplot of home prices versus average school ratings in the area reveals some interesting trends. While there isn't a distinct linear relationship, there seems to be a general tendency for home prices to increase with higher school ratings. This suggests that homes in areas with better-rated schools might be more expensive, reflecting the value placed on educational quality in real estate. However, the relationship does not appear strictly linear or strongly pronounced. There's considerable variability in prices at each level of school rating, indicating other factors also significantly influence home prices. Given the scatter and spread of data points, a second-order model could be explored to see if it captures the relationship more effectively than a linear model. A quadratic model might better represent the data if the relationship between price and school rating is non-linear, such as an increasing price increase rate with higher school ratings. 10

The scatterplot of home prices versus the crime rate per 100,000 people displays a pattern that suggests an inverse relationship; as the crime rate increases, there is a tendency for home prices to decrease. This aligns with common expectations that higher crime rates can negatively affect property values. The data points show quite a bit of spread, and the relationship is not tightly clustered around a line, indicating variability in how crime rates impact home prices or that other factors are also at play. A second-order model might be considered if the trend in the data suggests a non-linear pattern – for example, if the decrease in home prices accelerates as the crime rate goes up. However, the plot doesn't strongly suggest a pronounced curvature that would typically motivate a quadratic model. Reporting Results The general form and the prediction equation of a complete second-order model for price using the average school rating in the area and crime rate per 100,000 people as predictors are as follows:  General Form: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 2 + β 4 x 2 2 + β 5 x 1 x 2  Prediction Equation: ^ y = β 0 + β 1 school rating + β 2 crime + β 3 schoolrating 2 + β 4 crime 2 + β 5 ( school rating ∗ crime ) 11

plot, the second-order model seems reasonably well-specified, but it's important to investigate further with additional diagnostics. In a Q-Q plot, the points should fall approximately along the straight line if the residuals are normally distributed. In this plot, while many points align closely with the line, there are some deviations, particularly at the ends. This pattern suggests that the residuals may not follow a perfect normal distribution, indicating potential issues with normality. These deviations could be due to outliers or non-normality in the data. Minor deviations from normality are often not critical in larger datasets and may not significantly impact the model's performance. However, as suggested by the ends of this plot, substantial deviations could indicate that the model's assumptions are not fully met, and the predictions could be affected, especially for extreme values. It may be worth exploring data transformations or alternative modeling approaches to address these potential issues. Evaluating Significance of Model To determine if our second-order model is significant at a 5% level of significance, we can look at the results of the overall F-test:  Null Hypothesis (H0) : The null hypothesis for the F-test states that all the predictors (average school rating, crime rate, their squared terms, and the interaction term) have no effect on the response variable, which in this case is the house price. Essentially, it posits that the model with 13

these predictors is no better at explaining the variation in house prices than a model without them.  Alternative Hypothesis (H1) : The alternative hypothesis is that at least one of the predictors has a significant effect on the house price.  The p-value from the F-statistic is reported to be < 2.2e-16, which is essentially zero. Given that the p-value is significantly less than the 5% level of significance (0.05), we reject the null hypothesis. This means that we have strong evidence to conclude that at least one of the predictors in the model significantly affects house prices. The F-test confirms that the model, including the average school rating, crime rate, their squared terms, and the interaction term, is statistically significant. This suggests that the model with these predictors provides a better fit to the data than a model with no predictors and that these variables collectively have a significant impact on predicting the price of a home. Also, to determine which terms in our model are significant at a 5% level of significance, we performed individual beta tests for each predictor. The process involves examining the p-values associated with each term's coefficient:  Null Hypothesis (H0) : The coefficient of each term (linear and squared terms of school rating and crime rate, and the interaction term between them) is zero, suggesting that the term has no effect on house prices.  Alternative Hypothesis (H1) : The coefficient of each term is not zero, indicating that the term does have an effect on house prices. 1. School Rating (Linear Term) : o P-value: 0.000406 o Reject H0. The linear term of school rating is significant at a 5% level. 2. Crime Rate (Linear Term) : o P-value: 1.90e-09 o Reject H0. The linear term of crime rate is significant at a 5% level. 3. School Rating Squared : o P-value: < 2e-16 o Reject H0. The squared term of school rating is significant at a 5% level. 4. Crime Rate Squared : o P-value: < 2e-16 o Reject H0. The squared term of crime rate is significant at a 5% level. 5. School Rating and Crime Interaction : o P-value: 0.281513 o Fail to reject H0. The interaction term is not significant at a 5% level. The linear and squared terms for both school rating and crime rate are significant predictors of house price. However, the interaction term between school rating and crime rate is not significant, suggesting that the combined effect of these two predictors does not significantly affect the house price at the 5% level of significance. 14

The general form and the prediction equation of a first-order model for price using the average school rating in the area and crime rate per 100,000 people as predictors including the interaction term between average school rating and crime rate are as follows:  General Form: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2  Prediction Equation : ^ y = ^ β 0 + ^ β 1 school rating + ^ β 2 crime + ^ β 3 schoolrating ∗ crime  First Order Regression Model: ^ y =− 410233.37 + 155559.97 schoolrating + 2230.07 crime − 564.85 school rating ∗ crime Evaluating Significance of Model 1. To determine if our first-order regression model is significant at a 5% level of significance, we looked at the results of the overall F-test:  Null Hypothesis (H0) : The null hypothesis for the F-test is that all the coefficients of the predictors in the model (average school rating, crime rate, and their interaction) are zero. This implies that none of these variables significantly predict the house price.  Alternative Hypothesis (H1) : The alternative hypothesis is that at least one of the predictors has a non-zero coefficient, significantly predicting the house price.  The p-value from the model's F-statistic is reported as < 2.2e-16, which is essentially zero. Given such a small p-value, far below the 5% level of significance, we reject the null hypothesis. This indicates that we have strong evidence that at least one of the predictors in the model significantly affects house prices. The F-test confirms that the model, including average school rating, crime rate, and their interaction, is statistically significant. This means that the model with these predictors is better for predicting house prices than a model with no predictors. 2. We conducted individual beta tests for each coefficient in our first-order regression model to determine significance at a 5% level. The general null and alternative hypotheses for these tests are:  Null Hypothesis (H0) for all coefficients: The coefficient of the predictor (school rating, crime rate, or their interaction) is zero, suggesting that the predictor has no effect on house prices.  Alternative Hypothesis (H1) for all coefficients: The coefficient of the predictor is not zero, indicating that the predictor does affect house prices. 1. School Rating : o P-value: < 2e-16 o Reject H0. The school rating is a significant predictor. 16

2. Crime Rate : o P-value: < 2e-16 o Reject H0. The crime rate is a significant predictor. 3. School Rating and Crime Interaction : o P-value: < 2e-16 o Reject H0. The interaction term is a significant predictor. All coefficients in the model are significant at the 5% level. Each of these terms individually contributes to the model predicting house prices. Model Comparison When comparing two models in statistics, a "reduced model" and a "complete model" refer to two versions of a model that differ in complexity. The reduced model is the simpler of the two, containing fewer variables or terms. It represents a more fundamental relationship between the variables being studied. On the other hand, the complete model is more complex, including additional variables or terms (like interaction or squared terms) that the reduced model needs to be revised. The purpose of comparing these models is often to determine whether the additional complexity of the complete model significantly improves its predictive power or understanding of the underlying relationships compared to the simpler reduced model. The general form and prediction equation of the model that is the reduced model in this comparison are as follows:  General Form : y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2  Prediction Equation: ^ y = ^ β 0 + ^ β 1 school rating + ^ β 2 crime + ^ β 3 schoolrating ∗ crime The general form and prediction equation of the model that is the complete model in this comparison are as follows:  General Form: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 2 + β 4 x 2 2 + β 5 x 1 x 2  Prediction Equation: ^ y = β 0 + β 1 school rating + β 2 crime + β 3 schoolrating 2 + β 4 crime 2 + β 5 ( school rating ∗ crime ) Based on the ANOVA results, we performed the nested model F-test to evaluate if the quadratic terms are needed in the model. This test compares the complete model (with quadratic terms) against the reduced model (without quadratic terms). The null hypothesis states that the quadratic terms in the complete model do not significantly improve the model's fit. This implies that the reduced model is sufficient for explaining the data. It suggests that the coefficients of the quadratic terms in the complete model are effectively zero. 17