# Load the data
home <- read.csv("https://www.lock5stat.com/datasets3e/HomesForSale.csv")
# Filter for California
ca_data <- subset(home, State == "CA")Use the data only for California. How much does the size of a home influence its price?
Use the data only for California. How does the number of bedrooms of a home influence its price?
Use the data only for California. How does the number of bathrooms of a home influence its price?
Use the data only for California. How do the size, the number of bedrooms, and the number of bathrooms of a home jointly influence its price?
Are there significant differences in home prices among the four states (CA, NY, NJ, PA)? This will help you determine if the state in which a home is located has a significant impact on its price. All data should be used.
Understanding the factors that influence home prices is critical for buyers, sellers, and real estate professionals alike. Home prices are shaped by a variety of characteristics, including size, number of bedrooms, number of bathrooms, and location. This study aims to explore these factors using data on homes for sale in four states: California, New York, New Jersey, and Pennsylvania. By analyzing the data, we can determine which characteristics most strongly influence home prices and whether significant price differences exist across states.
To achieve these objectives, statistical methods such as regression analysis and ANOVA are employed. Regression analysis is used to evaluate the relationship between home prices and specific characteristics, both individually and collectively, while ANOVA is used to test for significant differences in prices among states. This study focuses on answering five key questions, including how size, bedrooms, and bathrooms influence home prices in California, and whether location significantly impacts prices across the four states. The findings of this study provide valuable insights into the real estate market, helping stakeholders better understand price dynamics and make informed decisions.
Use the data only for California. How much does the size of a home influence its price?
# Filter for California
ca_data <- subset(home, State == "CA")
# Regression: Size vs Price
size_model <- lm(Price ~ Size, data = ca_data)
summary(size_model)##
## Call:
## lm(formula = Price ~ Size, data = ca_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -462.55 -139.69 39.24 147.65 352.21
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -56.81675 154.68102 -0.367 0.716145
## Size 0.33919 0.08558 3.963 0.000463 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 219.3 on 28 degrees of freedom
## Multiple R-squared: 0.3594, Adjusted R-squared: 0.3365
## F-statistic: 15.71 on 1 and 28 DF, p-value: 0.0004634The regression analysis reveals a significant and positive relationship between the size of a home and its price in California. The slope of the model indicates that for every additional square foot of size, the price increases by approximately $339, demonstrating that larger homes tend to be more expensive. The p-value for the slope (\(<0.001\)) is highly significant, allowing us to confidently reject the null hypothesis that size has no impact on price. The \(R^2\) value of 35.94% indicates that the size of the home explains a substantial proportion of the variability in prices. While the intercept of \(-56.82\) has limited practical meaning in this context, the residual standard error of $219,300 shows the average deviation of actual prices from the modelβs predictions. Overall, the size of a home is a strong and meaningful predictor of its price in California.
Use the data only for California. How does the number of bedrooms of a home influence its price?
# Filter for California
ca_data <- subset(home, State == "CA")
# Regression: Bedrooms vs Price
beds_model <- lm(Price ~ Beds, data = ca_data)
# Summary of the regression model
summary(beds_model)##
## Call:
## lm(formula = Price ~ Beds, data = ca_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -413.83 -236.62 29.94 197.69 570.94
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 269.76 233.62 1.155 0.258
## Beds 84.77 72.91 1.163 0.255
##
## Residual standard error: 267.6 on 28 degrees of freedom
## Multiple R-squared: 0.04605, Adjusted R-squared: 0.01198
## F-statistic: 1.352 on 1 and 28 DF, p-value: 0.2548The regression analysis likely shows that the number of bedrooms has a weak and statistically insignificant relationship with home prices in California. This is indicated by a high p-value ( π > 0.05 p>0.05), suggesting that adding more bedrooms does not reliably increase the price of a home. The π 2 R 2 value (low, e.g., 4.6%) also supports this conclusion, showing that bedrooms explain only a small fraction of the variability in prices. Thus, the number of bedrooms is not a meaningful predictor of home prices compared to other factors like size or location.
How does the number of bathrooms influence home prices in California?
# Regression: Bathrooms vs Price
baths_model <- lm(Price ~ Baths, data = ca_data)
# Summary of the regression model
summary(baths_model)##
## Call:
## lm(formula = Price ~ Baths, data = ca_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -374.93 -181.56 -2.74 152.31 614.81
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 90.71 148.57 0.611 0.54641
## Baths 194.74 62.28 3.127 0.00409 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 235.8 on 28 degrees of freedom
## Multiple R-squared: 0.2588, Adjusted R-squared: 0.2324
## F-statistic: 9.779 on 1 and 28 DF, p-value: 0.004092Interpretation
The regression analysis shows that the number of bathrooms has a statistically significant and positive influence on home prices in California. The slope of the regression model indicates that for every additional bathroom, the price increases by approximately $194,740. The p-value for the slope ( π = 0.004 p=0.004) confirms the statistical significance of this relationship, allowing us to reject the null hypothesis that the number of bathrooms has no impact on price. The model explains 25.9% of the variability in home prices ( π 2 = 0.259 R 2 =0.259), suggesting that the number of bathrooms is a moderately strong predictor of home prices. However, other factors (e.g., size, location) may also contribute to price variations. Overall, the number of bathrooms is an important factor in determining the price of a home in California.
How do the size, the number of bedrooms, and the number of bathrooms of a home jointly influence its price?
# Multiple Regression: Size, Beds, Baths vs Price
multi_model <- lm(Price ~ Size + Beds + Baths, data = ca_data)
# Summary of the regression model
summary(multi_model)##
## Call:
## lm(formula = Price ~ Size + Beds + Baths, data = ca_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -415.47 -130.32 19.64 154.79 384.94
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -41.5608 210.3809 -0.198 0.8449
## Size 0.2811 0.1189 2.364 0.0259 *
## Beds -33.7036 67.9255 -0.496 0.6239
## Baths 83.9844 76.7530 1.094 0.2839
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 221.8 on 26 degrees of freedom
## Multiple R-squared: 0.3912, Adjusted R-squared: 0.3209
## F-statistic: 5.568 on 3 and 26 DF, p-value: 0.004353The multiple regression model reveals how the size, number of bedrooms, and number of bathrooms jointly influence the price of homes in California. The model shows that size has a statistically significant and positive influence on price ( π < 0.05 p<0.05), meaning that larger homes tend to be more expensive, even when controlling for the number of bedrooms and bathrooms. However, the coefficients for number of bedrooms and number of bathrooms are not statistically significant ( π > 0.05 p>0.05), suggesting that these variables do not reliably predict home prices when size is included in the model. The π 2 = 0.391 R 2 =0.391 indicates that 39.1% of the variability in home prices is explained by these three predictors. Overall, the size of a home emerges as the most important factor, while the number of bedrooms and bathrooms contribute less to explaining price variations.
Are there significant differences in home prices among the four states (CA, NY, NJ, PA)?
# ANOVA: Price differences among states
anova_model <- aov(Price ~ State, data = home)
# Summary of the ANOVA model
summary(anova_model)## Df Sum Sq Mean Sq F value Pr(>F)
## State 3 1198169 399390 7.355 0.000148 ***
## Residuals 116 6299266 54304
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The ANOVA analysis evaluates whether the state in which a home is located significantly affects its price. The results yield a statistically significant π p-value (e.g., π < 0.001 p<0.001), indicating that there are meaningful differences in average home prices among the four states (CA, NY, NJ, PA). This means that the state where a home is located has a significant impact on its price. To further explore which states differ from each other, post-hoc tests (e.g., Turkeyβs HSD) could be conducted. Overall, the location is a key factor influencing home prices.
This experiment provides valuable insights into the factors that influence home prices and highlights significant differences across states. The analysis reveals that the size of a home is the most significant and consistent predictor of price in California, with larger homes commanding higher prices. The number of bedrooms was found to have little to no significant influence on price, suggesting that buyers may prioritize overall square footage over the number of rooms. The number of bathrooms, while moderately correlated with price, was not a strong predictor when considered alongside size and bedrooms in a multiple regression model.
Additionally, an ANOVA analysis revealed significant differences in home prices across the four states (CA, NY, NJ, PA). This suggests that the state in which a home is located plays a major role in determining its price, likely due to regional differences in housing markets, cost of living, and demand.
In conclusion, the study emphasizes the importance of location and size in determining home prices, while features like the number of bedrooms and bathrooms have a lesser impact. This knowledge can guide buyers, sellers, and real estate professionals in making informed decisions about property valuation and investments. Future studies could expand on these findings by incorporating additional factors such as neighborhood characteristics, proximity to amenities, and economic conditions.