1. Overview

This report uses the HomesForSale dataset to study how home prices are related to:

size (square feet),
number of bedrooms,
number of bathrooms,

and to test whether prices differ across the four states (CA, NY, NJ, PA).

Questions 1–4 use only California homes. Question 5 uses all homes.

## # A tibble: 4 × 4
##   State     n mean_price sd_price
##   <chr> <int>      <dbl>    <dbl>
## 1 CA       30       535.     269.
## 2 NJ       30       328.     158 
## 3 NY       30       365.     318.
## 4 PA       30       266.     137.

2. Question 1 – Effect of Size on Price (California)

We first model Price as a function of Size for California homes.

mod1 <- lm(Price ~ Size, data = home_CA)
summary(mod1)

## 
## Call:
## lm(formula = Price ~ Size, data = home_CA)
## 
## 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: 0.0004634

Key findings

The fitted model is [ = _0 + _1 .]
The slope for Size ($\beta_1$) is positive, about 0.34, so each additional square foot is associated with roughly 0.34 units higher price (≈ $340 if price is in thousands).
The p-value for Size is very small (≪ 0.05), so there is strong evidence that size has a significant positive effect on price.
The model explains around 36% of the variation in California home prices (R² ≈ 0.36).

3. Question 2 – Effect of Bedrooms on Price (California)

Now we regress Price on Beds (number of bedrooms) for California homes.

mod2 <- lm(Price ~ Beds, data = home_CA)
summary(mod2)

## 
## Call:
## lm(formula = Price ~ Beds, data = home_CA)
## 
## 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: 0.2548

Key findings

The slope for Beds is positive, but the p-value is greater than 0.05.
There is no statistically significant evidence that the number of bedrooms alone has a linear effect on price in California.
The R² value is small, so bedrooms by themselves explain only a small portion of the variability in home prices.

4. Question 3 – Effect of Bathrooms on Price (California)

Next we regress Price on Baths (number of bathrooms).

mod3 <- lm(Price ~ Baths, data = home_CA)
summary(mod3)

## 
## Call:
## lm(formula = Price ~ Baths, data = home_CA)
## 
## 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: 0.004092

Key findings

The slope for Baths is clearly positive.
The p-value for Baths is less than 0.05, so there is significant evidence that the number of bathrooms is positively related to home price.
Bathrooms alone explain a moderate amount of the variation in prices (R² is around one quarter of the total variation).

5. Question 4 – Joint Effect of Size, Bedrooms, and Bathrooms (California)

We now fit a multiple regression with all three predictors:

[ = _0 + _1 + _2 + _3 + .]

mod4 <- lm(Price ~ Size + Beds + Baths, data = home_CA)
summary(mod4)

## 
## Call:
## lm(formula = Price ~ Size + Beds + Baths, data = home_CA)
## 
## 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: 0.004353

##                Estimate  Std. Error    t value   Pr(>|t|)
## (Intercept) -41.5608472 210.3809020 -0.1975505 0.84493313
## Size          0.2811026   0.1189322  2.3635534 0.02585657
## Beds        -33.7035665  67.9255137 -0.4961842 0.62393327
## Baths        83.9844096  76.7529751  1.0942170 0.28389382

Key findings

In the multiple regression, the p-value for Size is less than 0.05, so size remains a significant predictor of price even after controlling for bedrooms and bathrooms.
The p-values for Beds and Baths are greater than 0.05, so there is no evidence that bedrooms or bathrooms have additional linear effects on price once size is included in the model.
The multiple R² is close to 0.40, so the three predictors together explain about 40% of the variation in California home prices. Size appears to be the most important predictor among the three.

6. Question 5 – Do Home Prices Differ Among the Four States?

For this question we use all homes and compare prices across states (CA, NY, NJ, PA) with a one-way ANOVA.

anova_mod <- aov(Price ~ State, data = home)
summary(anova_mod)

##              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 ' ' 1

##       CA       NJ       NY       PA 
## 535.3667 328.5333 365.3333 265.5667

Key findings

The ANOVA table shows a very small p-value for State (much less than 0.05).
This provides strong evidence that mean home prices are not the same across the four states.
The mean prices by state indicate that California has the highest average home price, Pennsylvania has the lowest, and New York and New Jersey are in between.
This suggests that the state in which a home is located has a significant impact on its price.

7. Conclusion

Using regression and ANOVA on the HomesForSale dataset, we find that:

For California homes, size is the strongest predictor of price.
Bathrooms have a positive effect on price when considered alone, but their additional effect is not significant once size is in the model.
Bedrooms do not show a clear independent linear effect on price.
Across all four states, there are clear and statistically significant differences in average home prices, with California homes being the most expensive on average.

This analysis highlights how both home characteristics (especially size) and location (state) play important roles in determining home prices.