Background
This is a real dataset of house prices sold in Seattle, WA, USA between August and December 2022.
Data Description
The dataset consists of a dataframe with 1665 observations with the following 6 variables:
1.beds = Number of bedrooms in property 2. region = based on zip code, north, central or south Seattle. 3. baths = Number of bathrooms in property. Note 0.5 corresponds to a half-bath which has a sink and toilet but no tub or shower 4. size = Total floor area of property in square feet 5. lot_size = Total area of the land where the property is located on in square feet 6. price = Price the property was sold for (US dollars) ##THIS IS THE RESPONSE VARIABLE
Source: Kaggle
Load the data
No changes needed
# reading the dataset in to RStudio
# Call the data table price_data
#https://drive.google.com/file/d/1LGxmER8s0PiGtrxoJA0fLCGU5IvCB4mh/view?usp=sharing
# Set up the actual file ID from the share link
file_id <- "1LGxmER8s0PiGtrxoJA0fLCGU5IvCB4mh"
data_url <- paste0("https://docs.google.com/uc?export=download&id=", file_id)
#Read the data into R
price_data <- read.csv(data_url, header = TRUE)
# Show the first few rows of data
head(price_data)## beds Region baths size lot_size price
## 1 1 Central 1 704 500 645000
## 2 3 Central 3 1524 513 850000
## 3 3 Central 3 1524 513 875000
## 4 3 North 3 1450 525 750000
## 5 1 Central 1 480 560 275000
## 6 2 North 1 800 560 690000Exploratory Data Analysis
# Compute univariate numeric summary statistics
summary(as.data.frame(price_data))## beds Region baths size
## Min. : 1.000 Length:1665 Min. :0.500 Min. : 376
## 1st Qu.: 2.000 Class :character 1st Qu.:1.500 1st Qu.: 1260
## Median : 3.000 Mode :character Median :2.000 Median : 1720
## Mean : 3.126 Mean :2.298 Mean : 1896
## 3rd Qu.: 4.000 3rd Qu.:3.000 3rd Qu.: 2360
## Max. :15.000 Max. :9.000 Max. :11010
## lot_size price
## Min. : 500 Min. : 159488
## 1st Qu.: 2734 1st Qu.: 680000
## Median : 5000 Median : 865000
## Mean : 9673 Mean :1010483
## 3rd Qu.: 7350 3rd Qu.:1175000
## Max. :400752 Max. :6250000# Univariate Charts--histograms for the quantitative variables
# use hist(price_data$VARNAME)
par(mfrow = c(3, 2))
hist(price_data$beds)
hist(price_data$baths)
hist(price_data$size)
hist(price_data$lot_size)
hist(price_data$price)
Compute the correlation coefficient for each quantitative predicting variables against the response
#Create the correlation matrix for the quantitative variables CHANGE THE DATA TABLE NAME
round(cor(price_data [,-2]),2)## beds baths size lot_size price
## beds 1.00 0.59 0.73 -0.13 0.46
## baths 0.59 1.00 0.62 -0.08 0.54
## size 0.73 0.62 1.00 -0.07 0.74
## lot_size -0.13 -0.08 -0.07 1.00 -0.09
## price 0.46 0.54 0.74 -0.09 1.00Describe the strength and direction of the top 3 predicting variables that have the strongest linear relationships with the response.
Size has a 0.74 correlation with price, a strong positive correlation Baths has a 0.54 correlation with price, a moderate positive correlation Beds has a 0.46 correlation with price, a moderate positive correlation
Create a boxplot of the qualitative predicting variable versus the response variable.
# Box plot CHANGE THE VARIABLES AND DATA TABLE
boxplot(price_data$price ~ price_data$Region)
Explain the relationship between the qualitative predicting variable versus the response variable.
Central has the highest price overall, followed by North and South. All three have similar bottom ranges but the high end is higher for Central, then North, with South having the lowest high prices.
Create boxplots of the qualitative predicting variable versus the quantitative PREDICTOR variables
# Box plots by vehicle type CHANGE THE VARIABLES AND DATA TABLE
par(mfrow = c(2, 2))
boxplot(price_data$beds ~ price_data$Region)
boxplot(price_data$baths ~ price_data$Region)
boxplot(price_data$size ~ price_data$Region)
boxplot(price_data$lot_size ~ price_data$Region)
Explain the relationship between the variables.
Baths - South has less Baths on average than Central or North Beds - All regions have the same average bed count, but South has a lower Q3 Size - Central has a slightly higher average size than North or South, which have similar graphs Lot Size - There are many outliers for all regions, but North and South have the highest means, with South having the highest median
Create scatterplots of the response variable against each quantitative predicting variable Describe the general trend of each plot.
## Explore the relationships between all predictor variables and the response
## CHANGE THE VARIABLES AND DATA TABLE AND THE XLAB AND YLAB
par(mfrow = c(2, 3))
plot(price_data$beds,price_data$price,xlab = "beds", ylab = "Price Emissions")
plot(price_data$baths,price_data$price,xlab = "baths", ylab = "Price Emissions")
plot(price_data$size,price_data$price,xlab = "size", ylab = "Price Emissions")
plot(price_data$lot_size,price_data$price,xlab = "lot_size", ylab = "Price Emissions")
Describe the general trend of each plot.
beds - generally trending up, but with wide variance Region - baths - trending very slightly up, wide variance size - trending up, moderately correlated lot_size - lots of prices at first, settling into a low trend after
Regression Model Fitting and Interpretation
Fit a multiple linear regression model called model1 using with the response and the top 2 predicting variables with the strongest relationship with the response variable
# Fit a linear regression model CHANGE THE VARIABLES AND DATA TABLE
model1 <- lm(price ~ size + baths + beds, data = price_data )
summary(model1)##
## Call:
## lm(formula = price ~ size + baths + beds, data = price_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1841254 -181876 -23339 143997 4399224
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 172751.59 26670.93 6.477 1.23e-10 ***
## size 496.11 15.25 32.534 < 2e-16 ***
## baths 105351.90 11820.29 8.913 < 2e-16 ***
## beds -110288.68 11481.24 -9.606 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 367700 on 1661 degrees of freedom
## Multiple R-squared: 0.5872, Adjusted R-squared: 0.5864
## F-statistic: 787.5 on 3 and 1661 DF, p-value: < 2.2e-16Using alpha = 0.05, which of the estimated coefficients are statistically significant?
All coefficients are significant
Is the overall regression significant? And why or why not?
Yes, the p-value is under 0.05
What is the estimated coefficient for the intercept? Interpret this coefficient in the context of the dataset.
172751.59 - when all other variables are zero, this is the price
What is the estimated coefficient for the strongest correlated predictor variable? Interpret this coefficient in the context of the dataset.
the estimated coefficient of size is 496.11 - this means that for each increase of 1 in size, the price increases by 496.11
Goodness of Fit
# Extract the standardized residuals
#NO CHANGES NEEDED
resids = rstandard(model1)
fits = model1$fitted.values
# Constant Variance Assumption
plot(fits, resids,
xlab="Fitted Values",
ylab="Residuals",
main="")
abline(0, 0, lty=2, lwd=2)
Fit a linear regression model called model2 with response variable and the qualitataive variable as the predicting variable
# CHANGE THE VARIABLES AND DATA TABLE
model2 <- lm (price ~ Region, price_data )
summary(model2)##
## Call:
## lm(formula = price ~ Region, data = price_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -976039 -305368 -122368 149942 5023961
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1226039 26408 46.426 < 2e-16 ***
## RegionNorth -193671 33439 -5.792 8.31e-09 ***
## RegionSouth -425981 35779 -11.906 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 548900 on 1662 degrees of freedom
## Multiple R-squared: 0.07961, Adjusted R-squared: 0.0785
## F-statistic: 71.87 on 2 and 1662 DF, p-value: < 2.2e-16What is the model for each of the values of the qualitataive variable?
Central - 1226039 North - 1032368 South - 800058
Goodness of Fit
# Extract the standardized residuals
#NO CHANGES NEEDED
resids2 = rstandard(model2)
fits2 = model2$fitted.values
# Constant Variance Assumption
plot(fits2, resids2,
xlab="Fitted Values",
ylab="Residuals",
main="")
abline(0, 0, lty=2, lwd=2)
Does the fitted values vs. standardized residuals plot indicate constant variance?
Variance seems to increase slightly
Fit a multiple linear regression model called model3 using the response variable and your choice of predictor variables.
# CHANGE THE VARIABLES AND DATA TABLE
model3 <- lm (price ~ Region + size + baths + beds, price_data)
summary(model3)##
## Call:
## lm(formula = price ~ Region + size + baths + beds, data = price_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1739043 -171506 -26067 139679 4354258
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 306457.22 29239.64 10.481 < 2e-16 ***
## RegionNorth -107417.43 21669.17 -4.957 7.88e-07 ***
## RegionSouth -286054.93 23461.50 -12.193 < 2e-16 ***
## size 470.85 14.79 31.842 < 2e-16 ***
## baths 92190.41 11368.33 8.109 9.79e-16 ***
## beds -84877.34 11186.60 -7.587 5.41e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 351700 on 1659 degrees of freedom
## Multiple R-squared: 0.6228, Adjusted R-squared: 0.6216
## F-statistic: 547.8 on 5 and 1659 DF, p-value: < 2.2e-16Using alpha = 0.05, which of the estimated coefficients arestatistically significant in model2?
all coefficients are significant
Interpret the estimated coefficient for RegionNorth in the context of the dataset.
When the Region is North and all other variables are zero, the price is the intercept plus -107417.43
Is the overall regression (model3) significant at an alpha-level of 0.05? Explain how you determined the answer.
yes, the p-value is < 2.2e-16
Goodness of Fit
# Extract the standardized residuals
#NO CHANGES NEEDED
resids3 = rstandard(model3)
fits3 = model3$fitted.values
# Constant Variance Assumption
plot(fits3, resids3,
xlab="Fitted Values",
ylab="Residuals",
main="")
abline(0, 0, lty=2, lwd=2)
Does the fitted values vs. standardized residuals plot indicate constant variance?
Mostly, but it increases for greater values
What are the Adjusted R^2 values for model 1 and model 3?
# Return the R^2 values for model 1 and model 3
#NO CHANGES NEEDED
paste("Model 1 adjusted R^2:", round(summary(model1)$adj.r.squared,2))## [1] "Model 1 adjusted R^2: 0.59"paste("Model 3 adjusted R^2:",round(summary(model3)$adj.r.squared,2))## [1] "Model 3 adjusted R^2: 0.62"How would you describe the performance of model 1 and model 3 based on the R^2 value.
Model 3 has a slightly better performance, with 62% of the variation in price explained by the model, while model 1 explains 59% of the variation
Prediction
Using model3, predict the mean response for 2 sets of characteristics:
CHANGE THE VARIABLES AND THE VALUES TO MATCH YOUR DATA SET
Region=“Central”, beds=3, baths=4, size=2000
Region=“South”, beds=1, baths=0.5, size=250,
#new data
# CHANGE THE VARIABLES AND DATA TABLE AND VALUES
newvals1 <- data.frame(Region="Central", beds=3, baths=4, size=2000)
newvals2 <- data.frame(Region="South", beds=1, baths=0.5, size=250)
# Confidence Interval for the response variable
predict(model3,newvals1,interval='confidence',level=.95)## fit lwr upr
## 1 1362288 1312779 1411798predict(model3,newvals2,interval='confidence',level=.95)## fit lwr upr
## 1 99332.86 53958.28 144707.4What is the predicted mean response for set 1?
1362288
What is the predicted mean emissions for set 2?
99332.86
Comment on whether the plot shows constant variance.
Yes, the variance is constant