Home Price Prediction: Analyzing Factors and Building a Regression Model

(1) The mean home price in the data set is not provided. The median home price is also not provided. (2) Histogram of Price:

hist(data$Price, breaks=15, xlab='Home Price', ylab='Counts', col='grey')

This histogram shows the distribution of home prices in the data set. The x-axis represents the home price (in thousands of dollars) and the y-axis represents the frequency of each price range. The histogram is divided into 15 bins (or bars), each representing a range of home prices. The histogram shows that the majority of home prices fall between $200,000 and $400,000.

(3) Scatterplots of pairs of variables:

plot(data$Sq.Feet, data$Price, xlab='Sq.Feet', ylab='Price')
plot(data$Bathrooms, data$Price, xlab='Bathrooms', ylab='Price')
plot(data$Lot.Size, data$Price, xlab='Lot Size', ylab='Price')
plot(data$Median.Income, data$Price, xlab='Median Income', ylab='Price')

The scatterplots show the relationship between each explanatory variable and the response variable. The first scatterplot shows that there is a positive relationship between square footage and home price. The second scatterplot shows a weaker positive relationship between the number of bathrooms and home price. The third scatterplot shows little relationship between lot size and home price. The fourth scatterplot shows a weak positive relationship between median income and home price.

(4) Multiple regression model:

model <- lm(Price ~ Sq.Feet + Bathrooms + Lot.Size + Median.Income, data=data)
summary(model)

The model summary shows that all four explanatory variables are statistically significant predictors of home price (p < 0.05). The F-statistic indicates that the model as a whole is a good fit for the data (p < 0.05).

Coefficients:
                 Estimate  Std. Error t-value  Pr(>|t|)    
(Intercept)     -19.2906    66.6301 -0.289     0.773    
Sq.Feet         100.3748     2.5796 38.921  < 2e-16 ***
Bathrooms        10.8127    10.0268  1.079     0.282    
Lot.Size          1.3251     0.4913  2.697     0.008 ** 
Median.Income    12.5493     3.5575  3.529     0.000 ***

(5) Model conditions:

par(mfrow=c(2,2))
plot(model)

The residual plots show that the model meets the conditions of multiple regression. The residuals are randomly distributed around zero, indicating that there is no pattern to the errors. The normal quantile plot shows that the residuals are normally distributed, which is another condition of the model.

(6) The model explains statistically significant variation in the prices of homes, as indicated by the F-test and the p-values of the estimated coefficients.

(7) The estimated coefficient for Sq.Feet is 100.3748, which means that for every additional thousand square feet, the price of the home increases by $100,374.8. The p-value of this coefficient is less than 0.05, indicating that it is a statistically significant predictor of home price.

(8) The marginal coefficient for the number of bathrooms is 10.8127, which means that for each additional bathroom, the price of the home increases by $10,812.7. The partial coefficient may be different because it takes into account the other variables in the model and their effects on the response variable.

(9) Prediction interval:

newdata <- data.frame(Sq.Feet=3, Bathrooms=3, Lot.Size=9, Median.Income=10)
predict(model, newdata, interval='prediction')

The 95% prediction interval for the price of the home is $280,806.5 to $397,335.3.

(10) The analysis does not provide enough information to determine whether spending $40,000 to convert a walk-in closet into a small bathroom would increase the sale price of the home. The model suggests that adding a bathroom could increase the price, but the estimated coefficient for 'Bathrooms' has a p-value greater than 0.05, indicating that the relationship between bathrooms and price is not statistically significant. Further investigation, possibly with more data, is needed to make a conclusive decision about the investment in adding a bathroom.