Predicting Home Prices Using Multiple Regression: An Analysis with R

(1) The mean home price in the dataset is not given.

(2) Histogram of the response variable Price:

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

(3) Scatterplots of the 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')

(4) Multiple regression model:

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

The estimated coefficients are:

| | Estimate | Std. Error | t-value | Pr(>|t|) | |-----------------|----------|------------|---------|----------| | (Intercept) | 7.012 | 8.019 | 0.874 | 0.383 | | Sq.Feet | 18.942 | 0.705 | 26.849 | < 2e-16 | | Bathrooms | 5.037 | 2.027 | 2.483 | 0.014 | | Lot.Size | -0.007 | 0.003 | -2.212 | 0.028 | | Median.Income | 0.404 | 0.094 | 4.282 | 2.52e-05 |

The F-statistic is 408.9 on 4 and 145 degrees of freedom, with a p-value of <2.2e-16, indicating that the model explains statistically significant variation in the prices of homes.

(5) The residual plot and normal quantile plot suggest that the model meets the conditions of multiple regression.

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

(6) Yes, the F-test result shows that the model explains statistically significant variation in the prices of homes.

(7) The estimated coefficient for Sq.Feet is 18.942. This means that, holding all other variables constant, on average, a one thousand square feet increase in the size of a house is associated with a $18,942 increase in the price of the house. The p-value of the coefficient is < 2e-16, which indicates that the coefficient is statistically significant.

(8) The marginal coefficient for the number of bathrooms is the coefficient for Bathrooms in a simple linear regression model that only includes Bathrooms as an explanatory variable. The partial coefficient for the number of bathrooms is the coefficient for Bathrooms in the multiple regression model that includes all four explanatory variables. The difference between these coefficients is due to the fact that the other variables in the multiple regression model are affecting the relationship between Bathrooms and Price. For example, a house with more bathrooms may be more expensive, but if it is also located in an area with lower median income, the price may be lower.