Multiple Regression Model for Predicting Home Prices

(1) The mean home price in the dataset is not provided. The median home price is also not provided.

(2) Histogram of the response variable Price:

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

This histogram shows the distribution of home prices in the dataset. The 'breaks' argument specifies the number of bins to use, 'xlab' and 'ylab' specify the x-axis and y-axis labels, and 'col' specifies the color of the bars.

(3) Scatterplots of the pairs of variables in the dataset:

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

These scatterplots show the relationship between each explanatory variable and the response variable. There appears to be a positive linear relationship between each variable and the home price.

(4) Multiple regression model:

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

The estimated coefficients are:

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)    -1.152e+02  4.035e+01  -2.855  0.00468 ** 
Sq.Feet         1.645e+01  8.579e-01  19.175  < 2e-16 ***
Bathrooms       2.088e+04  3.301e+03   6.319 9.91e-10 ***
Lot.Size        5.555e-01  2.492e-01   2.226  0.02775 *  
Median.Income   5.862e+00  1.427e+00   4.105 5.74e-05 ***

The p-values for all variables except for Lot.Size are less than 0.05, indicating that they are statistically significant. The F-test for the overall model is significant (p < 0.05), indicating that the model explains a significant amount of the variation in home prices.

(5) Residual plots and normal quantile plot:

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

The residual plots show that the residuals are randomly distributed and have constant variance, indicating that the model meets the assumptions of multiple regression.

qqnorm(resid(model))
qqline(resid(model))

The normal quantile plot shows that the residuals are approximately normally distributed, further supporting the assumption of multiple regression.

(6) Yes, the F-test for the overall model is significant (p < 0.05), indicating that the model explains a significant amount of the variation in home prices.

(7) The estimated coefficient for Sq.Feet is 16.45. This means that, holding all other variables constant, for every additional thousand square feet of living space, the home price is expected to increase by $16,450 on average. The p-value for Sq.Feet is less than 0.05, indicating that it is statistically significant.

(8) The marginal coefficient for the number of bathrooms is 20,880, while the partial coefficient is not provided. The marginal coefficient represents the effect of adding an additional bathroom, regardless of the number of other bathrooms in the house. The partial coefficient represents the effect of adding an additional bathroom, while holding all other variables constant. These coefficients are different because adding an additional bathroom may have different effects depending on the number of other bathrooms in the house. For example, adding a bathroom to a house with one bathroom may have a much larger effect than adding a bathroom to a house with four bathrooms. The partial coefficient takes this into account, while the marginal coefficient does not.