In order to help clients determine the price at which their house is likely to sell, a realtor gathered a sample of 150 purchase transactions in her area during a recent three-month period. The price of the home is measured in thousands of dollars. The number of square feet is also expressed in thousands, and the number of bathrooms is just that. Fit the multiple regression of Price on Square Feet and Bathrooms.
(a) Thinking marginally for a moment, should there be a correlation between the square feet and the number of bathrooms in a home?
(b) One of the two explanatory variables in this model does not explain statistically significant variation in the price. Had the two explanatory variables been uncorrelated (and produced these estimates), would this variable have been statistically significant? Use the VIF to find your answer.
(c) We can see the effects of collinearity by constructing a plot that shows the slope of the multiple regression. To do this, we remove the effect of one of the explanatory variables from the other variables. Here’s how to make a so-called partial regression leverage plot for these data. First, regress Price on Square Feet and save the residuals. Second, regress Bathrooms on Square Feet and save these residuals. Now, make a scatterplot of the residuals from the regression of Price on Square Feet on the residuals from the regression of Bathrooms on Square Feet. Fit the simple regression for this scatterplot, and compare the slope in this ft to the partial slope for Bathrooms in the multiple regression. Are they different?
(d) Compare the scatterplot of Price on Bathrooms to the partial regression plot constructed in part (c). What has changed?