For this exercise, you’re a real-estate developer. You’re planning a suburban housing development outside Philadelphia. The design calls for 25 homes that you expect to sell for about $450,000 each. If all goes as planned, you’ll make a proft of $50,000 per house, or $1.25 million overall.
If you add a security wall around the development, you might be able to sell each home for more. Gates convey safety and low crime rates to potential homebuyers. The crime rate in the area where you are building the development is already low, about 15 incidents per 1,000 residents. A security consultant claims a gate and fence would reduce this further to 10 per 1,000.
If this consultant is right, is it worth adding the gate and wall? The builders say that it will cost you about $875,000 ($35,000 more per house) to add the gate and fence to the development—if you do it now while construction is starting. If you wait until people move in, the costs will rise sharply.
You have some data to help you decide. The data include the median selling price of homes in communities in the Philadelphia area. The data also include the crime rate in these communities, expressed in incidents per 1,000 residents. This analysis will use the reciprocal of this rate. These data appeared in the April 1996 issue of Philadelphia Magazine. Because the values of homes have increased a lot since 1996, let’s assume that prices have doubled since these data were measured. (These data also appear in an exercise in Chapter 20. That exercise focuses on the transformation. For this exercise, we will focus on the use of the model with the transformation.)
Motivation
(a) Assume that the addition of a gate and wall has the effect of convincing potential buyers that the crime rate of this development will “feel” like 10 crimes per 1,000 rather than 15. How much does this have to increase the value of these homes (on average) in order for building the security fence to be cost effective?
(b) If the regression model identifies a statistically significant association between the price of housing and the number of people per crime (the reciprocal of the crime rate), will this prove that lowering the crime rate will pay for the cost of constructing the security wall?
Method
(c) Plot the selling prices of homes in these communities versus 1,000 divided by the crime rate. Does the plot seem straight enough to continue? (The variable created by 1,000 divided by the crime rate is the number of residents per crime.)
(d) Fit the linear equation to the scatterplot in part (c). If you accept the ft of this equation, what do you think about building the wall? Be sure to take the doubling of home prices into account.
Mechanics
(e) Which communities are leveraged in this analysis? What distinguishes these communities from the others?
(f) Which communities are outliers with unusually positive or negative residuals? Identify these in the plot of the residuals on the explanatory variable.
(g) Does this model meet the conditions needed for using the SRM for inference about the parameters? What about prediction intervals?
(h) If we ignore any problems noted in the form of this model, would the usual inferences lead us to tell the developer to build the wall? (Again, re-member to take account of the doubling of prices since 1996 into account.)
Message
(i) How would you answer the question for the developer? Should the developer proceed with the wall?
(j) What could you do to improve the analysis? State your suggestions in a form that the developer would understand.